Subgroups — C₆₇×#21234 #30700485

Order 17152.

1518 conjugacy classes of subgroups.

The subgroup lattice has 576 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	C₂×#428×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2311	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2310	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1971	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1969	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#137×C₆₇	—	2	4	no	—
15	C₂×#137×C₆₇	—	2	4	no	—
16	C₂×#137×C₆₇	C₄	1	8	yes	complemented
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	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	#428×C₆₇	—	2	4	no	—
25	C₂×#107×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	2	yes	—
27	#416×C₆₇	—	2	4	no	—
28	C₂×#103×C₆₇	—	2	4	no	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
35	#405×C₆₇	—	2	4	no	—
36	C₂×#97×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	—	2	2	no	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#391×C₆₇	C₂²	1	2	yes	—
41	#391×C₆₇	C₂²	1	2	yes	—
42	#390×C₆₇	C₂²	1	2	yes	—
43	#386×C₆₇	—	2	2	no	—
44	#381×C₆₇	—	2	2	no	—
45	#380×C₆₇	—	2	2	no	—
46	C₂×#41×C₆₇	—	2	2	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
52	C₂²×Q₈×C₆₇	—	2	1	no	—
53	C₂²×Q₈×C₆₇	—	2	2	no	—
54	C₂²×Q₈×C₆₇	—	2	2	no	—
55	C₂²×Q₈×C₆₇	—	2	2	no	—
56	C₂²×Q₈×C₆₇	—	2	2	no	—
57	C₂²×Q₈×C₆₇	—	2	2	no	—
58	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	—	2	1	no	—
64	C₂³×C₄×C₆₇	—	2	1	no	—
65	C₂³×C₄×C₆₇	—	2	1	no	—
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	#137×C₆₇	—	2	32	no	—
68	C₂×Q₁₆×C₆₇	—	2	4	no	—
69	C₂×Q₁₆×C₆₇	—	2	4	no	—
70	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
71	C₂×SD₁₆×C₆₇	—	2	4	no	—
72	C₂×SD₁₆×C₆₇	—	2	4	no	—
73	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
74	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
75	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
76	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
77	C₂²×C₈×C₆₇	—	2	2	no	—
78	C₂²×C₈×C₆₇	—	2	2	no	—
79	C₂²×C₈×C₆₇	D₄	1	4	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
83	#117×C₆₇	—	2	2	no	—
84	#117×C₆₇	—	2	2	no	—
85	#117×C₆₇	—	2	2	no	—
86	#117×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
97	C₂×C₄²×C₆₇	—	2	1	no	—
98	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
99	#107×C₆₇	—	2	4	no	—
100	#106×C₆₇	—	2	4	no	—
101	#103×C₆₇	—	2	4	no	—
102	#103×C₆₇	—	2	4	no	—
103	#103×C₆₇	—	2	2	no	—
104	#103×C₆₇	—	2	2	no	—
105	#102×C₆₇	—	2	2	no	—
106	#102×C₆₇	—	2	2	no	—
107	#97×C₆₇	—	2	4	no	—
108	C₄×C₈×C₆₇	—	2	4	no	—
109	#95×C₆₇	—	2	4	no	—
110	#95×C₆₇	—	2	4	no	—
111	#95×C₆₇	—	2	4	no	—
112	C₂⁴×C₆₇	—	2	1	no	—
113	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
114	#41×C₆₇	—	2	8	no	—
115	#41×C₆₇	—	2	8	no	—
116	#41×C₆₇	—	4	4	no	—
117	#41×C₆₇	—	4	4	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	2	1	no	—
121	C₂×Q₈×C₆₇	—	2	8	no	—
122	C₂×Q₈×C₆₇	—	2	8	no	—
123	C₂×Q₈×C₆₇	—	2	2	no	—
124	C₂×Q₈×C₆₇	—	2	1	no	—
125	C₂×Q₈×C₆₇	—	2	2	no	—
126	C₂×Q₈×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	2	2	no	—
130	C₂×Q₈×C₆₇	—	2	1	no	—
131	C₂×Q₈×C₆₇	—	4	2	no	—
132	C₂×Q₈×C₆₇	—	4	4	no	—
133	C₂×Q₈×C₆₇	—	4	4	no	—
134	C₂×Q₈×C₆₇	—	4	2	no	—
135	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	—	2	1	no	—
140	C₂×D₄×C₆₇	—	2	1	no	—
141	C₂×D₄×C₆₇	—	2	1	no	—
142	C₂×D₄×C₆₇	—	2	1	no	—
143	C₂×D₄×C₆₇	—	4	2	no	—
144	C₂×D₄×C₆₇	—	4	2	no	—
145	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	4	2	no	—
154	C₂²×C₄×C₆₇	—	4	2	no	—
155	C₂²×C₄×C₆₇	—	4	2	no	—
156	C₂²×C₄×C₆₇	—	4	2	no	—
157	C₂²×C₄×C₆₇	—	4	2	no	—
158	C₂²×C₄×C₆₇	—	8	2	no	—
159	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₆₇	#41	1	2	yes	—
166	C₂²×C₄×C₆₇	#41	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₆₇	—	2	16	no	—
169	SD₁₆×C₆₇	—	2	16	no	—
170	M₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
171	C₂×C₈×C₆₇	—	2	8	no	—
172	C₂×C₈×C₆₇	—	2	1	no	—
173	C₂×C₈×C₆₇	—	2	1	no	—
174	C₂×C₈×C₆₇	—	2	1	no	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
177	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
178	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
180	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	1	no	—
188	C₄⋊C₄×C₆₇	—	4	2	no	—
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	4	2	no	—
191	C₂²⋊C₄×C₆₇	—	8	4	no	—
192	C₄²×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	—	4	2	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	—	4	2	no	—
201	C₂³×C₆₇	—	4	2	no	—
202	C₂³×C₆₇	C₄×D₄	1	2	yes	—
203	C₂³×C₆₇	#121	1	2	yes	—
204	C₂³×C₆₇	#127	1	2	yes	—
205	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₆₇	—	2	4	no	—
207	Q₈×C₆₇	—	2	2	no	—
208	Q₈×C₆₇	—	2	4	no	—
209	Q₈×C₆₇	—	2	2	no	—
210	Q₈×C₆₇	—	4	2	no	—
211	Q₈×C₆₇	—	4	1	no	—
212	Q₈×C₆₇	—	4	2	no	—
213	Q₈×C₆₇	—	4	1	no	—
214	Q₈×C₆₇	—	4	8	no	—
215	D₄×C₆₇	—	2	2	no	—
216	D₄×C₆₇	—	2	2	no	—
217	D₄×C₆₇	—	4	8	no	—
218	D₄×C₆₇	—	4	1	no	—
219	D₄×C₆₇	—	4	1	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	4	1	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	1	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	8	no	—
230	C₂×C₄×C₆₇	—	4	8	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	4	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	8	4	no	—
238	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
239	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
240	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
241	C₂×C₄×C₆₇	#121	1	4	yes	—
242	C₂×C₄×C₆₇	#121	1	2	yes	—
243	C₂×C₄×C₆₇	#124	1	4	yes	—
244	C₂×C₄×C₆₇	#127	1	2	yes	—
245	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
251	C₈×C₆₇	—	2	2	no	—
252	C₈×C₆₇	—	2	2	no	—
253	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	4	no	—
257	C₂²×C₆₇	—	2	4	no	—
258	C₂²×C₆₇	—	4	4	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	8	1	no	—
261	C₂²×C₆₇	—	8	1	no	—
262	C₂²×C₆₇	—	8	2	no	—
263	C₂²×C₆₇	#390	1	4	yes	—
264	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
265	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
266	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
267	C₂²×C₆₇	#577	1	2	yes	—
268	C₂²×C₆₇	#578	1	2	yes	—
269	C₄×C₆₇	—	4	2	no	—
270	C₄×C₆₇	—	4	4	no	—
271	C₄×C₆₇	—	4	4	no	—
272	C₄×C₆₇	—	8	1	no	—
273	C₄×C₆₇	—	8	2	no	—
274	C₄×C₆₇	—	8	1	no	—
275	C₄×C₆₇	#390	1	4	yes	—
276	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
277	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
278	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
279	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
280	#21234	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
281	C₂×C₆₇	—	2	4	no	—
282	C₂×C₆₇	—	8	2	no	—
283	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
284	C₂×C₆₇	#2947	1	2	yes	—
285	C₂×C₆₇	#3063	1	2	yes	—
286	C₂×C₆₇	#3154	1	1	yes	characteristic
287	C₂×C₆₇	#3156	1	1	yes	characteristic
288	C₂²×#137	C₂×C₆₇	1	1	yes	characteristic
289	C₂×#428	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₂×#416	C₂×C₆₇	1	1	yes	characteristic
291	C₂²×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
292	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
293	#2311	C₂×C₆₇	1	1	yes	characteristic
294	#2310	C₂×C₆₇	1	1	yes	characteristic, complemented
295	#1973	C₂×C₆₇	1	2	yes	complemented
296	#1971	C₂×C₆₇	1	2	yes	complemented
297	#1969	C₂×C₆₇	1	2	yes	complemented
298	#1968	C₂×C₆₇	1	2	yes	complemented
299	C₆₇	#21234	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
301	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×#137	—	2	4	no	—
303	C₂×#137	—	2	4	no	—
304	C₂×#137	C₄×C₆₇	1	8	yes	complemented
305	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
306	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
307	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
308	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
311	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
312	#428	—	2	4	no	—
313	C₂×#107	C₂²×C₆₇	1	2	yes	—
314	C₂×#106	C₂²×C₆₇	1	2	yes	—
315	#416	—	2	4	no	—
316	C₂×#103	—	2	4	no	—
317	C₂×#103	C₂²×C₆₇	1	2	yes	—
318	C₂×#103	C₂²×C₆₇	1	2	yes	—
319	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
320	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
323	#405	—	2	4	no	—
324	C₂×#97	C₂²×C₆₇	1	2	yes	—
325	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
326	#391	—	2	2	no	—
327	#391	C₂²×C₆₇	1	2	yes	—
328	#391	C₂²×C₆₇	1	2	yes	—
329	#391	C₂²×C₆₇	1	2	yes	—
330	#390	C₂²×C₆₇	1	2	yes	—
331	#386	—	2	2	no	—
332	#381	—	2	2	no	—
333	#380	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	2	2	no	—
337	C₂×#41	—	2	2	no	—
338	C₂×#41	—	2	2	no	—
339	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
340	C₂²×Q₈	—	2	1	no	—
341	C₂²×Q₈	—	2	2	no	—
342	C₂²×Q₈	—	2	2	no	—
343	C₂²×Q₈	—	2	2	no	—
344	C₂²×Q₈	—	2	2	no	—
345	C₂²×Q₈	—	2	2	no	—
346	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
347	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
348	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
349	C₂²×D₄	—	2	1	no	—
350	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
351	C₂³×C₄	—	2	1	no	—
352	C₂³×C₄	—	2	1	no	—
353	C₂³×C₄	—	2	1	no	—
354	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
355	#137	—	2	32	no	—
356	C₂×Q₁₆	—	2	4	no	—
357	C₂×Q₁₆	—	2	4	no	—
358	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
359	C₂×SD₁₆	—	2	4	no	—
360	C₂×SD₁₆	—	2	4	no	—
361	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
362	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
363	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
364	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
365	C₂²×C₈	—	2	2	no	—
366	C₂²×C₈	—	2	2	no	—
367	C₂²×C₈	D₄×C₆₇	1	4	yes	—
368	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
371	#117	—	2	2	no	—
372	#117	—	2	2	no	—
373	#117	—	2	2	no	—
374	#117	—	2	2	no	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	—	2	4	no	—
384	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
387	#107	—	2	4	no	—
388	#106	—	2	4	no	—
389	#103	—	2	4	no	—
390	#103	—	2	4	no	—
391	#103	—	2	2	no	—
392	#103	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	#97	—	2	4	no	—
396	C₄×C₈	—	2	4	no	—
397	#95	—	2	4	no	—
398	#95	—	2	4	no	—
399	#95	—	2	4	no	—
400	C₂⁴	—	2	1	no	—
401	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
402	#41	—	2	8	no	—
403	#41	—	2	8	no	—
404	#41	—	4	4	no	—
405	#41	—	4	4	no	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	8	no	—
410	C₂×Q₈	—	2	8	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	4	2	no	—
420	C₂×Q₈	—	4	4	no	—
421	C₂×Q₈	—	4	4	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
424	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	4	2	no	—
432	C₂×D₄	—	4	2	no	—
433	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	8	2	no	—
447	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
448	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
449	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
450	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
453	C₂²×C₄	#41×C₆₇	1	2	yes	—
454	C₂²×C₄	#41×C₆₇	1	2	yes	—
455	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
456	Q₁₆	—	2	16	no	—
457	SD₁₆	—	2	16	no	—
458	M₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
459	C₂×C₈	—	2	8	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
465	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
466	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
467	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
468	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	4	2	no	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	4	2	no	—
479	C₂²⋊C₄	—	8	4	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	4	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	4	1	no	—
487	C₂³	—	4	1	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	4	2	no	—
490	C₂³	C₄×D₄×C₆₇	1	2	yes	—
491	C₂³	#121×C₆₇	1	2	yes	—
492	C₂³	#127×C₆₇	1	2	yes	—
493	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
494	Q₈	—	2	4	no	—
495	Q₈	—	2	2	no	—
496	Q₈	—	2	4	no	—
497	Q₈	—	2	2	no	—
498	Q₈	—	4	2	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	8	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	1	no	—
507	D₄	—	4	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	8	4	no	—
526	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
527	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
528	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
529	C₂×C₄	#121×C₆₇	1	4	yes	—
530	C₂×C₄	#121×C₆₇	1	2	yes	—
531	C₂×C₄	#124×C₆₇	1	4	yes	—
532	C₂×C₄	#127×C₆₇	1	2	yes	—
533	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
537	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
538	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
539	C₈	—	2	2	no	—
540	C₈	—	2	2	no	—
541	C₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
542	C₂²	—	2	4	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	4	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	4	no	—
548	C₂²	—	8	2	no	—
549	C₂²	—	8	1	no	—
550	C₂²	—	8	1	no	—
551	C₂²	#390×C₆₇	1	4	yes	—
552	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
553	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
554	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
555	C₂²	#577×C₆₇	1	2	yes	—
556	C₂²	#578×C₆₇	1	2	yes	—
557	C₄	—	4	2	no	—
558	C₄	—	4	4	no	—
559	C₄	—	4	4	no	—
560	C₄	—	8	2	no	—
561	C₄	—	8	1	no	—
562	C₄	—	8	1	no	—
563	C₄	#390×C₆₇	1	4	yes	—
564	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
565	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
566	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
567	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
568	C₂	—	2	4	no	—
569	C₂	—	8	2	no	—
570	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
571	C₂	C₆₇×#2947	1	2	yes	—
572	C₂	C₆₇×#3063	1	2	yes	—
573	C₂	C₆₇×#3154	1	1	yes	characteristic
574	C₂	C₆₇×#3156	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₆₇	C₂	semidirect
4. C₂²×#103×C₆₇	C₂	semidirect
5. C₂×#405×C₆₇	C₂	semidirect
7. C₆₇×#2310	C₂	semidirect
8. C₆₇×#1973	C₂	semidirect
9. C₆₇×#1971	C₂	semidirect
10. C₆₇×#1969	C₂	semidirect
11. C₆₇×#1968	C₂	semidirect
16. C₂×#137×C₆₇	C₄	semidirect
70. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
73. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
170. M₁₆×C₆₇	C₂²⋊C₄	semidirect
280. #21234	C₆₇	direct (N × H)
289. C₂×#428	C₂×C₆₇	semidirect
291. C₂²×#103	C₂×C₆₇	semidirect
292. C₂×#405	C₂×C₆₇	semidirect
294. #2310	C₂×C₆₇	semidirect
295. #1973	C₂×C₆₇	semidirect
296. #1971	C₂×C₆₇	semidirect
297. #1969	C₂×C₆₇	semidirect
298. #1968	C₂×C₆₇	semidirect
299. C₆₇	#21234	direct (N × H)
304. C₂×#137	C₄×C₆₇	semidirect
358. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
361. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
458. M₁₆	C₂²⋊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
C₂×#428×C₆₇	2
C₂×#416×C₆₇	2
C₂²×#103×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2311	2
C₆₇×#2310	2
C₆₇×#1973	2
C₆₇×#1971	2
C₆₇×#1969	2
C₆₇×#1968	2
#21234	67

Subgroups — C67×#21234 #30700485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21234 #30700485