Subgroups — #31013466

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	#22572631	C₂	1	1	yes	maximal, characteristic
2	#22572601	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562419	C₂	1	2	yes	maximal, complemented
4	#22562391	C₂	1	2	yes	maximal, complemented
5	#22562358	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562353	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562323	C₂	1	2	yes	maximal, complemented
8	C₄×#14523988	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562042	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561994	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864112	—	2	1	no	—
13	#16864107	—	2	1	no	—
14	C₂×#14523988	C₂²	1	1	yes	characteristic
15	C₂×#14523982	C₂²	1	2	yes	—
16	#16863733	—	2	2	no	—
17	#16863723	—	2	2	no	—
18	#16863688	—	2	1	no	—
19	#16863683	—	2	1	no	—
20	#16863638	C₂²	1	2	yes	—
21	#16863633	C₂²	1	2	yes	—
22	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
23	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
24	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
25	#16863433	C₂²	1	2	yes	—
26	#107×D₆₇	C₂²	1	2	yes	—
27	#16863224	C₂²	1	1	yes	characteristic
28	#16863223	C₂²	1	2	yes	—
29	#16863221	C₂²	1	2	yes	—
30	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
31	#16863201	C₂²	1	1	yes	characteristic
32	#16863199	C₂²	1	2	yes	—
33	#16863198	C₂²	1	1	yes	characteristic
34	#16863196	C₂²	1	2	yes	—
35	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
36	#500×C₆₇	C₂²	1	1	yes	characteristic
37	#500×C₆₇	C₂²	1	1	yes	characteristic
38	#366×C₆₇	C₂²	1	2	yes	—
39	#366×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
42	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₆₇	C₂³	1	2	yes	—
44	#14524034	—	4	1	no	—
45	#14524032	—	4	2	no	—
46	#14524023	—	2	2	no	—
47	#14524019	—	4	2	no	—
48	#14524018	—	2	2	no	—
49	#14524018	—	4	1	no	—
50	C₄×Dic₁₃₄	—	2	1	no	—
51	C₄×D₂₆₈	—	2	1	no	—
52	C₄×D₂₆₈	D₄	1	1	yes	characteristic
53	#14524005	—	4	1	no	—
54	#14524001	—	2	2	no	—
55	#14524001	—	4	1	no	—
56	C₄⋊C₄×D₆₇	—	2	2	no	—
57	#14523988	—	2	2	no	—
58	#14523982	—	2	4	no	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
60	#14523977	C₂³	1	1	yes	characteristic
61	#14523977	C₂³	1	1	yes	characteristic
62	#14523976	—	2	2	no	—
63	#14523974	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
65	#14523968	—	2	1	no	—
66	#14523966	—	2	2	no	—
67	#14523965	—	2	1	no	—
68	#14523963	—	2	2	no	—
69	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
70	C₄⋊Q₈×C₆₇	—	2	1	no	—
71	C₄⋊Q₈×C₆₇	—	2	1	no	—
72	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	—	2	1	no	—
75	#107×C₆₇	—	2	1	no	—
76	#107×C₆₇	C₂³	1	2	yes	—
77	#107×C₆₇	C₂³	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	2	1	no	—
80	C₂×D₂₆₈	—	2	1	no	—
81	C₂×D₂₆₈	—	4	2	no	—
82	C₂×D₂₆₈	D₈	1	1	yes	characteristic
83	C₂×C₄×D₆₇	—	4	1	no	—
84	C₂×C₄×D₆₇	—	4	2	no	—
85	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
86	Dic₂₆₈	—	2	1	no	—
87	Dic₂₆₈	—	2	1	no	—
88	D₅₃₆	—	2	1	no	—
89	D₅₃₆	—	2	1	no	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₈×D₆₇	—	2	2	no	—
92	C₈×D₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	—	2	1	no	—
94	C₄⋊Dic₆₇	—	2	1	no	—
95	C₄⋊Dic₆₇	—	4	2	no	—
96	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
97	Dic₆₇⋊C₄	—	4	1	no	—
98	Dic₆₇⋊C₄	—	4	2	no	—
99	(C₂×D₆₇)⋊C₄	—	4	2	no	—
100	(C₂×D₆₇)⋊C₄	—	4	2	no	—
101	(C₂×D₆₇)⋊C₄	—	4	1	no	—
102	(C₂×D₆₇)⋊C₄	—	4	2	no	—
103	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
104	C₄×Dic₆₇	—	4	1	no	—
105	C₄×Dic₆₇	—	4	2	no	—
106	C₆₇⋊C₁₆	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	4	1	no	—
108	C₂×Q₈×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₆₇	—	2	1	no	—
117	C₁₆×C₆₇	#41	1	2	yes	—
118	C₂²×D₆₇	—	4	1	no	—
119	C₂²×D₆₇	C₂×D₈	1	2	yes	—
120	Q₈×C₆₇	—	16	1	no	—
121	Q₈×C₆₇	—	16	1	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₆₇	—	4	1	no	—
128	C₂×Dic₆₇	—	4	1	no	—
129	C₂×Dic₆₇	—	4	1	no	—
130	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
131	D₂₆₈	—	2	2	no	—
132	D₂₆₈	—	4	1	no	—
133	D₂₆₈	—	4	1	no	—
134	D₂₆₈	—	16	2	no	—
135	C₄×D₆₇	—	2	2	no	—
136	C₄×D₆₇	—	2	2	no	—
137	C₄×D₆₇	—	8	2	no	—
138	C₄×D₆₇	—	16	2	no	—
139	Dic₁₃₄	—	4	1	no	—
140	Dic₁₃₄	—	4	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₆₇⋊C₈	—	2	2	no	—
145	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₆₇	—	2	2	no	—
147	C₂×D₆₇	—	2	2	no	—
148	C₂×D₆₇	—	2	2	no	—
149	C₂×D₆₇	—	8	1	no	—
150	C₂×D₆₇	—	8	1	no	—
151	C₂×D₆₇	—	8	1	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	#510	1	2	yes	—
156	C₄×C₆₇	#547	1	1	yes	characteristic
157	C₄×C₆₇	#578	1	1	yes	characteristic
158	Dic₆₇	—	2	2	no	—
159	Dic₆₇	—	8	1	no	—
160	Dic₆₇	—	8	1	no	—
161	Dic₆₇	—	8	1	no	—
162	#33924	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	#3334	1	1	yes	characteristic
164	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
165	C₂×C₆₇	#3415	1	1	yes	characteristic
166	D₆₇	—	4	2	no	—
167	D₆₇	—	16	1	no	—
168	#3341	—	67	1	no	—
169	#3338	—	67	1	no	—
170	#2252	D₆₇	1	1	yes	characteristic, complemented
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	—	134	1	no	—
182	C₂×SD₃₂	—	67	1	no	—
183	C₂×M₃₂	—	67	2	no	—
184	#500	C₂×D₆₇	1	1	yes	characteristic
185	#500	C₂×D₆₇	1	1	yes	characteristic
186	#484	—	134	2	no	—
187	#482	—	134	2	no	—
188	#477	—	134	1	no	—
189	#476	—	134	1	no	—
190	#467	—	67	2	no	—
191	#466	—	67	2	no	—
192	C₄×Q₁₆	—	67	1	no	—
193	C₄×D₈	—	67	1	no	—
194	C₈×D₄	—	67	1	no	—
195	#427	—	67	2	no	—
196	C₂×#107	—	67	2	no	—
197	#367	—	67	1	no	—
198	#367	—	67	2	no	—
199	#366	—	67	2	no	—
200	#366	C₂×D₆₇	1	2	yes	—
201	#366	C₂×D₆₇	1	2	yes	—
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	C₂×D₆₇	1	1	yes	characteristic
209	C₂×Q₁₆	—	67	1	no	—
210	C₂×D₈	—	67	1	no	—
211	C₂²×C₈	—	67	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	#125	—	268	1	no	—
215	#124	—	268	2	no	—
216	C₂²⋊Q₈	—	268	2	no	—
217	C₂²⋊Q₈	—	134	2	no	—
218	#121	—	268	1	no	—
219	#121	—	134	2	no	—
220	C₄×Q₈	—	134	1	no	—
221	C₄×D₄	—	268	1	no	—
222	C₄×D₄	—	67	1	no	—
223	C₄×D₄	—	134	1	no	—
224	#117	—	268	1	no	—
225	#117	—	134	2	no	—
226	C₂×C₄⋊C₄	—	134	2	no	—
227	SD₃₂	—	134	2	no	—
228	M₃₂	—	134	4	no	—
229	C₂×C₁₆	—	67	2	no	—
230	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	#107	—	67	1	no	—
235	#107	—	67	1	no	—
236	#107	—	134	2	no	—
237	#107	C₂²×D₆₇	1	2	yes	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#106	—	134	2	no	—
240	C₄⋊C₈	—	67	1	no	—
241	#103	—	134	2	no	—
242	#103	—	134	1	no	—
243	#102	—	134	2	no	—
244	#102	—	134	1	no	—
245	C₂²⋊C₈	—	67	2	no	—
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	134	1	no	—
250	C₂×D₄	—	268	2	no	—
251	C₂×D₄	—	67	1	no	—
252	C₂×D₄	—	134	1	no	—
253	C₂²×C₄	—	268	2	no	—
254	C₂²×C₄	—	268	1	no	—
255	C₂²×C₄	—	67	2	no	—
256	Q₁₆	—	134	1	no	—
257	Q₁₆	—	134	1	no	—
258	D₈	—	134	1	no	—
259	D₈	—	134	1	no	—
260	C₂×C₈	—	67	2	no	—
261	C₂×C₈	—	134	2	no	—
262	C₂×C₈	—	134	2	no	—
263	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	268	2	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	1	no	—
272	C₄⋊C₄	—	67	1	no	—
273	C₄⋊C₄	—	134	1	no	—
274	C₄⋊C₄	—	134	1	no	—
275	C₂²⋊C₄	—	268	2	no	—
276	C₂²⋊C₄	—	268	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₂²⋊C₄	—	268	1	no	—
279	C₂²⋊C₄	—	67	2	no	—
280	C₄²	—	268	2	no	—
281	C₄²	—	268	1	no	—
282	C₄²	D₄×D₆₇	1	1	yes	characteristic
283	C₁₆	—	2	1	no	—
284	C₁₆	—	134	2	no	—
285	C₁₆	#11761629	1	2	yes	—
286	C₂³	—	268	1	no	—
287	C₂³	—	67	2	no	—
288	Q₈	—	268	1	no	—
289	Q₈	—	268	1	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	16	1	no	—
292	D₄	—	268	1	no	—
293	D₄	—	268	1	no	—
294	D₄	—	1072	2	no	—
295	D₄	—	134	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	268	1	no	—
299	C₂×C₄	—	268	1	no	—
300	C₂×C₄	—	268	1	no	—
301	C₂×C₄	—	536	2	no	—
302	C₂×C₄	—	1072	2	no	—
303	C₂×C₄	—	67	2	no	—
304	C₂×C₄	—	134	2	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
307	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	134	2	no	—
311	C₈	C₂×#11761629	1	1	yes	characteristic
312	C₈	#14524091	1	1	yes	characteristic
313	C₂²	—	536	1	no	—
314	C₂²	—	536	1	no	—
315	C₂²	—	536	1	no	—
316	C₂²	—	134	2	no	—
317	C₂²	—	134	2	no	—
318	C₂²	—	134	2	no	—
319	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	536	1	no	—
324	C₄	—	536	1	no	—
325	C₄	—	536	1	no	—
326	C₄	—	134	2	no	—
327	C₄	#16863838	1	2	yes	—
328	C₄	#16864049	1	1	yes	characteristic
329	C₄	#16864209	1	1	yes	characteristic
330	C₂	—	268	2	no	—
331	C₂	—	1072	1	no	—
332	C₂	#22572556	1	1	yes	characteristic
333	C₂	C₂×#16863838	1	1	yes	characteristic
334	C₂	#22573346	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2252	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22572631	C₂
hypercenter	#2252	D₆₇
hypocenter	C₆₇	#33924
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #22572601	C₂	semidirect
3. #22562419	C₂	semidirect
4. #22562391	C₂	semidirect
5. #22562358	C₂	semidirect
6. #22562353	C₂	semidirect
7. #22562323	C₂	semidirect
8. C₄×#14523988	C₂	semidirect
9. #22562042	C₂	semidirect
10. #22561994	C₂	semidirect
11. C₆₇×#2252	C₂	semidirect
170. #2252	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
#22572631	2
#22572601	2
#22562419	2
#22562391	2
#22562358	2
#22562353	2
#22562323	2
C₄×#14523988	2
#22562042	2
#22561994	2
C₆₇×#2252	2
#33924	67