Subgroups — C₆₇×#33951 #30713202

Order 17152.

498 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2255	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2228	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2226	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2177	C₂	1	2	yes	maximal
11	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#554×C₆₇	—	2	1	no	—
13	D₄×Q₈×C₆₇	—	2	1	no	—
14	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#500×C₆₇	C₂²	1	1	yes	characteristic
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#500×C₆₇	C₂²	1	2	yes	—
18	#499×C₆₇	C₂²	1	2	yes	—
19	#478×C₆₇	—	2	2	no	—
20	#476×C₆₇	—	2	1	no	—
21	#474×C₆₇	—	2	1	no	—
22	#474×C₆₇	—	2	2	no	—
23	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₆₇	C₂²	1	2	yes	—
26	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₇	C₂²	1	2	yes	—
28	#367×C₆₇	C₂²	1	2	yes	—
29	#367×C₆₇	C₂²	1	2	yes	—
30	#367×C₆₇	C₂²	1	2	yes	—
31	#366×C₆₇	C₂²	1	1	yes	characteristic
32	#366×C₆₇	C₂²	1	2	yes	—
33	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
35	#357×C₆₇	C₂²	1	1	yes	characteristic
36	#357×C₆₇	C₂²	1	1	yes	characteristic
37	#357×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	2	yes	—
39	#346×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₆₇	—	4	1	no	—
42	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	—	2	1	no	—
46	C₄⋊Q₈×C₆₇	—	2	1	no	—
47	#125×C₆₇	—	2	2	no	—
48	C₂²⋊Q₈×C₆₇	—	4	1	no	—
49	C₂²⋊Q₈×C₆₇	—	4	1	no	—
50	C₂²⋊Q₈×C₆₇	—	4	2	no	—
51	C₂²⋊Q₈×C₆₇	—	4	2	no	—
52	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄×D₄×C₆₇	—	2	2	no	—
54	C₄×D₄×C₆₇	—	2	1	no	—
55	C₄×D₄×C₆₇	—	2	1	no	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
58	D₁₆×C₆₇	Q₈	1	4	yes	complemented
59	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
62	#107×C₆₇	—	2	2	no	—
63	#107×C₆₇	—	2	1	no	—
64	#107×C₆₇	—	2	1	no	—
65	#107×C₆₇	C₂³	1	1	yes	characteristic
66	#107×C₆₇	C₂³	1	2	yes	—
67	#107×C₆₇	C₂³	1	2	yes	—
68	#107×C₆₇	C₂³	1	1	yes	characteristic
69	#106×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
71	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
72	#102×C₆₇	—	2	2	no	—
73	#102×C₆₇	—	2	1	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	1	no	—
76	C₄×C₈×C₆₇	C₂³	1	2	yes	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₆₇	—	4	2	no	—
79	C₂×Q₈×C₆₇	—	4	1	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	—	8	4	no	—
82	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
83	C₂×D₄×C₆₇	—	2	1	no	—
84	C₂×D₄×C₆₇	—	2	1	no	—
85	C₂²×C₄×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	—	4	2	no	—
89	D₈×C₆₇	—	2	1	no	—
90	D₈×C₆₇	—	2	1	no	—
91	D₈×C₆₇	C₂×Q₈	1	2	yes	—
92	D₈×C₆₇	C₂×Q₈	1	2	yes	complemented
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₄⋊C₄×C₆₇	—	4	1	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
106	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
107	C₂²⋊C₄×C₆₇	—	4	1	no	—
108	C₂²⋊C₄×C₆₇	—	4	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	1	no	—
110	C₂²⋊C₄×C₆₇	—	4	2	no	—
111	C₄²×C₆₇	C₂×D₄	1	2	yes	—
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₆₇	—	2	1	no	—
114	C₁₆×C₆₇	—	2	2	no	—
115	C₁₆×C₆₇	C₂×Q₈	1	2	yes	complemented
116	C₂³×C₆₇	—	4	1	no	—
117	C₂³×C₆₇	—	4	1	no	—
118	Q₈×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	—	8	4	no	—
120	Q₈×C₆₇	—	8	2	no	—
121	Q₈×C₆₇	—	16	1	no	—
122	D₄×C₆₇	—	2	2	no	—
123	D₄×C₆₇	—	2	2	no	—
124	D₄×C₆₇	—	4	1	no	—
125	D₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	8	4	no	—
131	C₂×C₄×C₆₇	—	8	2	no	—
132	C₂×C₄×C₆₇	—	8	2	no	—
133	C₂×C₄×C₆₇	—	16	2	no	—
134	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
142	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	—	8	1	no	—
146	C₂²×C₆₇	—	8	1	no	—
147	C₂²×C₆₇	—	8	1	no	—
148	C₂²×C₆₇	—	8	1	no	—
149	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	8	1	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	—	8	1	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	#509	1	2	yes	—
158	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
159	C₄×C₆₇	#578	1	1	yes	characteristic
160	#33951	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₆₇	—	8	2	no	—
162	C₂×C₆₇	—	8	2	no	—
163	C₂×C₆₇	Q₈×D₈	1	1	yes	characteristic
164	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
165	C₂×C₆₇	#3416	1	1	yes	characteristic
166	#3380	C₂×C₆₇	1	1	yes	characteristic, complemented
167	Q₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
168	#2255	C₂×C₆₇	1	2	yes	complemented
169	#2254	C₂×C₆₇	1	1	yes	characteristic, complemented
170	#2228	C₂×C₆₇	1	2	yes	complemented
171	#2226	C₂×C₆₇	1	2	yes	complemented
172	#2226	C₂×C₆₇	1	1	yes	characteristic, complemented
173	#2224	C₂×C₆₇	1	1	yes	characteristic, complemented
174	#2183	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#2177	C₂×C₆₇	1	2	yes	—
176	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic
177	C₆₇	#33951	1	1	yes	characteristic, direct product factor, Sylow
178	#554	—	2	1	no	—
179	D₄×Q₈	—	2	1	no	—
180	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
181	#500	C₂²×C₆₇	1	1	yes	characteristic
182	#500	C₂²×C₆₇	1	2	yes	—
183	#500	C₂²×C₆₇	1	1	yes	characteristic
184	#499	C₂²×C₆₇	1	2	yes	—
185	#478	—	2	2	no	—
186	#476	—	2	1	no	—
187	#474	—	2	1	no	—
188	#474	—	2	2	no	—
189	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
190	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
191	C₄×D₈	C₂²×C₆₇	1	2	yes	—
192	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₆₇	1	2	yes	—
194	#367	C₂²×C₆₇	1	2	yes	—
195	#367	C₂²×C₆₇	1	2	yes	—
196	#367	C₂²×C₆₇	1	2	yes	—
197	#366	C₂²×C₆₇	1	1	yes	characteristic
198	#366	C₂²×C₆₇	1	2	yes	—
199	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
201	#357	C₂²×C₆₇	1	1	yes	characteristic
202	#357	C₂²×C₆₇	1	1	yes	characteristic
203	#357	C₂²×C₆₇	1	2	yes	—
204	#357	C₂²×C₆₇	1	2	yes	—
205	#346	C₂²×C₆₇	1	2	yes	—
206	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
207	C₂²×Q₈	—	4	1	no	—
208	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
209	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#125	—	2	2	no	—
214	C₂²⋊Q₈	—	4	1	no	—
215	C₂²⋊Q₈	—	4	1	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	4	1	no	—
224	D₁₆	Q₈×C₆₇	1	4	yes	complemented
225	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
226	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
227	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
228	#107	—	2	2	no	—
229	#107	—	2	1	no	—
230	#107	—	2	1	no	—
231	#107	C₂³×C₆₇	1	1	yes	characteristic
232	#107	C₂³×C₆₇	1	1	yes	characteristic
233	#107	C₂³×C₆₇	1	2	yes	—
234	#107	C₂³×C₆₇	1	2	yes	—
235	#106	—	2	2	no	—
236	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
237	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
238	#102	—	2	2	no	—
239	#102	—	2	1	no	—
240	#102	—	2	2	no	—
241	#102	—	2	1	no	—
242	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₇	1	2	yes	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	8	4	no	—
248	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Q₈×C₆₇	1	2	yes	—
258	D₈	C₂×Q₈×C₆₇	1	2	yes	complemented
259	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
272	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	C₂×D₄×C₆₇	1	2	yes	—
278	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	C₂×Q₈×C₆₇	1	2	yes	complemented
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	8	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	1	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	16	2	no	—
300	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
302	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
303	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
308	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	#509×C₆₇	1	2	yes	—
324	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
325	C₄	#578×C₆₇	1	1	yes	characteristic
326	C₂	—	8	2	no	—
327	C₂	—	8	2	no	—
328	C₂	Q₈×D₈×C₆₇	1	1	yes	characteristic
329	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
330	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×D₁₆×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3380	C₂	semidirect
2. Q₈×D₈×C₆₇	C₂	semidirect
3. C₆₇×#2255	C₂	semidirect
4. C₆₇×#2254	C₂	semidirect
5. C₆₇×#2228	C₂	semidirect
6. C₆₇×#2226	C₂	semidirect
7. C₆₇×#2226	C₂	semidirect
8. C₆₇×#2224	C₂	semidirect
9. C₆₇×#2183	C₂	semidirect
58. D₁₆×C₆₇	Q₈	semidirect
92. D₈×C₆₇	C₂×Q₈	semidirect
115. C₁₆×C₆₇	C₂×Q₈	semidirect
160. #33951	C₆₇	direct (N × H)
166. #3380	C₂×C₆₇	semidirect
167. Q₈×D₈	C₂×C₆₇	semidirect
168. #2255	C₂×C₆₇	semidirect
169. #2254	C₂×C₆₇	semidirect
170. #2228	C₂×C₆₇	semidirect
171. #2226	C₂×C₆₇	semidirect
172. #2226	C₂×C₆₇	semidirect
173. #2224	C₂×C₆₇	semidirect
174. #2183	C₂×C₆₇	semidirect
177. C₆₇	#33951	direct (N × H)
224. D₁₆	Q₈×C₆₇	semidirect
258. D₈	C₂×Q₈×C₆₇	semidirect
281. C₁₆	C₂×Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3380	2
Q₈×D₈×C₆₇	2
C₆₇×#2255	2
C₆₇×#2254	2
C₆₇×#2228	2
C₆₇×#2226	2
C₆₇×#2224	2
C₆₇×#2183	2
C₆₇×#2177	2
C₄×D₁₆×C₆₇	2
#33951	67

Subgroups — C67×#33951 #30713202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33951 #30713202