Subgroups — C₆₇×#61091 #30740342

Order 17152.

2650 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3499	C₂	1	2	yes	maximal, complemented
3	D₄×Q₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3283	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3276	C₂	1	8	yes	maximal, complemented
6	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#3059	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3039	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3011	C₂	1	4	yes	maximal, complemented
10	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	—
13	#578×C₆₇	—	2	16	no	—
14	C₂×#137×C₆₇	C₂²	1	4	yes	—
15	C₂×#137×C₆₇	C₂²	1	8	yes	—
16	C₂×#135×C₆₇	C₂²	1	4	yes	—
17	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
19	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×#131×C₆₇	C₂²	1	2	yes	—
22	D₄×Q₈×C₆₇	C₂²	1	8	yes	—
23	#547×C₆₇	—	2	8	no	—
24	#547×C₆₇	C₂²	1	8	yes	—
25	#536×C₆₇	—	2	4	no	—
26	#536×C₆₇	C₂²	1	4	yes	—
27	#532×C₆₇	—	2	2	no	—
28	#532×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
30	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
31	C₄×#41×C₆₇	C₂²	1	2	yes	—
32	#494×C₆₇	C₂²	1	4	yes	—
33	#494×C₆₇	C₂²	1	4	yes	—
34	#470×C₆₇	C₂²	1	8	yes	—
35	#467×C₆₇	C₂²	1	8	yes	—
36	#462×C₆₇	C₂²	1	8	yes	—
37	#461×C₆₇	C₂²	1	8	yes	—
38	#452×C₆₇	C₂²	1	16	yes	—
39	#451×C₆₇	C₂²	1	16	yes	—
40	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
41	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
42	C₄×D₈×C₆₇	C₂²	1	2	yes	—
43	#436×C₆₇	C₂²	1	4	yes	—
44	C₈×D₄×C₆₇	C₂²	1	4	yes	—
45	#427×C₆₇	C₂²	1	1	yes	characteristic
46	#423×C₆₇	C₂²	1	1	yes	characteristic
47	#419×C₆₇	C₂²	1	4	yes	—
48	#416×C₆₇	C₂²	1	2	yes	—
49	#408×C₆₇	C₂²	1	2	yes	—
50	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
51	2₋¹⁺⁴×C₆₇	—	2	16	no	—
52	2₋¹⁺⁴×C₆₇	—	4	8	no	—
53	C₂×#41×C₆₇	—	2	4	no	—
54	C₂×#41×C₆₇	—	2	8	no	—
55	C₂×#41×C₆₇	C₂³	1	8	yes	—
56	C₂×#41×C₆₇	C₂³	1	2	yes	—
57	C₂×#41×C₆₇	C₂³	1	2	yes	—
58	C₂²×Q₈×C₆₇	—	2	4	no	—
59	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
60	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
61	#137×C₆₇	—	2	16	no	—
62	#137×C₆₇	—	2	32	no	—
63	#135×C₆₇	—	2	16	no	—
64	#135×C₆₇	D₄	1	8	yes	complemented
65	C₂×Q₁₆×C₆₇	—	2	16	no	—
66	C₂×Q₁₆×C₆₇	—	2	8	no	—
67	C₂×Q₁₆×C₆₇	—	4	4	no	—
68	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
69	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
70	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
71	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
72	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
76	#131×C₆₇	—	2	8	no	—
77	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
78	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₆₇	—	2	4	no	—
83	C₄⋊Q₈×C₆₇	—	2	8	no	—
84	C₄⋊Q₈×C₆₇	—	2	2	no	—
85	C₄⋊Q₈×C₆₇	—	2	2	no	—
86	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
88	#127×C₆₇	D₄	1	1	yes	characteristic
89	#124×C₆₇	—	2	4	no	—
90	#124×C₆₇	D₄	1	2	yes	—
91	#123×C₆₇	—	2	8	no	—
92	#123×C₆₇	—	2	16	no	—
93	C₂²⋊Q₈×C₆₇	—	2	16	no	—
94	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
95	#121×C₆₇	—	2	8	no	—
96	#121×C₆₇	D₄	1	4	yes	—
97	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
98	C₄×D₄×C₆₇	—	2	8	no	—
99	C₄×D₄×C₆₇	—	2	4	no	—
100	C₄×D₄×C₆₇	C₂³	1	4	yes	—
101	C₄×D₄×C₆₇	C₂³	1	4	yes	—
102	#117×C₆₇	—	2	2	no	—
103	#117×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
106	C₂×C₄²×C₆₇	—	2	2	no	—
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	#107×C₆₇	C₂³	1	2	yes	—
109	#106×C₆₇	C₂³	1	2	yes	—
110	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
111	#103×C₆₇	C₂³	1	4	yes	—
112	#103×C₆₇	C₂³	1	16	yes	—
113	#102×C₆₇	C₂³	1	4	yes	—
114	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
115	C₄×C₈×C₆₇	C₂³	1	2	yes	—
116	C₄×C₈×C₆₇	C₂³	1	2	yes	—
117	#41×C₆₇	—	2	4	no	—
118	#41×C₆₇	—	2	4	no	—
119	#41×C₆₇	—	2	4	no	—
120	#41×C₆₇	—	2	16	no	—
121	#41×C₆₇	—	4	8	no	—
122	#41×C₆₇	—	4	16	no	—
123	#41×C₆₇	—	4	32	no	—
124	#41×C₆₇	C₂×D₄	1	8	yes	—
125	C₂×Q₈×C₆₇	—	2	8	no	—
126	C₂×Q₈×C₆₇	—	2	4	no	—
127	C₂×Q₈×C₆₇	—	2	16	no	—
128	C₂×Q₈×C₆₇	—	2	8	no	—
129	C₂×Q₈×C₆₇	—	2	8	no	—
130	C₂×Q₈×C₆₇	—	4	8	no	—
131	C₂×Q₈×C₆₇	—	4	16	no	—
132	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
134	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	8	no	—
137	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
138	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	8	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	8	no	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	Q₁₆×C₆₇	—	2	16	no	—
150	Q₁₆×C₆₇	—	2	8	no	—
151	Q₁₆×C₆₇	—	2	8	no	—
152	Q₁₆×C₆₇	—	4	8	no	—
153	Q₁₆×C₆₇	C₂×D₄	1	4	yes	—
154	SD₁₆×C₆₇	—	2	16	no	—
155	SD₁₆×C₆₇	C₂×D₄	1	8	yes	—
156	D₈×C₆₇	C₂×D₄	1	4	yes	—
157	M₁₆×C₆₇	—	2	4	no	—
158	M₁₆×C₆₇	—	2	8	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	8	no	—
161	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
162	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
163	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
164	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₆₇	—	2	8	no	—
167	C₄⋊C₄×C₆₇	—	2	8	no	—
168	C₄⋊C₄×C₆₇	—	2	16	no	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
171	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
172	C₂²⋊C₄×C₆₇	—	2	2	no	—
173	C₂²⋊C₄×C₆₇	—	2	16	no	—
174	C₂²⋊C₄×C₆₇	—	2	4	no	—
175	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
176	C₄²×C₆₇	—	2	2	no	—
177	C₄²×C₆₇	—	2	4	no	—
178	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₆₇	C₂⁴	1	2	yes	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
183	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	Q₈×C₆₇	—	2	4	no	—
185	Q₈×C₆₇	—	2	2	no	—
186	Q₈×C₆₇	—	2	8	no	—
187	Q₈×C₆₇	—	2	8	no	—
188	Q₈×C₆₇	—	4	16	no	—
189	Q₈×C₆₇	—	4	8	no	—
190	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
191	D₄×C₆₇	—	2	8	no	—
192	D₄×C₆₇	—	2	8	no	—
193	D₄×C₆₇	—	2	4	no	—
194	D₄×C₆₇	—	2	8	no	—
195	D₄×C₆₇	—	2	2	no	—
196	D₄×C₆₇	—	4	16	no	—
197	D₄×C₆₇	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	8	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	4	16	no	—
206	C₂×C₄×C₆₇	—	4	16	no	—
207	C₂×C₄×C₆₇	—	4	8	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
215	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
216	C₈×C₆₇	—	2	4	no	—
217	C₈×C₆₇	—	2	2	no	—
218	C₈×C₆₇	C₂²×D₄	1	2	yes	—
219	C₈×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂²×C₆₇	—	2	2	no	—
221	C₂²×C₆₇	—	2	4	no	—
222	C₂²×C₆₇	—	2	4	no	—
223	C₂²×C₆₇	—	2	8	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	D₄²	1	2	yes	—
226	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
227	C₄×C₆₇	—	2	4	no	—
228	C₄×C₆₇	—	2	4	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	4	8	no	—
231	C₄×C₆₇	—	4	4	no	—
232	C₄×C₆₇	D₄²	1	2	yes	—
233	C₄×C₆₇	#578	1	4	yes	complemented
234	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
235	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
236	#61091	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂×C₆₇	—	2	2	no	—
238	C₂×C₆₇	—	4	4	no	—
239	C₂×C₆₇	—	4	4	no	—
240	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
241	C₂×C₆₇	C₂×#578	1	2	yes	—
242	C₂×#578	C₂×C₆₇	1	2	yes	complemented
243	#3499	C₂×C₆₇	1	2	yes	complemented
244	D₄×Q₁₆	C₂×C₆₇	1	4	yes	complemented
245	#3283	C₂×C₆₇	1	4	yes	complemented
246	#3276	C₂×C₆₇	1	8	yes	complemented
247	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#3059	C₂×C₆₇	1	2	yes	complemented
249	#3039	C₂×C₆₇	1	2	yes	complemented
250	#3011	C₂×C₆₇	1	4	yes	complemented
251	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₆₇	#61091	1	1	yes	characteristic, direct product factor, Sylow
254	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	—
255	#578	—	2	16	no	—
256	C₂×#137	C₂²×C₆₇	1	4	yes	—
257	C₂×#137	C₂²×C₆₇	1	8	yes	—
258	C₂×#135	C₂²×C₆₇	1	4	yes	—
259	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
261	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
262	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
263	C₂×#131	C₂²×C₆₇	1	2	yes	—
264	D₄×Q₈	C₂²×C₆₇	1	8	yes	—
265	#547	—	2	8	no	—
266	#547	C₂²×C₆₇	1	8	yes	—
267	#536	—	2	4	no	—
268	#536	C₂²×C₆₇	1	4	yes	—
269	#532	—	2	2	no	—
270	#532	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄⋊Q₈	—	2	2	no	—
272	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
273	C₄×#41	C₂²×C₆₇	1	2	yes	—
274	#494	C₂²×C₆₇	1	4	yes	—
275	#494	C₂²×C₆₇	1	4	yes	—
276	#470	C₂²×C₆₇	1	8	yes	—
277	#467	C₂²×C₆₇	1	8	yes	—
278	#462	C₂²×C₆₇	1	8	yes	—
279	#461	C₂²×C₆₇	1	8	yes	—
280	#452	C₂²×C₆₇	1	16	yes	—
281	#451	C₂²×C₆₇	1	16	yes	—
282	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
283	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
284	C₄×D₈	C₂²×C₆₇	1	2	yes	—
285	#436	C₂²×C₆₇	1	4	yes	—
286	C₈×D₄	C₂²×C₆₇	1	4	yes	—
287	#427	C₂²×C₆₇	1	1	yes	characteristic
288	#423	C₂²×C₆₇	1	1	yes	characteristic
289	#419	C₂²×C₆₇	1	4	yes	—
290	#416	C₂²×C₆₇	1	2	yes	—
291	#408	C₂²×C₆₇	1	2	yes	—
292	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
293	2₋¹⁺⁴	—	2	16	no	—
294	2₋¹⁺⁴	—	4	8	no	—
295	C₂×#41	—	2	4	no	—
296	C₂×#41	—	2	8	no	—
297	C₂×#41	C₂³×C₆₇	1	8	yes	—
298	C₂×#41	C₂³×C₆₇	1	2	yes	—
299	C₂×#41	C₂³×C₆₇	1	2	yes	—
300	C₂²×Q₈	—	2	4	no	—
301	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
302	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
303	#137	—	2	16	no	—
304	#137	—	2	32	no	—
305	#135	—	2	16	no	—
306	#135	D₄×C₆₇	1	8	yes	complemented
307	C₂×Q₁₆	—	2	16	no	—
308	C₂×Q₁₆	—	2	8	no	—
309	C₂×Q₁₆	—	4	4	no	—
310	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
311	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
312	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
313	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
314	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
315	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
316	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
317	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
318	#131	—	2	8	no	—
319	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
320	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
321	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
322	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
323	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
324	C₄⋊Q₈	—	2	4	no	—
325	C₄⋊Q₈	—	2	8	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
329	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
330	#127	D₄×C₆₇	1	1	yes	characteristic
331	#124	—	2	4	no	—
332	#124	D₄×C₆₇	1	2	yes	—
333	#123	—	2	8	no	—
334	#123	—	2	16	no	—
335	C₂²⋊Q₈	—	2	16	no	—
336	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
337	#121	—	2	8	no	—
338	#121	D₄×C₆₇	1	4	yes	—
339	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
340	C₄×D₄	—	2	8	no	—
341	C₄×D₄	—	2	4	no	—
342	C₄×D₄	C₂³×C₆₇	1	4	yes	—
343	C₄×D₄	C₂³×C₆₇	1	4	yes	—
344	#117	—	2	2	no	—
345	#117	C₂³×C₆₇	1	2	yes	—
346	C₂×C₄⋊C₄	—	2	4	no	—
347	C₂×C₄⋊C₄	—	2	8	no	—
348	C₂×C₄²	—	2	2	no	—
349	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
350	#107	C₂³×C₆₇	1	2	yes	—
351	#106	C₂³×C₆₇	1	2	yes	—
352	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
353	#103	C₂³×C₆₇	1	4	yes	—
354	#103	C₂³×C₆₇	1	16	yes	—
355	#102	C₂³×C₆₇	1	4	yes	—
356	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
357	C₄×C₈	C₂³×C₆₇	1	2	yes	—
358	C₄×C₈	C₂³×C₆₇	1	2	yes	—
359	#41	—	2	4	no	—
360	#41	—	2	4	no	—
361	#41	—	2	4	no	—
362	#41	—	2	16	no	—
363	#41	—	4	8	no	—
364	#41	—	4	16	no	—
365	#41	—	4	32	no	—
366	#41	C₂×D₄×C₆₇	1	8	yes	—
367	C₂×Q₈	—	2	8	no	—
368	C₂×Q₈	—	2	4	no	—
369	C₂×Q₈	—	2	16	no	—
370	C₂×Q₈	—	2	8	no	—
371	C₂×Q₈	—	2	8	no	—
372	C₂×Q₈	—	4	8	no	—
373	C₂×Q₈	—	4	16	no	—
374	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
375	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
376	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	8	no	—
379	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
380	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
381	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
382	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
390	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
391	Q₁₆	—	2	16	no	—
392	Q₁₆	—	2	8	no	—
393	Q₁₆	—	2	8	no	—
394	Q₁₆	—	4	8	no	—
395	Q₁₆	C₂×D₄×C₆₇	1	4	yes	—
396	SD₁₆	—	2	16	no	—
397	SD₁₆	C₂×D₄×C₆₇	1	8	yes	—
398	D₈	C₂×D₄×C₆₇	1	4	yes	—
399	M₁₆	—	2	4	no	—
400	M₁₆	—	2	8	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	8	no	—
403	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
404	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
405	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
406	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
407	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	16	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
413	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	16	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	4	no	—
420	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
422	C₄²	C₂⁴×C₆₇	1	2	yes	—
423	C₂³	—	2	2	no	—
424	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
425	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
426	Q₈	—	2	4	no	—
427	Q₈	—	2	2	no	—
428	Q₈	—	2	8	no	—
429	Q₈	—	2	8	no	—
430	Q₈	—	4	16	no	—
431	Q₈	—	4	8	no	—
432	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
433	D₄	—	2	8	no	—
434	D₄	—	2	8	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	8	no	—
437	D₄	—	2	2	no	—
438	D₄	—	4	16	no	—
439	D₄	C₂²×D₄×C₆₇	1	4	yes	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	4	16	no	—
448	C₂×C₄	—	4	16	no	—
449	C₂×C₄	—	4	8	no	—
450	C₂×C₄	—	4	8	no	—
451	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
453	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
455	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
456	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
457	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
458	C₈	—	2	4	no	—
459	C₈	—	2	2	no	—
460	C₈	C₂²×D₄×C₆₇	1	2	yes	—
461	C₈	C₂²×D₄×C₆₇	1	2	yes	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	8	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	4	4	no	—
467	C₂²	D₄²×C₆₇	1	2	yes	—
468	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
469	C₄	—	2	2	no	—
470	C₄	—	2	4	no	—
471	C₄	—	2	4	no	—
472	C₄	—	4	4	no	—
473	C₄	—	4	8	no	—
474	C₄	D₄²×C₆₇	1	2	yes	—
475	C₄	#578×C₆₇	1	4	yes	complemented
476	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
477	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
478	C₂	—	2	2	no	—
479	C₂	—	4	4	no	—
480	C₂	—	4	4	no	—
481	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
482	C₂	C₂×#578×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #532×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#578×C₆₇	C₂	semidirect
2. C₆₇×#3499	C₂	semidirect
3. D₄×Q₁₆×C₆₇	C₂	semidirect
4. C₆₇×#3283	C₂	semidirect
5. C₆₇×#3276	C₂	semidirect
6. C₂×#494×C₆₇	C₂	semidirect
7. C₆₇×#3059	C₂	semidirect
8. C₆₇×#3039	C₂	semidirect
9. C₆₇×#3011	C₂	semidirect
10. C₄×#135×C₆₇	C₂	semidirect
11. C₆₇×#2931	C₂	semidirect
64. #135×C₆₇	D₄	semidirect
233. C₄×C₆₇	#578	semidirect
236. #61091	C₆₇	direct (N × H)
242. C₂×#578	C₂×C₆₇	semidirect
243. #3499	C₂×C₆₇	semidirect
244. D₄×Q₁₆	C₂×C₆₇	semidirect
245. #3283	C₂×C₆₇	semidirect
246. #3276	C₂×C₆₇	semidirect
247. C₂×#494	C₂×C₆₇	semidirect
248. #3059	C₂×C₆₇	semidirect
249. #3039	C₂×C₆₇	semidirect
250. #3011	C₂×C₆₇	semidirect
251. C₄×#135	C₂×C₆₇	semidirect
252. #2931	C₂×C₆₇	semidirect
253. C₆₇	#61091	direct (N × H)
306. #135	D₄×C₆₇	semidirect
475. C₄	#578×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#578×C₆₇	2
C₆₇×#3499	2
D₄×Q₁₆×C₆₇	2
C₆₇×#3283	2
C₆₇×#3276	2
C₂×#494×C₆₇	2
C₆₇×#3059	2
C₆₇×#3039	2
C₆₇×#3011	2
C₄×#135×C₆₇	2
C₆₇×#2931	2
#61091	67

Subgroups — C67×#61091 #30740342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61091 #30740342