Subgroups — #31013467

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572632	C₂	1	1	yes	maximal, characteristic
2	#22562482	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562414	C₂	1	2	yes	maximal, complemented
4	#22562389	C₂	1	2	yes	maximal, complemented
5	#22562361	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562353	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562324	C₂	1	2	yes	maximal, complemented
8	C₄×#14523986	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562041	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561995	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3338	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864111	—	2	1	no	—
13	C₂×#14523986	C₂²	1	1	yes	characteristic
14	C₂×#14523983	C₂²	1	2	yes	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863721	—	2	2	no	—
17	#16863687	—	2	1	no	—
18	#16863632	C₂²	1	2	yes	—
19	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
20	#16863434	C₂²	1	2	yes	—
21	#16863223	C₂²	1	1	yes	characteristic
22	#16863223	C₂²	1	2	yes	—
23	#16863221	C₂²	1	2	yes	—
24	#16863221	C₂²	1	2	yes	—
25	#16863221	C₂²	1	2	yes	—
26	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
27	#16863200	C₂²	1	1	yes	characteristic
28	#16863200	C₂²	1	2	yes	—
29	#16863196	C₂²	1	1	yes	characteristic
30	#16863196	C₂²	1	2	yes	—
31	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	#554×C₆₇	—	2	1	no	—
34	#500×C₆₇	C₂²	1	1	yes	characteristic
35	#484×C₆₇	—	2	2	no	—
36	#477×C₆₇	—	2	1	no	—
37	#467×C₆₇	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#107×C₆₇	C₂²	1	2	yes	—
41	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
42	#14524043	—	2	1	no	—
43	#14524029	—	4	2	no	—
44	#14524017	—	2	2	no	—
45	#14524014	—	4	1	no	—
46	C₄×D₂₆₈	—	2	1	no	—
47	#14524002	—	2	2	no	—
48	#14524001	—	4	1	no	—
49	#14523986	—	2	2	no	—
50	#14523983	—	2	4	no	—
51	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	#14523977	—	2	1	no	—
55	#14523977	C₂³	1	2	yes	—
56	#14523977	C₂³	1	1	yes	characteristic
57	#14523975	—	2	2	no	—
58	#14523965	—	2	2	no	—
59	#14523965	—	2	1	no	—
60	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₆₇	—	2	1	no	—
63	#125×C₆₇	—	4	1	no	—
64	C₂²⋊Q₈×C₆₇	—	2	2	no	—
65	C₂²⋊Q₈×C₆₇	—	4	2	no	—
66	C₄×Q₈×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	—	4	1	no	—
68	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
70	#107×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	C₂³	1	2	yes	—
73	#107×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
75	#103×C₆₇	—	2	2	no	—
76	#103×C₆₇	—	2	1	no	—
77	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	4	1	no	—
80	C₂×D₂₆₈	—	2	1	no	—
81	C₂×C₂²⋊D₆₇	—	4	2	no	—
82	C₂×C₄×D₆₇	—	4	1	no	—
83	D₅₃₆	—	2	1	no	—
84	D₅₃₆	—	2	1	no	—
85	C₄⋊Dic₆₇	—	2	1	no	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	—	4	1	no	—
88	C₂²⋊Dic₆₇	—	4	2	no	—
89	(C₂×D₆₇)⋊C₄	—	4	2	no	—
90	(C₂×D₆₇)⋊C₄	—	4	1	no	—
91	C₄×Dic₆₇	—	4	2	no	—
92	C₆₇⋊C₁₆	—	2	2	no	—
93	C₆₇⋊C₁₆	—	2	1	no	—
94	C₆₇⋊C₁₆	#41	1	2	yes	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	4	1	no	—
97	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	—	4	2	no	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	Q₁₆×C₆₇	—	2	1	no	—
101	Q₁₆×C₆₇	—	2	1	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	4	2	no	—
110	C₄⋊C₄×C₆₇	—	4	1	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
114	C₂²⋊C₄×C₆₇	—	4	2	no	—
115	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
116	C₄²×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₆₇	C₂×D₈	1	2	yes	—
119	C₂²×D₆₇	—	4	1	no	—
120	Q₈×C₆₇	—	4	1	no	—
121	Q₈×C₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	16	1	no	—
123	D₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	16	2	no	—
130	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₆₇	—	16	2	no	—
135	C₂×Dic₆₇	—	4	1	no	—
136	C₂×Dic₆₇	—	4	2	no	—
137	D₂₆₈	—	4	1	no	—
138	D₂₆₈	—	4	1	no	—
139	C₄×D₆₇	—	8	2	no	—
140	Dic₁₃₄	—	16	1	no	—
141	C₈×C₆₇	—	2	2	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	—	2	2	no	—
147	C₂²×C₆₇	—	2	2	no	—
148	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₆₇	—	8	1	no	—
150	C₂×D₆₇	—	8	1	no	—
151	C₂×D₆₇	—	8	1	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	8	1	no	—
155	C₄×C₆₇	—	8	1	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	#510	1	2	yes	—
158	C₄×C₆₇	#547	1	1	yes	characteristic
159	C₄×C₆₇	#578	1	1	yes	characteristic
160	Dic₆₇	—	8	1	no	—
161	Dic₆₇	—	8	2	no	—
162	#33924	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	4	2	no	—
164	C₂×C₆₇	#3334	1	1	yes	characteristic
165	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
166	C₂×C₆₇	#3415	1	1	yes	characteristic
167	D₆₇	—	16	1	no	—
168	#3341	—	67	1	no	—
169	#3338	D₆₇	1	1	yes	characteristic, complemented
170	#2252	—	67	1	no	—
171	#2237	—	67	2	no	—
172	#2232	—	67	2	no	—
173	#2227	—	67	1	no	—
174	#2226	—	67	1	no	—
175	#2220	—	67	2	no	—
176	C₄×SD₃₂	—	67	1	no	—
177	#2169	—	67	1	no	—
178	#2159	—	67	2	no	—
179	C₆₇	#33924	1	1	yes	characteristic, complemented, Sylow
180	#555	—	134	1	no	—
181	#554	—	2	1	no	—
182	C₂×SD₃₂	—	67	1	no	—
183	C₂×M₃₂	—	67	2	no	—
184	#500	—	67	1	no	—
185	#500	C₂×D₆₇	1	1	yes	characteristic
186	#484	—	2	2	no	—
187	#482	—	134	2	no	—
188	#477	—	2	1	no	—
189	#476	—	134	1	no	—
190	#467	C₂×D₆₇	1	2	yes	—
191	#466	—	67	2	no	—
192	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
193	C₄×D₈	—	67	1	no	—
194	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
195	#427	—	67	2	no	—
196	C₂×#107	C₂×D₆₇	1	2	yes	—
197	#367	—	67	1	no	—
198	#367	—	67	2	no	—
199	#366	—	67	2	no	—
200	#366	—	67	2	no	—
201	#366	—	67	2	no	—
202	C₄⋊C₁₆	—	67	1	no	—
203	#358	—	67	1	no	—
204	#358	—	67	2	no	—
205	#357	—	67	1	no	—
206	#357	—	67	2	no	—
207	C₂²⋊C₁₆	—	67	2	no	—
208	C₄×C₁₆	—	67	1	no	—
209	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
210	C₂×D₈	—	67	1	no	—
211	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	134	1	no	—
214	#125	—	4	1	no	—
215	#124	—	268	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	#121	—	268	1	no	—
219	#121	—	134	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×D₄	—	4	1	no	—
222	C₄×D₄	—	134	1	no	—
223	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
224	#117	—	268	1	no	—
225	#117	—	134	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	SD₃₂	—	134	2	no	—
228	M₃₂	—	134	4	no	—
229	C₂×C₁₆	—	67	1	no	—
230	C₂×C₁₆	—	67	1	no	—
231	C₂×C₁₆	—	67	2	no	—
232	#107	—	2	2	no	—
233	#107	—	2	1	no	—
234	#107	—	67	2	no	—
235	#107	—	67	1	no	—
236	#107	—	134	1	no	—
237	#107	C₂²×D₆₇	1	1	yes	characteristic
238	#107	C₂²×D₆₇	1	2	yes	—
239	#106	—	134	2	no	—
240	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
241	#103	—	2	1	no	—
242	#103	—	2	2	no	—
243	#102	—	134	1	no	—
244	#102	—	134	2	no	—
245	C₂²⋊C₈	C₂²⋊D₆₇	1	2	yes	—
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	268	1	no	—
250	C₂×D₄	—	268	2	no	—
251	C₂×D₄	—	134	1	no	—
252	C₂×D₄	#11761613	1	1	yes	characteristic
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	268	1	no	—
255	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	D₈	—	134	1	no	—
259	D₈	—	134	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
264	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	268	1	no	—
272	C₄⋊C₄	—	134	1	no	—
273	C₄⋊C₄	—	134	2	no	—
274	C₄⋊C₄	#11761613	1	1	yes	characteristic
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	268	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₂²⋊C₄	—	268	1	no	—
279	C₂²⋊C₄	#11761613	1	2	yes	—
280	C₄²	—	4	1	no	—
281	C₄²	—	268	2	no	—
282	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
283	C₁₆	—	67	2	no	—
284	C₁₆	—	134	2	no	—
285	C₁₆	—	134	1	no	—
286	C₂³	—	268	1	no	—
287	C₂³	C₂×#11761613	1	2	yes	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	1072	1	no	—
292	D₄	—	2	2	no	—
293	D₄	—	268	1	no	—
294	D₄	—	268	1	no	—
295	D₄	—	1072	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	268	1	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	16	2	no	—
304	C₂×C₄	—	536	2	no	—
305	C₂×C₄	C₂×#11761613	1	2	yes	—
306	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
307	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	#41×D₆₇	1	1	yes	characteristic
312	C₈	#14524090	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	536	1	no	—
317	C₂²	—	536	1	no	—
318	C₂²	—	536	1	no	—
319	C₂²	C₂²×#11761613	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	536	1	no	—
326	C₄	—	536	2	no	—
327	C₄	#16863837	1	2	yes	—
328	C₄	#16864045	1	1	yes	characteristic
329	C₄	#16864211	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	1072	1	no	—
332	C₂	#22572552	1	1	yes	characteristic
333	C₂	C₂×#16863837	1	1	yes	characteristic
334	C₂	#22573348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3338	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22572632	C₂
hypercenter	#3338	D₆₇
hypocenter	C₆₇	#33924
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3338

Complements

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

N	H ≅ G/N	type
2. #22562482	C₂	semidirect
3. #22562414	C₂	semidirect
4. #22562389	C₂	semidirect
5. #22562361	C₂	semidirect
6. #22562353	C₂	semidirect
7. #22562324	C₂	semidirect
8. C₄×#14523986	C₂	semidirect
9. #22562041	C₂	semidirect
10. #22561995	C₂	semidirect
11. C₆₇×#3338	C₂	semidirect
169. #3338	D₆₇	semidirect
179. C₆₇	#33924	semidirect

Maximal subgroups

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

subgroup	index
#22572632	2
#22562482	2
#22562414	2
#22562389	2
#22562361	2
#22562353	2
#22562324	2
C₄×#14523986	2
#22562041	2
#22561995	2
C₆₇×#3338	2
#33924	67