Subgroups — C₆₇×#60169 #30739420

Order 17152.

2666 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3428	C₂	1	1	yes	maximal, characteristic
3	C₂×#519×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2950	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2949	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2947	C₂	1	8	yes	maximal, complemented
7	C₄×#137×C₆₇	C₂	1	8	yes	maximal, complemented
8	C₆₇×#2911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2896	C₂	1	1	yes	maximal, characteristic
10	C₂×#419×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#137×C₆₇	C₄	1	16	yes	complemented
15	C₂×#137×C₆₇	C₂²	1	4	yes	—
16	C₂×#137×C₆₇	C₂²	1	4	yes	—
17	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	#520×C₆₇	—	2	2	no	—
22	#520×C₆₇	C₂²	1	4	yes	—
23	#519×C₆₇	—	2	2	no	—
24	#519×C₆₇	—	2	2	no	—
25	#519×C₆₇	C₂²	1	4	yes	—
26	#519×C₆₇	C₂²	1	2	yes	—
27	#519×C₆₇	C₂²	1	4	yes	complemented
28	#518×C₆₇	—	2	2	no	—
29	#518×C₆₇	C₂²	1	2	yes	—
30	C₄×#41×C₆₇	—	2	2	no	—
31	C₄×#41×C₆₇	C₂²	1	4	yes	—
32	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
33	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
34	C₂×#117×C₆₇	—	2	2	no	—
35	C₂×#117×C₆₇	—	2	2	no	—
36	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
39	#441×C₆₇	C₂²	1	16	yes	complemented
40	#440×C₆₇	C₂²	1	16	yes	complemented
41	C₄×Q₁₆×C₆₇	C₂²	1	16	yes	complemented
42	C₄×SD₁₆×C₆₇	C₂²	1	16	yes	complemented
43	#428×C₆₇	C₂²	1	4	yes	complemented
44	#428×C₆₇	C₂²	1	4	yes	complemented
45	#427×C₆₇	C₂²	1	4	yes	—
46	#419×C₆₇	C₂²	1	4	yes	—
47	#419×C₆₇	C₂²	1	2	yes	—
48	#419×C₆₇	C₂²	1	4	yes	complemented
49	#418×C₆₇	C₂²	1	2	yes	—
50	#417×C₆₇	C₂²	1	4	yes	—
51	#416×C₆₇	C₂²	1	4	yes	—
52	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
53	#405×C₆₇	C₂²	1	4	yes	—
54	#405×C₆₇	C₂²	1	4	yes	complemented
55	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
56	C₂×#41×C₆₇	—	2	1	no	—
57	C₂×#41×C₆₇	—	2	1	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
60	C₂×#41×C₆₇	C₂³	1	2	yes	—
61	C₂×#41×C₆₇	C₂³	1	2	yes	—
62	C₂²×Q₈×C₆₇	—	2	1	no	—
63	C₂²×Q₈×C₆₇	—	2	1	no	—
64	C₂²×Q₈×C₆₇	—	2	1	no	—
65	C₂²×Q₈×C₆₇	—	2	4	no	—
66	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
67	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×D₄×C₆₇	—	2	1	no	—
72	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	—	2	1	no	—
74	C₂³×C₄×C₆₇	—	2	1	no	—
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	#137×C₆₇	C₂×C₄	1	32	yes	complemented
77	#137×C₆₇	C₂×C₄	1	32	yes	complemented
78	C₂×Q₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
79	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
80	C₂×SD₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
81	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
82	C₂×M₁₆×C₆₇	—	2	4	no	—
83	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
84	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
85	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
86	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
87	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
89	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
90	C₄×Q₈×C₆₇	—	2	8	no	—
91	C₄×Q₈×C₆₇	—	2	8	no	—
92	C₄×Q₈×C₆₇	—	2	4	no	—
93	C₄×Q₈×C₆₇	—	2	8	no	—
94	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
95	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
96	C₄×D₄×C₆₇	—	2	8	no	—
97	C₄×D₄×C₆₇	—	2	4	no	—
98	C₄×D₄×C₆₇	—	2	8	no	—
99	C₄×D₄×C₆₇	C₂³	1	8	yes	—
100	#117×C₆₇	—	2	1	no	—
101	#117×C₆₇	—	2	1	no	—
102	#117×C₆₇	—	2	1	no	—
103	#117×C₆₇	—	2	1	no	—
104	#117×C₆₇	—	2	4	no	—
105	#117×C₆₇	—	2	4	no	—
106	#117×C₆₇	—	2	8	no	—
107	#117×C₆₇	—	2	8	no	—
108	#117×C₆₇	—	2	4	no	—
109	#117×C₆₇	D₄	1	2	yes	—
110	#117×C₆₇	D₄	1	2	yes	—
111	#117×C₆₇	C₂³	1	2	yes	—
112	#117×C₆₇	C₂³	1	2	yes	—
113	#117×C₆₇	C₂³	1	2	yes	—
114	#117×C₆₇	C₂³	1	2	yes	—
115	#117×C₆₇	C₂³	1	2	yes	—
116	#117×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
126	C₂×C₄²×C₆₇	—	2	4	no	—
127	C₂×C₄²×C₆₇	—	2	1	no	—
128	C₂×C₄²×C₆₇	—	2	1	no	—
129	C₂×C₄²×C₆₇	—	2	4	no	—
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	#107×C₆₇	C₂³	1	8	yes	—
132	#106×C₆₇	C₂³	1	8	yes	—
133	#103×C₆₇	C₂³	1	16	yes	—
134	#103×C₆₇	C₂³	1	8	yes	—
135	#102×C₆₇	C₂³	1	8	yes	—
136	#97×C₆₇	C₂³	1	8	yes	—
137	C₄×C₈×C₆₇	C₂³	1	8	yes	—
138	C₂⁴×C₆₇	—	2	1	no	—
139	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
140	#41×C₆₇	—	2	8	no	—
141	#41×C₆₇	—	2	8	no	—
142	#41×C₆₇	—	2	4	no	—
143	#41×C₆₇	—	2	4	no	—
144	#41×C₆₇	C₂²×C₄	1	8	yes	—
145	#41×C₆₇	C₂²×C₄	1	8	yes	—
146	C₂×Q₈×C₆₇	—	2	8	no	—
147	C₂×Q₈×C₆₇	—	2	8	no	—
148	C₂×Q₈×C₆₇	—	2	8	no	—
149	C₂×Q₈×C₆₇	—	2	2	no	—
150	C₂×Q₈×C₆₇	—	2	2	no	—
151	C₂×Q₈×C₆₇	—	2	2	no	—
152	C₂×Q₈×C₆₇	—	2	1	no	—
153	C₂×Q₈×C₆₇	—	2	4	no	—
154	C₂×Q₈×C₆₇	—	2	4	no	—
155	C₂×Q₈×C₆₇	—	2	4	no	—
156	C₂×Q₈×C₆₇	—	2	2	no	—
157	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
158	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
159	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
160	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
161	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
162	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
163	C₂×D₄×C₆₇	—	2	8	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	2	no	—
166	C₂×D₄×C₆₇	—	2	1	no	—
167	C₂×D₄×C₆₇	—	2	2	no	—
168	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
169	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₇	—	2	8	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	—	2	2	no	—
174	C₂²×C₄×C₆₇	—	2	2	no	—
175	C₂²×C₄×C₆₇	—	2	8	no	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	2	2	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	2	2	no	—
180	C₂²×C₄×C₆₇	—	2	8	no	—
181	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	#41	1	4	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₆₇	C₂²×C₄	1	32	yes	—
191	SD₁₆×C₆₇	C₂²×C₄	1	32	yes	—
192	M₁₆×C₆₇	—	2	4	no	—
193	M₁₆×C₆₇	—	2	4	no	—
194	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
195	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
196	C₂×C₈×C₆₇	—	2	4	no	—
197	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
198	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
199	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₆₇	—	2	2	no	—
201	C₄⋊C₄×C₆₇	—	2	8	no	—
202	C₄⋊C₄×C₆₇	—	2	16	no	—
203	C₄⋊C₄×C₆₇	—	2	2	no	—
204	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
205	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
206	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
207	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
208	C₂²⋊C₄×C₆₇	—	2	2	no	—
209	C₂²⋊C₄×C₆₇	—	2	2	no	—
210	C₂²⋊C₄×C₆₇	—	2	2	no	—
211	C₂²⋊C₄×C₆₇	—	2	2	no	—
212	C₂²⋊C₄×C₆₇	—	2	8	no	—
213	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
214	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
215	C₄²×C₆₇	—	2	2	no	—
216	C₄²×C₆₇	—	2	8	no	—
217	C₄²×C₆₇	—	2	16	no	—
218	C₄²×C₆₇	—	2	2	no	—
219	C₄²×C₆₇	—	2	2	no	—
220	C₄²×C₆₇	—	2	2	no	—
221	C₄²×C₆₇	C₂×D₄	1	4	yes	—
222	C₄²×C₆₇	C₂⁴	1	4	yes	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	4	no	—
225	C₂³×C₆₇	—	4	2	no	—
226	C₂³×C₆₇	—	8	2	no	—
227	C₂³×C₆₇	C₄×D₄	1	4	yes	—
228	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
229	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
231	Q₈×C₆₇	—	2	8	no	—
232	Q₈×C₆₇	—	2	8	no	—
233	Q₈×C₆₇	—	2	8	no	—
234	Q₈×C₆₇	—	2	4	no	—
235	Q₈×C₆₇	—	2	4	no	—
236	Q₈×C₆₇	—	2	2	no	—
237	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
238	Q₈×C₆₇	C₂³×C₄	1	4	yes	—
239	D₄×C₆₇	—	2	8	no	—
240	D₄×C₆₇	—	2	8	no	—
241	D₄×C₆₇	—	2	4	no	—
242	D₄×C₆₇	—	2	2	no	—
243	D₄×C₆₇	C₂³×C₄	1	4	yes	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	8	no	—
247	C₂×C₄×C₆₇	—	2	8	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	8	no	—
251	C₂×C₄×C₆₇	—	2	8	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	4	4	no	—
255	C₂×C₄×C₆₇	—	4	4	no	—
256	C₂×C₄×C₆₇	—	8	8	no	—
257	C₂×C₄×C₆₇	—	8	2	no	—
258	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
259	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
260	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
261	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
262	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
263	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
265	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
269	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
270	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
271	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
272	C₈×C₆₇	—	2	4	no	—
273	C₈×C₆₇	C₂³×C₄	1	8	yes	—
274	C₂²×C₆₇	—	2	2	no	—
275	C₂²×C₆₇	—	2	2	no	—
276	C₂²×C₆₇	—	2	2	no	—
277	C₂²×C₆₇	—	2	4	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	4	4	no	—
280	C₂²×C₆₇	—	8	4	no	—
281	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
282	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
283	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
284	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
285	C₄×C₆₇	—	2	8	no	—
286	C₄×C₆₇	—	2	4	no	—
287	C₄×C₆₇	—	2	8	no	—
288	C₄×C₆₇	—	4	4	no	—
289	C₄×C₆₇	—	4	4	no	—
290	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
291	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
292	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
293	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
294	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
295	#60169	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	2	2	no	—
297	C₂×C₆₇	—	2	2	no	—
298	C₂×C₆₇	—	2	2	no	—
299	C₂×C₆₇	—	4	4	no	—
300	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
301	C₂×C₆₇	#3586	1	2	yes	—
302	C₂²×#137	C₂×C₆₇	1	1	yes	characteristic
303	#3428	C₂×C₆₇	1	1	yes	characteristic
304	C₂×#519	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#2950	C₂×C₆₇	1	4	yes	complemented
306	#2949	C₂×C₆₇	1	4	yes	complemented
307	#2947	C₂×C₆₇	1	8	yes	complemented
308	C₄×#137	C₂×C₆₇	1	8	yes	complemented
309	#2911	C₂×C₆₇	1	1	yes	characteristic, complemented
310	#2896	C₂×C₆₇	1	1	yes	characteristic
311	C₂×#419	C₂×C₆₇	1	1	yes	characteristic, complemented
312	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
313	C₆₇	#60169	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
315	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#137	C₄×C₆₇	1	16	yes	complemented
317	C₂×#137	C₂²×C₆₇	1	4	yes	—
318	C₂×#137	C₂²×C₆₇	1	4	yes	—
319	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
320	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
321	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
323	#520	—	2	2	no	—
324	#520	C₂²×C₆₇	1	4	yes	—
325	#519	—	2	2	no	—
326	#519	—	2	2	no	—
327	#519	C₂²×C₆₇	1	4	yes	—
328	#519	C₂²×C₆₇	1	2	yes	—
329	#519	C₂²×C₆₇	1	4	yes	complemented
330	#518	—	2	2	no	—
331	#518	C₂²×C₆₇	1	2	yes	—
332	C₄×#41	—	2	2	no	—
333	C₄×#41	C₂²×C₆₇	1	4	yes	—
334	C₂×C₄×Q₈	—	2	2	no	—
335	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
336	C₂×#117	—	2	2	no	—
337	C₂×#117	—	2	2	no	—
338	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
340	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
341	#441	C₂²×C₆₇	1	16	yes	complemented
342	#440	C₂²×C₆₇	1	16	yes	complemented
343	C₄×Q₁₆	C₂²×C₆₇	1	16	yes	complemented
344	C₄×SD₁₆	C₂²×C₆₇	1	16	yes	complemented
345	#428	C₂²×C₆₇	1	4	yes	complemented
346	#428	C₂²×C₆₇	1	4	yes	complemented
347	#427	C₂²×C₆₇	1	4	yes	—
348	#419	C₂²×C₆₇	1	4	yes	—
349	#419	C₂²×C₆₇	1	2	yes	—
350	#419	C₂²×C₆₇	1	4	yes	complemented
351	#418	C₂²×C₆₇	1	2	yes	—
352	#417	C₂²×C₆₇	1	4	yes	—
353	#416	C₂²×C₆₇	1	4	yes	—
354	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
355	#405	C₂²×C₆₇	1	4	yes	—
356	#405	C₂²×C₆₇	1	4	yes	complemented
357	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
358	C₂×#41	—	2	1	no	—
359	C₂×#41	—	2	1	no	—
360	C₂×#41	—	2	4	no	—
361	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
362	C₂×#41	C₂³×C₆₇	1	2	yes	—
363	C₂×#41	C₂³×C₆₇	1	2	yes	—
364	C₂²×Q₈	—	2	1	no	—
365	C₂²×Q₈	—	2	1	no	—
366	C₂²×Q₈	—	2	1	no	—
367	C₂²×Q₈	—	2	4	no	—
368	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
369	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
370	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
371	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
372	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
373	C₂²×D₄	—	2	1	no	—
374	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
375	C₂³×C₄	—	2	1	no	—
376	C₂³×C₄	—	2	1	no	—
377	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
378	#137	C₂×C₄×C₆₇	1	32	yes	complemented
379	#137	C₂×C₄×C₆₇	1	32	yes	complemented
380	C₂×Q₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
381	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
382	C₂×SD₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
383	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
384	C₂×M₁₆	—	2	4	no	—
385	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
386	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
387	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
388	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
389	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
390	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
391	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
392	C₄×Q₈	—	2	8	no	—
393	C₄×Q₈	—	2	8	no	—
394	C₄×Q₈	—	2	4	no	—
395	C₄×Q₈	—	2	8	no	—
396	C₄×Q₈	C₂³×C₆₇	1	16	yes	—
397	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
398	C₄×D₄	—	2	8	no	—
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	—	2	8	no	—
401	C₄×D₄	C₂³×C₆₇	1	8	yes	—
402	#117	—	2	1	no	—
403	#117	—	2	1	no	—
404	#117	—	2	1	no	—
405	#117	—	2	1	no	—
406	#117	—	2	4	no	—
407	#117	—	2	4	no	—
408	#117	—	2	8	no	—
409	#117	—	2	8	no	—
410	#117	—	2	4	no	—
411	#117	D₄×C₆₇	1	2	yes	—
412	#117	D₄×C₆₇	1	2	yes	—
413	#117	C₂³×C₆₇	1	2	yes	—
414	#117	C₂³×C₆₇	1	2	yes	—
415	#117	C₂³×C₆₇	1	2	yes	—
416	#117	C₂³×C₆₇	1	2	yes	—
417	#117	C₂³×C₆₇	1	2	yes	—
418	#117	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	—	2	4	no	—
420	C₂×C₄⋊C₄	—	2	4	no	—
421	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	4	no	—
427	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
428	C₂×C₄²	—	2	4	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	—	2	1	no	—
431	C₂×C₄²	—	2	4	no	—
432	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
433	#107	C₂³×C₆₇	1	8	yes	—
434	#106	C₂³×C₆₇	1	8	yes	—
435	#103	C₂³×C₆₇	1	16	yes	—
436	#103	C₂³×C₆₇	1	8	yes	—
437	#102	C₂³×C₆₇	1	8	yes	—
438	#97	C₂³×C₆₇	1	8	yes	—
439	C₄×C₈	C₂³×C₆₇	1	8	yes	—
440	C₂⁴	—	2	1	no	—
441	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
442	#41	—	2	8	no	—
443	#41	—	2	8	no	—
444	#41	—	2	4	no	—
445	#41	—	2	4	no	—
446	#41	C₂²×C₄×C₆₇	1	8	yes	—
447	#41	C₂²×C₄×C₆₇	1	8	yes	—
448	C₂×Q₈	—	2	8	no	—
449	C₂×Q₈	—	2	8	no	—
450	C₂×Q₈	—	2	8	no	—
451	C₂×Q₈	—	2	2	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	—	2	2	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	4	no	—
456	C₂×Q₈	—	2	4	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	2	2	no	—
459	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
460	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
461	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
462	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
463	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
464	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
465	C₂×D₄	—	2	8	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
471	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	8	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
485	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
488	C₂²×C₄	#41×C₆₇	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
492	Q₁₆	C₂²×C₄×C₆₇	1	32	yes	—
493	SD₁₆	C₂²×C₄×C₆₇	1	32	yes	—
494	M₁₆	—	2	4	no	—
495	M₁₆	—	2	4	no	—
496	M₁₆	C₂²×C₄×C₆₇	1	8	yes	—
497	M₁₆	C₂²×C₄×C₆₇	1	8	yes	—
498	C₂×C₈	—	2	4	no	—
499	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
500	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
501	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	16	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
507	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
508	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
509	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	8	no	—
515	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
516	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	8	no	—
519	C₄²	—	2	16	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	2	2	no	—
523	C₄²	C₂×D₄×C₆₇	1	4	yes	—
524	C₄²	C₂⁴×C₆₇	1	4	yes	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	8	2	no	—
529	C₂³	C₄×D₄×C₆₇	1	4	yes	—
530	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
531	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
532	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
533	Q₈	—	2	8	no	—
534	Q₈	—	2	8	no	—
535	Q₈	—	2	8	no	—
536	Q₈	—	2	4	no	—
537	Q₈	—	2	4	no	—
538	Q₈	—	2	2	no	—
539	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
540	Q₈	C₂³×C₄×C₆₇	1	4	yes	—
541	D₄	—	2	8	no	—
542	D₄	—	2	8	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	2	no	—
545	D₄	C₂³×C₄×C₆₇	1	4	yes	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	8	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	8	8	no	—
559	C₂×C₄	—	8	2	no	—
560	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
561	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
562	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
563	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
567	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
569	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
571	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
572	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
573	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
574	C₈	—	2	4	no	—
575	C₈	C₂³×C₄×C₆₇	1	8	yes	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	8	4	no	—
583	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
584	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
585	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
586	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
587	C₄	—	2	8	no	—
588	C₄	—	2	4	no	—
589	C₄	—	2	8	no	—
590	C₄	—	4	4	no	—
591	C₄	—	4	4	no	—
592	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
593	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
594	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
595	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
596	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
597	C₂	—	2	2	no	—
598	C₂	—	2	2	no	—
599	C₂	—	2	2	no	—
600	C₂	—	4	4	no	—
601	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
602	C₂	C₆₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×#519×C₆₇	C₂	semidirect
4. C₆₇×#2950	C₂	semidirect
5. C₆₇×#2949	C₂	semidirect
6. C₆₇×#2947	C₂	semidirect
7. C₄×#137×C₆₇	C₂	semidirect
8. C₆₇×#2911	C₂	semidirect
10. C₂×#419×C₆₇	C₂	semidirect
11. C₆₇×#2873	C₂	semidirect
14. C₂×#137×C₆₇	C₄	semidirect
27. #519×C₆₇	C₂²	semidirect
33. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
39. #441×C₆₇	C₂²	semidirect
40. #440×C₆₇	C₂²	semidirect
41. C₄×Q₁₆×C₆₇	C₂²	semidirect
42. C₄×SD₁₆×C₆₇	C₂²	semidirect
43. #428×C₆₇	C₂²	semidirect
44. #428×C₆₇	C₂²	semidirect
48. #419×C₆₇	C₂²	semidirect
52. C₂×#103×C₆₇	C₂²	semidirect
54. #405×C₆₇	C₂²	semidirect
55. C₄×M₁₆×C₆₇	C₂²	semidirect
66. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
76. #137×C₆₇	C₂×C₄	semidirect
77. #137×C₆₇	C₂×C₄	semidirect
78. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
80. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
83. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
295. #60169	C₆₇	direct (N × H)
304. C₂×#519	C₂×C₆₇	semidirect
305. #2950	C₂×C₆₇	semidirect
306. #2949	C₂×C₆₇	semidirect
307. #2947	C₂×C₆₇	semidirect
308. C₄×#137	C₂×C₆₇	semidirect
309. #2911	C₂×C₆₇	semidirect
311. C₂×#419	C₂×C₆₇	semidirect
312. #2873	C₂×C₆₇	semidirect
313. C₆₇	#60169	direct (N × H)
316. C₂×#137	C₄×C₆₇	semidirect
329. #519	C₂²×C₆₇	semidirect
335. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
341. #441	C₂²×C₆₇	semidirect
342. #440	C₂²×C₆₇	semidirect
343. C₄×Q₁₆	C₂²×C₆₇	semidirect
344. C₄×SD₁₆	C₂²×C₆₇	semidirect
345. #428	C₂²×C₆₇	semidirect
346. #428	C₂²×C₆₇	semidirect
350. #419	C₂²×C₆₇	semidirect
354. C₂×#103	C₂²×C₆₇	semidirect
356. #405	C₂²×C₆₇	semidirect
357. C₄×M₁₆	C₂²×C₆₇	semidirect
368. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
378. #137	C₂×C₄×C₆₇	semidirect
379. #137	C₂×C₄×C₆₇	semidirect
380. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
382. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
385. C₂×M₁₆	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#137×C₆₇	2
C₆₇×#3428	2
C₂×#519×C₆₇	2
C₆₇×#2950	2
C₆₇×#2949	2
C₆₇×#2947	2
C₄×#137×C₆₇	2
C₆₇×#2911	2
C₆₇×#2896	2
C₂×#419×C₆₇	2
C₆₇×#2873	2
#60169	67

Subgroups — C67×#60169 #30739420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60169 #30739420