Subgroups — #30955674

Order 17152.

810 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	C₄×#14524043	C₂	1	1	yes	maximal, characteristic
2	#22561347	C₂	1	4	yes	maximal
3	#22561010	C₂	1	2	yes	maximal
4	#22560336	C₂	1	2	yes	maximal
5	#22560211	C₂	1	4	yes	maximal
6	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524043	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×#14523966	C₂²	1	8	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
13	#16863302	C₂²	1	2	yes	—
14	#16863294	—	2	1	no	—
15	#16863294	—	2	1	no	—
16	#16863266	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊Dic₆₇	—	2	1	no	—
18	C₄×C₄⋊Dic₆₇	—	2	1	no	—
19	#16863158	C₂²	1	8	yes	—
20	#16863151	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
23	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
24	#384×C₆₇	—	2	1	no	—
25	#384×C₆₇	—	2	1	no	—
26	#384×C₆₇	C₂²	1	2	yes	—
27	#384×C₆₇	C₂²	1	2	yes	—
28	C₄³×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₃₄	C₂³	1	2	yes	—
30	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
31	#14524043	—	2	2	no	—
32	#14524043	Q₈	1	4	yes	complemented
33	C₄×Dic₁₃₄	—	2	8	no	—
34	C₂×C₄⋊Dic₆₇	—	2	1	no	—
35	C₂×C₄⋊Dic₆₇	—	2	1	no	—
36	C₂×C₄⋊Dic₆₇	—	2	1	no	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
39	C₂×Dic₆₇⋊C₄	—	2	2	no	—
40	C₂×C₄×Dic₆₇	—	2	2	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
42	#14523966	—	4	8	no	—
43	C₄×C₆₇⋊C₈	—	2	4	no	—
44	C₆₇⋊#95	—	2	2	no	—
45	C₆₇⋊#95	—	2	2	no	—
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	—	4	4	no	—
48	C₄⋊Q₈×C₆₇	—	4	4	no	—
49	C₄⋊Q₈×C₆₇	—	4	4	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₃₄	—	2	4	no	—
64	C₂×Dic₁₃₄	—	2	4	no	—
65	C₂×Dic₁₃₄	—	2	2	no	—
66	C₂×Dic₁₃₄	C₂×Q₈	1	4	yes	—
67	C₂²×Dic₆₇	—	2	2	no	—
68	C₂²×Dic₆₇	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	2	4	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₄⋊Dic₆₇	—	2	4	no	—
73	C₄⋊Dic₆₇	—	2	4	no	—
74	C₄⋊Dic₆₇	—	2	4	no	—
75	C₄⋊Dic₆₇	—	4	2	no	—
76	C₄⋊Dic₆₇	—	4	2	no	—
77	Dic₆₇⋊C₄	—	4	4	no	—
78	C₄×Dic₆₇	—	4	4	no	—
79	C₂×Q₈×C₆₇	—	4	4	no	—
80	C₂×Q₈×C₆₇	—	4	4	no	—
81	C₂×Q₈×C₆₇	—	4	4	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₆₇	—	2	8	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₄⋊C₄×C₆₇	—	4	4	no	—
93	C₄⋊C₄×C₆₇	—	4	4	no	—
94	C₄⋊C₄×C₆₇	—	4	4	no	—
95	C₄⋊C₄×C₆₇	—	4	4	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	SD₁₆	1	4	yes	—
100	C₄²×C₆₇	Q₁₆	1	4	yes	—
101	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
102	C₄²×C₆₇	#41	1	2	yes	—
103	C₄²×C₆₇	#41	1	4	yes	—
104	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₆₇	—	8	8	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₆₇	#124	1	2	yes	—
118	C₂×C₄×C₆₇	#124	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₆₇	#137	1	2	yes	—
127	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
131	C₂×Dic₆₇	—	2	4	no	—
132	C₂×Dic₆₇	—	4	2	no	—
133	C₂×Dic₆₇	—	4	2	no	—
134	C₂×Dic₆₇	—	4	2	no	—
135	C₂×Dic₆₇	—	4	2	no	—
136	C₂×Dic₆₇	—	4	2	no	—
137	Dic₁₃₄	—	2	8	no	—
138	Dic₁₃₄	—	4	4	no	—
139	C₆₇⋊C₈	—	4	4	no	—
140	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
142	C₂²×C₆₇	#535	1	1	yes	characteristic
143	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	2	2	no	—
149	C₄×C₆₇	—	2	2	no	—
150	C₄×C₆₇	—	8	4	no	—
151	C₄×C₆₇	#475	1	4	yes	—
152	C₄×C₆₇	#477	1	4	yes	—
153	C₄×C₆₇	#487	1	4	yes	—
154	C₄×C₆₇	#554	1	2	yes	—
155	C₄×C₆₇	#555	1	2	yes	—
156	Dic₆₇	—	4	4	no	—
157	Dic₆₇	—	8	2	no	—
158	#29141	—	67	1	no	maximal, Sylow
159	C₂×C₆₇	#2563	1	1	yes	characteristic
160	C₂×C₆₇	C₂×#475	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#477	1	1	yes	characteristic
162	C₂×C₆₇	C₂×#487	1	1	yes	characteristic
163	C₂×C₆₇	#3141	1	1	yes	characteristic
164	C₂×C₆₇	#3208	1	1	yes	characteristic
165	C₂×C₆₇	#3209	1	1	yes	characteristic
166	#2868	D₆₇	1	1	yes	characteristic
167	C₄×C₄⋊Q₈	—	67	1	no	—
168	#2024	—	67	4	no	—
169	#1956	—	67	2	no	—
170	#1834	—	67	2	no	—
171	#1803	—	67	4	no	—
172	C₄×C₄⋊C₈	—	67	1	no	—
173	C₆₇	#29141	1	1	yes	characteristic, complemented, Sylow
174	C₂×C₄⋊Q₈	—	2	1	no	—
175	C₂×C₄⋊Q₈	—	2	1	no	—
176	C₂×C₄⋊Q₈	—	67	1	no	—
177	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
178	C₂×C₄×Q₈	—	67	2	no	—
179	C₂×C₄⋊C₈	—	67	2	no	—
180	C₂×#103	—	67	8	no	—
181	C₂×C₄×C₈	—	67	2	no	—
182	#391	—	67	2	no	—
183	#389	—	134	1	no	—
184	#389	—	134	1	no	—
185	#384	—	2	1	no	—
186	#384	—	2	1	no	—
187	#384	—	67	1	no	—
188	#384	C₂×D₆₇	1	2	yes	—
189	#384	C₂×D₆₇	1	2	yes	—
190	C₄×C₄⋊C₄	—	134	1	no	—
191	C₄×C₄⋊C₄	—	134	1	no	—
192	C₄³	C₂×D₆₇	1	1	yes	characteristic
193	#340	—	67	8	no	—
194	#336	—	67	2	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	67	2	no	—
197	C₂²×C₈	—	67	4	no	—
198	C₄⋊Q₈	—	4	4	no	—
199	C₄⋊Q₈	—	4	4	no	—
200	C₄⋊Q₈	—	4	4	no	—
201	C₄⋊Q₈	—	67	4	no	—
202	C₄⋊Q₈	—	134	2	no	—
203	C₄×Q₈	—	134	8	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	67	2	no	—
210	C₂×C₄⋊C₄	—	134	2	no	—
211	C₂×C₄⋊C₄	—	134	1	no	—
212	C₂×C₄⋊C₄	—	134	1	no	—
213	C₂×C₄⋊C₄	—	134	1	no	—
214	C₂×C₄⋊C₄	—	134	1	no	—
215	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
216	C₂×C₄²	—	134	2	no	—
217	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
224	C₄⋊C₈	—	134	4	no	—
225	#103	—	268	8	no	—
226	C₄×C₈	—	134	4	no	—
227	#95	—	134	2	no	—
228	#95	—	134	2	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	—	4	4	no	—
232	C₂×Q₈	—	67	4	no	—
233	C₂×Q₈	—	134	4	no	—
234	C₂×Q₈	—	134	4	no	—
235	C₂×Q₈	—	134	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	134	2	no	—
238	C₂²×C₄	—	134	2	no	—
239	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
246	C₂×C₈	—	134	4	no	—
247	C₂×C₈	—	134	4	no	—
248	C₂×C₈	—	134	4	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	268	4	no	—
256	C₄⋊C₄	—	268	2	no	—
257	C₄⋊C₄	—	268	2	no	—
258	C₄⋊C₄	—	134	4	no	—
259	C₄⋊C₄	—	134	4	no	—
260	C₄⋊C₄	—	134	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	268	4	no	—
265	C₄²	#11761616	1	4	yes	—
266	C₄²	#11761618	1	4	yes	—
267	C₄²	Q₈×D₆₇	1	2	yes	—
268	C₄²	#11761629	1	2	yes	—
269	C₄²	#11761629	1	4	yes	—
270	C₂³	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
271	Q₈	—	8	8	no	—
272	Q₈	—	268	4	no	—
273	Q₈	—	134	8	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	268	2	no	—
284	C₂×C₄	—	268	2	no	—
285	C₂×C₄	—	268	2	no	—
286	C₂×C₄	—	268	2	no	—
287	C₂×C₄	—	268	2	no	—
288	C₂×C₄	—	134	4	no	—
289	C₂×C₄	#14524019	1	2	yes	—
290	C₂×C₄	#14524019	1	2	yes	—
291	C₂×C₄	#14524032	1	2	yes	—
292	C₂×C₄	#14524032	1	2	yes	—
293	C₂×C₄	C₂×#11761616	1	2	yes	—
294	C₂×C₄	C₂×#11761616	1	2	yes	—
295	C₂×C₄	C₂×#11761616	1	2	yes	—
296	C₂×C₄	C₂×#11761618	1	2	yes	—
297	C₂×C₄	C₂×#11761618	1	2	yes	—
298	C₂×C₄	C₂×#11761618	1	2	yes	—
299	C₂×C₄	#14524069	1	2	yes	—
300	C₂×C₄	#14524074	1	2	yes	—
301	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×#11761629	1	2	yes	—
303	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
304	C₂×C₄	#14524088	1	2	yes	—
305	C₈	—	268	4	no	—
306	C₂²	C₂×#14524019	1	1	yes	characteristic
307	C₂²	C₂×#14524032	1	1	yes	characteristic
308	C₂²	#16863963	1	1	yes	characteristic
309	C₂²	C₂²×#11761616	1	1	yes	characteristic
310	C₂²	C₂²×#11761618	1	1	yes	characteristic
311	C₂²	C₂×#14524069	1	1	yes	characteristic
312	C₂²	C₂×#14524074	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	8	4	no	—
317	C₄	—	268	4	no	—
318	C₄	—	536	2	no	—
319	C₄	#16863678	1	4	yes	—
320	C₄	#16863688	1	4	yes	—
321	C₄	#16863748	1	4	yes	—
322	C₄	#16864107	1	2	yes	—
323	C₄	#16864112	1	2	yes	—
324	C₂	#22565383	1	1	yes	characteristic
325	C₂	C₂×#16863678	1	1	yes	characteristic
326	C₂	C₂×#16863688	1	1	yes	characteristic
327	C₂	C₂×#16863748	1	1	yes	characteristic
328	C₂	#22570605	1	1	yes	characteristic
329	C₂	#22571150	1	1	yes	characteristic
330	C₂	#22571157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₆₇
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2868	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
hypercenter	#2868	D₆₇
hypocenter	C₆₇	#29141
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
32. #14524043	Q₈	semidirect
173. C₆₇	#29141	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524043	2
#22561347	2
#22561010	2
#22560336	2
#22560211	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₆₇×#2868	2
#29141	67