Subgroups — C₆₇×#61662 #30740913

Order 17152.

2144 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	C₆₇×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3498	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3324	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3304	C₂	1	4	yes	maximal, complemented
5	D₄×SD₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3278	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3208	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3200	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3107	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3009	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
13	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
14	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
15	#563×C₆₇	—	2	2	no	—
16	#559×C₆₇	—	2	1	no	—
17	Q₈²×C₆₇	—	2	1	no	—
18	#556×C₆₇	C₂²	1	4	yes	complemented
19	#550×C₆₇	C₂²	1	4	yes	—
20	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
21	D₄²×C₆₇	—	2	4	no	—
22	D₄²×C₆₇	C₂²	1	4	yes	—
23	#542×C₆₇	—	2	4	no	—
24	#536×C₆₇	C₂²	1	4	yes	complemented
25	#535×C₆₇	—	2	2	no	—
26	#535×C₆₇	C₂²	1	2	yes	—
27	#533×C₆₇	C₂²	1	2	yes	complemented
28	#532×C₆₇	—	2	1	no	—
29	#532×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#127×C₆₇	—	2	1	no	—
31	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
33	#492×C₆₇	C₂²	1	4	yes	complemented
34	#486×C₆₇	C₂²	1	4	yes	complemented
35	#485×C₆₇	C₂²	1	4	yes	complemented
36	#482×C₆₇	C₂²	1	8	yes	complemented
37	#475×C₆₇	C₂²	1	2	yes	complemented
38	#474×C₆₇	C₂²	1	2	yes	complemented
39	#468×C₆₇	C₂²	1	8	yes	complemented
40	#465×C₆₇	C₂²	1	8	yes	complemented
41	#462×C₆₇	C₂²	1	2	yes	complemented
42	#461×C₆₇	C₂²	1	2	yes	complemented
43	#460×C₆₇	C₂²	1	4	yes	complemented
44	#460×C₆₇	C₂²	1	4	yes	complemented
45	#452×C₆₇	C₂²	1	4	yes	—
46	#452×C₆₇	C₂²	1	8	yes	complemented
47	#450×C₆₇	C₂²	1	8	yes	complemented
48	#450×C₆₇	C₂²	1	8	yes	complemented
49	#448×C₆₇	C₂²	1	8	yes	complemented
50	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
51	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
52	#436×C₆₇	C₂²	1	4	yes	complemented
53	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
54	#431×C₆₇	C₂²	1	2	yes	complemented
55	#418×C₆₇	C₂²	1	4	yes	complemented
56	#416×C₆₇	C₂²	1	4	yes	complemented
57	#408×C₆₇	C₂²	1	4	yes	complemented
58	H(5,𝔽₂)×C₆₇	—	4	4	no	—
59	H(5,𝔽₂)×C₆₇	—	4	4	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×#41×C₆₇	C₂³	1	4	yes	—
63	C₂×#41×C₆₇	C₂³	1	2	yes	—
64	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	—	2	4	no	—
67	C₂²×D₄×C₆₇	—	2	4	no	—
68	C₂²×D₄×C₆₇	—	2	2	no	—
69	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
71	#137×C₆₇	—	4	8	no	—
72	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
73	C₂×SD₁₆×C₆₇	—	4	8	no	—
74	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
76	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
77	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
79	C₄⋊Q₈×C₆₇	—	2	4	no	—
80	C₄⋊Q₈×C₆₇	—	2	4	no	—
81	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
83	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
84	#127×C₆₇	—	2	2	no	—
85	#127×C₆₇	—	2	4	no	—
86	#127×C₆₇	—	2	1	no	—
87	#127×C₆₇	—	2	1	no	—
88	#127×C₆₇	D₄	1	1	yes	characteristic
89	#127×C₆₇	C₂³	1	4	yes	—
90	#126×C₆₇	—	2	8	no	—
91	#125×C₆₇	—	2	2	no	—
92	#124×C₆₇	—	2	2	no	—
93	#124×C₆₇	—	2	4	no	—
94	#124×C₆₇	—	2	4	no	—
95	#124×C₆₇	D₄	1	2	yes	—
96	#123×C₆₇	—	2	4	no	—
97	#123×C₆₇	—	2	4	no	—
98	C₂²⋊Q₈×C₆₇	—	2	2	no	—
99	C₂²⋊Q₈×C₆₇	—	2	8	no	—
100	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
101	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
102	#121×C₆₇	—	2	4	no	—
103	#121×C₆₇	—	2	8	no	—
104	#121×C₆₇	D₄	1	4	yes	—
105	#121×C₆₇	C₂³	1	8	yes	—
106	#120×C₆₇	—	2	4	no	—
107	#120×C₆₇	—	2	8	no	—
108	C₄×Q₈×C₆₇	—	2	4	no	—
109	C₄×Q₈×C₆₇	—	2	2	no	—
110	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
111	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
112	C₄×D₄×C₆₇	—	2	4	no	—
113	C₄×D₄×C₆₇	—	2	2	no	—
114	C₄×D₄×C₆₇	—	2	4	no	—
115	C₄×D₄×C₆₇	C₂³	1	2	yes	—
116	C₄×D₄×C₆₇	C₂³	1	4	yes	—
117	#117×C₆₇	—	2	2	no	—
118	#117×C₆₇	—	2	1	no	—
119	#117×C₆₇	C₂³	1	4	yes	—
120	#117×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	—	2	1	no	—
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	#107×C₆₇	C₂³	1	4	yes	—
124	#106×C₆₇	C₂³	1	4	yes	—
125	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
126	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
127	#103×C₆₇	C₂³	1	4	yes	—
128	#103×C₆₇	C₂³	1	4	yes	—
129	#103×C₆₇	C₂³	1	4	yes	—
130	#102×C₆₇	C₂³	1	4	yes	—
131	#102×C₆₇	C₂³	1	8	yes	—
132	#102×C₆₇	C₂³	1	8	yes	—
133	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
134	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
135	C₄×C₈×C₆₇	C₂³	1	4	yes	—
136	C₂⁴×C₆₇	—	2	4	no	—
137	C₂⁴×C₆₇	—	2	2	no	—
138	#41×C₆₇	—	2	2	no	—
139	#41×C₆₇	—	2	2	no	—
140	#41×C₆₇	—	2	8	no	—
141	#41×C₆₇	—	4	2	no	—
142	#41×C₆₇	—	4	2	no	—
143	#41×C₆₇	—	4	4	no	—
144	#41×C₆₇	—	4	8	no	—
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	—	2	4	no	—
147	C₂×Q₈×C₆₇	—	4	4	no	—
148	C₂×Q₈×C₆₇	—	4	4	no	—
149	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
150	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
151	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	8	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	8	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	2	no	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	—	4	4	no	—
161	C₂×D₄×C₆₇	—	4	2	no	—
162	C₂×D₄×C₆₇	—	4	2	no	—
163	C₂×D₄×C₆₇	—	4	8	no	—
164	C₂×D₄×C₆₇	—	4	16	no	—
165	C₂×D₄×C₆₇	—	8	4	no	—
166	C₂×D₄×C₆₇	—	8	8	no	—
167	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
168	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
169	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
170	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	2	no	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
179	Q₁₆×C₆₇	—	2	8	no	—
180	SD₁₆×C₆₇	—	2	8	no	—
181	SD₁₆×C₆₇	—	4	8	no	—
182	SD₁₆×C₆₇	—	4	8	no	—
183	M₁₆×C₆₇	—	4	4	no	—
184	C₂×C₈×C₆₇	—	4	4	no	—
185	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
186	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	8	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	1	no	—
199	C₂²⋊C₄×C₆₇	—	2	4	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₂²⋊C₄×C₆₇	—	2	2	no	—
202	C₂²⋊C₄×C₆₇	—	2	8	no	—
203	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
204	C₄²×C₆₇	—	2	1	no	—
205	C₄²×C₆₇	—	2	2	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂⁴	1	2	yes	—
211	C₂³×C₆₇	—	2	1	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	4	8	no	—
214	C₂³×C₆₇	—	4	4	no	—
215	C₂³×C₆₇	—	8	4	no	—
216	C₂³×C₆₇	—	8	8	no	—
217	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
218	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
219	Q₈×C₆₇	—	2	1	no	—
220	Q₈×C₆₇	—	2	4	no	—
221	Q₈×C₆₇	—	2	4	no	—
222	Q₈×C₆₇	—	4	2	no	—
223	Q₈×C₆₇	—	4	4	no	—
224	Q₈×C₆₇	—	8	4	no	—
225	Q₈×C₆₇	H(5,𝔽₂)	1	2	yes	—
226	D₄×C₆₇	—	2	4	no	—
227	D₄×C₆₇	—	2	8	no	—
228	D₄×C₆₇	—	2	1	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	4	2	no	—
232	D₄×C₆₇	—	4	4	no	—
233	D₄×C₆₇	—	4	8	no	—
234	D₄×C₆₇	—	4	8	no	—
235	D₄×C₆₇	—	4	4	no	—
236	D₄×C₆₇	—	4	8	no	—
237	D₄×C₆₇	—	8	4	no	—
238	D₄×C₆₇	—	8	8	no	—
239	D₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	4	8	no	—
251	C₂×C₄×C₆₇	—	4	4	no	—
252	C₂×C₄×C₆₇	—	8	4	no	—
253	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
258	C₈×C₆₇	—	2	4	no	—
259	C₈×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	2	4	no	—
261	C₂²×C₆₇	—	2	1	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	8	no	—
265	C₂²×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	—	4	4	no	—
267	C₂²×C₆₇	—	4	8	no	—
268	C₂²×C₆₇	—	16	8	no	—
269	C₂²×C₆₇	—	16	4	no	—
270	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
271	C₄×C₆₇	—	2	1	no	—
272	C₄×C₆₇	—	2	2	no	—
273	C₄×C₆₇	—	2	4	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	4	4	no	—
276	C₄×C₆₇	—	4	2	no	—
277	C₄×C₆₇	—	4	4	no	—
278	C₄×C₆₇	#577	1	4	yes	—
279	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
280	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
281	#61662	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
282	C₂×C₆₇	—	4	4	no	—
283	C₂×C₆₇	—	4	1	no	—
284	C₂×C₆₇	—	4	2	no	—
285	C₂×C₆₇	—	8	4	no	—
286	C₂×C₆₇	#3484	1	1	yes	characteristic
287	C₂×C₆₇	C₂×#577	1	2	yes	—
288	#3550	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#3498	C₂×C₆₇	1	1	yes	characteristic
290	#3324	C₂×C₆₇	1	4	yes	complemented
291	#3304	C₂×C₆₇	1	4	yes	complemented
292	D₄×SD₁₆	C₂×C₆₇	1	4	yes	complemented
293	#3278	C₂×C₆₇	1	4	yes	complemented
294	#3208	C₂×C₆₇	1	2	yes	complemented
295	#3200	C₂×C₆₇	1	4	yes	complemented
296	#3107	C₂×C₆₇	1	2	yes	complemented
297	#3009	C₂×C₆₇	1	4	yes	complemented
298	#2979	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₆₇	#61662	1	1	yes	characteristic, direct product factor, Sylow
300	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
301	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
302	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
303	#563	—	2	2	no	—
304	#559	—	2	1	no	—
305	Q₈²	—	2	1	no	—
306	#556	C₂²×C₆₇	1	4	yes	complemented
307	#550	C₂²×C₆₇	1	4	yes	—
308	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
309	D₄²	—	2	4	no	—
310	D₄²	C₂²×C₆₇	1	4	yes	—
311	#542	—	2	4	no	—
312	#536	C₂²×C₆₇	1	4	yes	complemented
313	#535	—	2	2	no	—
314	#535	C₂²×C₆₇	1	2	yes	—
315	#533	C₂²×C₆₇	1	2	yes	complemented
316	#532	—	2	1	no	—
317	#532	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×#127	—	2	1	no	—
319	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
320	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
321	#492	C₂²×C₆₇	1	4	yes	complemented
322	#486	C₂²×C₆₇	1	4	yes	complemented
323	#485	C₂²×C₆₇	1	4	yes	complemented
324	#482	C₂²×C₆₇	1	8	yes	complemented
325	#475	C₂²×C₆₇	1	2	yes	complemented
326	#474	C₂²×C₆₇	1	2	yes	complemented
327	#468	C₂²×C₆₇	1	8	yes	complemented
328	#465	C₂²×C₆₇	1	8	yes	complemented
329	#462	C₂²×C₆₇	1	2	yes	complemented
330	#461	C₂²×C₆₇	1	2	yes	complemented
331	#460	C₂²×C₆₇	1	4	yes	complemented
332	#460	C₂²×C₆₇	1	4	yes	complemented
333	#452	C₂²×C₆₇	1	4	yes	—
334	#452	C₂²×C₆₇	1	8	yes	complemented
335	#450	C₂²×C₆₇	1	8	yes	complemented
336	#450	C₂²×C₆₇	1	8	yes	complemented
337	#448	C₂²×C₆₇	1	8	yes	complemented
338	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
339	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
340	#436	C₂²×C₆₇	1	4	yes	complemented
341	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
342	#431	C₂²×C₆₇	1	2	yes	complemented
343	#418	C₂²×C₆₇	1	4	yes	complemented
344	#416	C₂²×C₆₇	1	4	yes	complemented
345	#408	C₂²×C₆₇	1	4	yes	complemented
346	H(5,𝔽₂)	—	4	4	no	—
347	H(5,𝔽₂)	—	4	4	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
350	C₂×#41	C₂³×C₆₇	1	4	yes	—
351	C₂×#41	C₂³×C₆₇	1	2	yes	—
352	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	2	4	no	—
355	C₂²×D₄	—	2	4	no	—
356	C₂²×D₄	—	2	2	no	—
357	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
358	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
359	#137	—	4	8	no	—
360	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
361	C₂×SD₁₆	—	4	8	no	—
362	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
363	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
364	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
365	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
366	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
367	C₄⋊Q₈	—	2	4	no	—
368	C₄⋊Q₈	—	2	4	no	—
369	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
370	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
371	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
372	#127	—	2	2	no	—
373	#127	—	2	4	no	—
374	#127	—	2	1	no	—
375	#127	—	2	1	no	—
376	#127	D₄×C₆₇	1	1	yes	characteristic
377	#127	C₂³×C₆₇	1	4	yes	—
378	#126	—	2	8	no	—
379	#125	—	2	2	no	—
380	#124	—	2	2	no	—
381	#124	—	2	4	no	—
382	#124	—	2	4	no	—
383	#124	D₄×C₆₇	1	2	yes	—
384	#123	—	2	4	no	—
385	#123	—	2	4	no	—
386	C₂²⋊Q₈	—	2	2	no	—
387	C₂²⋊Q₈	—	2	8	no	—
388	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
389	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
390	#121	—	2	4	no	—
391	#121	—	2	8	no	—
392	#121	D₄×C₆₇	1	4	yes	—
393	#121	C₂³×C₆₇	1	8	yes	—
394	#120	—	2	4	no	—
395	#120	—	2	8	no	—
396	C₄×Q₈	—	2	4	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
399	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
400	C₄×D₄	—	2	4	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	4	no	—
403	C₄×D₄	C₂³×C₆₇	1	2	yes	—
404	C₄×D₄	C₂³×C₆₇	1	4	yes	—
405	#117	—	2	2	no	—
406	#117	—	2	1	no	—
407	#117	C₂³×C₆₇	1	1	yes	characteristic
408	#117	C₂³×C₆₇	1	4	yes	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
411	#107	C₂³×C₆₇	1	4	yes	—
412	#106	C₂³×C₆₇	1	4	yes	—
413	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
414	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
415	#103	C₂³×C₆₇	1	4	yes	—
416	#103	C₂³×C₆₇	1	4	yes	—
417	#103	C₂³×C₆₇	1	4	yes	—
418	#102	C₂³×C₆₇	1	8	yes	—
419	#102	C₂³×C₆₇	1	4	yes	—
420	#102	C₂³×C₆₇	1	8	yes	—
421	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
422	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
423	C₄×C₈	C₂³×C₆₇	1	4	yes	—
424	C₂⁴	—	2	4	no	—
425	C₂⁴	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	2	8	no	—
429	#41	—	4	2	no	—
430	#41	—	4	2	no	—
431	#41	—	4	4	no	—
432	#41	—	4	8	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	4	4	no	—
436	C₂×Q₈	—	4	4	no	—
437	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
438	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
439	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	4	4	no	—
449	C₂×D₄	—	4	2	no	—
450	C₂×D₄	—	4	2	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	—	4	16	no	—
453	C₂×D₄	—	8	4	no	—
454	C₂×D₄	—	8	8	no	—
455	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
456	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
457	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
458	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
459	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
467	Q₁₆	—	2	8	no	—
468	SD₁₆	—	2	8	no	—
469	SD₁₆	—	4	8	no	—
470	SD₁₆	—	4	8	no	—
471	M₁₆	—	4	4	no	—
472	C₂×C₈	—	4	4	no	—
473	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
474	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	8	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
483	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
484	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	4	no	—
495	C₄²	—	2	4	no	—
496	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
497	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
498	C₄²	C₂⁴×C₆₇	1	2	yes	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	4	8	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	8	4	no	—
504	C₂³	—	8	8	no	—
505	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
506	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
507	Q₈	—	2	1	no	—
508	Q₈	—	2	4	no	—
509	Q₈	—	2	4	no	—
510	Q₈	—	4	2	no	—
511	Q₈	—	4	4	no	—
512	Q₈	—	8	4	no	—
513	Q₈	H(5,𝔽₂)×C₆₇	1	2	yes	—
514	D₄	—	2	4	no	—
515	D₄	—	2	8	no	—
516	D₄	—	2	1	no	—
517	D₄	—	2	4	no	—
518	D₄	—	2	4	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	4	no	—
521	D₄	—	4	8	no	—
522	D₄	—	4	8	no	—
523	D₄	—	4	4	no	—
524	D₄	—	4	8	no	—
525	D₄	—	8	4	no	—
526	D₄	—	8	8	no	—
527	D₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	8	4	no	—
541	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
543	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
544	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
545	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
546	C₈	—	2	4	no	—
547	C₈	—	2	4	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	8	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	8	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	16	8	no	—
557	C₂²	—	16	4	no	—
558	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
559	C₄	—	2	2	no	—
560	C₄	—	2	1	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	4	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	4	no	—
565	C₄	—	4	4	no	—
566	C₄	#577×C₆₇	1	4	yes	—
567	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
568	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
569	C₂	—	4	1	no	—
570	C₂	—	4	4	no	—
571	C₂	—	4	2	no	—
572	C₂	—	8	4	no	—
573	C₂	C₆₇×#3484	1	1	yes	characteristic
574	C₂	C₂×#577×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3498	C₂
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. C₆₇×#3550	C₂	semidirect
3. C₆₇×#3324	C₂	semidirect
4. C₆₇×#3304	C₂	semidirect
5. D₄×SD₁₆×C₆₇	C₂	semidirect
6. C₆₇×#3278	C₂	semidirect
7. C₆₇×#3208	C₂	semidirect
8. C₆₇×#3200	C₂	semidirect
9. C₆₇×#3107	C₂	semidirect
10. C₆₇×#3009	C₂	semidirect
11. C₆₇×#2979	C₂	semidirect
13. C₂×#137×C₆₇	C₂²	semidirect
14. C₂²×SD₁₆×C₆₇	C₂²	semidirect
18. #556×C₆₇	C₂²	semidirect
20. D₄×Q₈×C₆₇	C₂²	semidirect
24. #536×C₆₇	C₂²	semidirect
27. #533×C₆₇	C₂²	semidirect
33. #492×C₆₇	C₂²	semidirect
34. #486×C₆₇	C₂²	semidirect
35. #485×C₆₇	C₂²	semidirect
36. #482×C₆₇	C₂²	semidirect
37. #475×C₆₇	C₂²	semidirect
38. #474×C₆₇	C₂²	semidirect
39. #468×C₆₇	C₂²	semidirect
40. #465×C₆₇	C₂²	semidirect
41. #462×C₆₇	C₂²	semidirect
42. #461×C₆₇	C₂²	semidirect
43. #460×C₆₇	C₂²	semidirect
44. #460×C₆₇	C₂²	semidirect
46. #452×C₆₇	C₂²	semidirect
47. #450×C₆₇	C₂²	semidirect
48. #450×C₆₇	C₂²	semidirect
49. #448×C₆₇	C₂²	semidirect
50. C₄×Q₁₆×C₆₇	C₂²	semidirect
51. C₄×SD₁₆×C₆₇	C₂²	semidirect
52. #436×C₆₇	C₂²	semidirect
53. C₈×D₄×C₆₇	C₂²	semidirect
54. #431×C₆₇	C₂²	semidirect
55. #418×C₆₇	C₂²	semidirect
56. #416×C₆₇	C₂²	semidirect
57. #408×C₆₇	C₂²	semidirect
281. #61662	C₆₇	direct (N × H)
288. #3550	C₂×C₆₇	semidirect
290. #3324	C₂×C₆₇	semidirect
291. #3304	C₂×C₆₇	semidirect
292. D₄×SD₁₆	C₂×C₆₇	semidirect
293. #3278	C₂×C₆₇	semidirect
294. #3208	C₂×C₆₇	semidirect
295. #3200	C₂×C₆₇	semidirect
296. #3107	C₂×C₆₇	semidirect
297. #3009	C₂×C₆₇	semidirect
298. #2979	C₂×C₆₇	semidirect
299. C₆₇	#61662	direct (N × H)
301. C₂×#137	C₂²×C₆₇	semidirect
302. C₂²×SD₁₆	C₂²×C₆₇	semidirect
306. #556	C₂²×C₆₇	semidirect
308. D₄×Q₈	C₂²×C₆₇	semidirect
312. #536	C₂²×C₆₇	semidirect
315. #533	C₂²×C₆₇	semidirect
321. #492	C₂²×C₆₇	semidirect
322. #486	C₂²×C₆₇	semidirect
323. #485	C₂²×C₆₇	semidirect
324. #482	C₂²×C₆₇	semidirect
325. #475	C₂²×C₆₇	semidirect
326. #474	C₂²×C₆₇	semidirect
327. #468	C₂²×C₆₇	semidirect
328. #465	C₂²×C₆₇	semidirect
329. #462	C₂²×C₆₇	semidirect
330. #461	C₂²×C₆₇	semidirect
331. #460	C₂²×C₆₇	semidirect
332. #460	C₂²×C₆₇	semidirect
334. #452	C₂²×C₆₇	semidirect
335. #450	C₂²×C₆₇	semidirect
336. #450	C₂²×C₆₇	semidirect
337. #448	C₂²×C₆₇	semidirect
338. C₄×Q₁₆	C₂²×C₆₇	semidirect
339. C₄×SD₁₆	C₂²×C₆₇	semidirect
340. #436	C₂²×C₆₇	semidirect
341. C₈×D₄	C₂²×C₆₇	semidirect
342. #431	C₂²×C₆₇	semidirect
343. #418	C₂²×C₆₇	semidirect
344. #416	C₂²×C₆₇	semidirect
345. #408	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3550	2
C₆₇×#3498	2
C₆₇×#3324	2
C₆₇×#3304	2
D₄×SD₁₆×C₆₇	2
C₆₇×#3278	2
C₆₇×#3208	2
C₆₇×#3200	2
C₆₇×#3107	2
C₆₇×#3009	2
C₆₇×#2979	2
#61662	67

Subgroups — C67×#61662 #30740913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61662 #30740913