Subgroups — #31013489

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573195	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573078	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562439	C₂	1	2	yes	maximal, complemented
4	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562376	C₂	1	2	yes	maximal, complemented
6	#22562263	C₂	1	2	yes	maximal
7	#22562255	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562111	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562060	C₂	1	2	yes	maximal, complemented
10	C₄×#14523984	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864116	—	2	1	no	—
13	#16864107	—	2	1	no	—
14	C₂×#14523988	C₂²	1	2	yes	—
15	C₂×#14523984	C₂²	1	1	yes	characteristic
16	#16863797	C₂²	1	2	yes	—
17	#16863779	C₂²	1	2	yes	—
18	#16863776	C₂²	1	1	yes	characteristic
19	#16863699	—	2	2	no	—
20	#16863683	—	2	1	no	—
21	#16863617	—	2	2	no	—
22	#16863601	—	2	1	no	—
23	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
24	#16863490	C₂²	1	2	yes	—
25	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
26	#16863478	C₂²	1	2	yes	—
27	#16863221	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	#16863201	C₂²	1	2	yes	—
30	#16863199	C₂²	1	2	yes	—
31	#16863199	C₂²	1	2	yes	—
32	#16863197	C₂²	1	1	yes	characteristic
33	#16863196	C₂²	1	1	yes	characteristic
34	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₆₇	C₂²	1	2	yes	—
36	#500×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
38	#367×C₆₇	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	#357×C₆₇	C₂²	1	2	yes	—
41	C₂×Dic₂₆₈	—	2	1	no	—
42	C₂×Dic₂₆₈	C₂³	1	2	yes	—
43	C₂×#11761616	—	2	2	no	—
44	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
45	#14524043	—	2	1	no	—
46	#14524036	—	4	2	no	—
47	#14524034	—	2	2	no	—
48	#14524030	—	2	2	no	—
49	#14524023	—	4	1	no	—
50	#14524022	—	4	1	no	—
51	#14524019	—	4	2	no	—
52	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
53	#14524008	—	2	2	no	—
54	C₄×D₂₆₈	—	2	1	no	—
55	#14524005	—	2	2	no	—
56	#14524002	—	4	1	no	—
57	C₄⋊C₄×D₆₇	—	4	1	no	—
58	#14523988	—	2	4	no	—
59	#14523984	—	2	2	no	—
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	#14523977	C₂³	1	1	yes	characteristic
63	#14523974	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
65	Dic₆₇⋊C₈	D₄	1	2	yes	—
66	#14523966	—	2	2	no	—
67	#14523965	—	2	1	no	—
68	#14523963	—	2	2	no	—
69	C₈×Dic₆₇	C₂³	1	2	yes	—
70	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₇	—	2	1	no	—
72	C₄×D₄×C₆₇	—	2	1	no	—
73	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	—	2	1	no	—
75	#107×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	C₂³	1	2	yes	—
77	#102×C₆₇	—	2	1	no	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	2	2	no	—
80	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
81	C₂×D₂₆₈	—	2	1	no	—
82	C₂×C₄×D₆₇	—	4	1	no	—
83	C₂×C₄×D₆₇	—	4	2	no	—
84	Dic₂₆₈	—	2	2	no	—
85	Dic₂₆₈	—	2	2	no	—
86	Dic₂₆₈	—	8	1	no	—
87	#11761616	—	8	2	no	—
88	D₅₃₆	—	2	1	no	—
89	D₅₃₆	#41	1	2	yes	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₆₇	—	2	1	no	—
92	C₄⋊Dic₆₇	—	4	1	no	—
93	C₄⋊Dic₆₇	—	4	1	no	—
94	C₄⋊Dic₆₇	—	4	2	no	—
95	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
96	Dic₆₇⋊C₄	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	D₈	1	2	yes	—
99	C₂²⋊Dic₆₇	—	4	2	no	—
100	C₂²⋊Dic₆₇	—	4	2	no	—
101	(C₂×D₆₇)⋊C₄	—	4	1	no	—
102	(C₂×D₆₇)⋊C₄	—	4	2	no	—
103	C₄×Dic₆₇	—	4	2	no	—
104	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
105	C₆₇⋊C₁₆	—	2	1	no	—
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₂×Q₈×C₆₇	—	4	1	no	—
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	4	1	no	—
110	D₈×C₆₇	—	2	1	no	—
111	D₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₂²⋊C₄×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₆₇	—	2	2	no	—
120	C₂³×C₆₇	—	4	1	no	—
121	C₂²×D₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	16	1	no	—
123	D₄×C₆₇	—	4	1	no	—
124	D₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	8	2	no	—
128	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	—	4	1	no	—
133	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
134	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
135	D₂₆₈	—	2	2	no	—
136	D₂₆₈	—	4	1	no	—
137	C₄×D₆₇	—	8	2	no	—
138	C₄×D₆₇	—	16	2	no	—
139	Dic₁₃₄	—	2	2	no	—
140	Dic₁₃₄	—	4	2	no	—
141	Dic₁₃₄	—	4	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	C₆₇⋊C₈	—	2	2	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	C₂×D₆₇	—	4	2	no	—
151	C₂×D₆₇	—	8	1	no	—
152	C₂×D₆₇	—	8	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	8	1	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	#508	1	2	yes	—
157	C₄×C₆₇	#548	1	1	yes	characteristic
158	C₄×C₆₇	#577	1	1	yes	characteristic
159	Dic₆₇	—	2	2	no	—
160	Dic₆₇	—	2	2	no	—
161	Dic₆₇	—	8	2	no	—
162	Dic₆₇	—	8	1	no	—
163	#33927	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	—	16	1	no	—
165	C₂×C₆₇	#3358	1	1	yes	characteristic
166	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
167	C₂×C₆₇	#3416	1	1	yes	characteristic
168	D₆₇	—	8	2	no	—
169	#3393	—	67	1	no	—
170	#3380	—	67	1	no	—
171	#2242	—	67	2	no	—
172	#2241	—	67	1	no	—
173	#2229	—	67	2	no	—
174	#2226	D₆₇	1	1	yes	characteristic, complemented
175	#2210	—	67	2	no	—
176	#2208	—	67	1	no	—
177	Q₈×C₁₆	—	67	1	no	—
178	C₄×SD₃₂	—	67	2	no	—
179	C₄×D₁₆	—	67	1	no	—
180	C₆₇	#33927	1	1	yes	characteristic, complemented, Sylow
181	#556	—	134	1	no	—
182	#554	—	134	1	no	—
183	C₂×SD₃₂	—	67	2	no	—
184	C₂×D₁₆	—	67	1	no	—
185	#500	C₂×D₆₇	1	1	yes	characteristic
186	#499	—	67	2	no	—
187	#495	—	67	2	no	—
188	#494	—	67	1	no	—
189	#478	—	134	2	no	—
190	#476	—	134	1	no	—
191	#464	—	134	2	no	—
192	#461	—	134	1	no	—
193	C₈×Q₈	—	67	1	no	—
194	C₄×Q₁₆	—	67	2	no	—
195	C₄×D₈	—	67	2	no	—
196	C₄×D₈	—	67	1	no	—
197	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
198	#367	C₂×D₆₇	1	2	yes	—
199	#366	—	67	1	no	—
200	C₄⋊C₁₆	—	67	2	no	—
201	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
202	#358	—	67	2	no	—
203	#358	—	67	2	no	—
204	#358	—	67	2	no	—
205	#357	—	67	1	no	—
206	#357	—	67	1	no	—
207	#357	C₂×D₆₇	1	2	yes	—
208	C₄×C₁₆	—	67	2	no	—
209	C₄×C₁₆	—	67	1	no	—
210	C₂×Q₁₆	—	67	2	no	—
211	C₂×Q₁₆	—	134	1	no	—
212	C₂×SD₁₆	—	134	2	no	—
213	C₂×D₈	—	67	1	no	—
214	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	134	1	no	—
217	#126	—	268	2	no	—
218	#125	—	134	2	no	—
219	#124	—	134	2	no	—
220	C₂²⋊Q₈	—	268	1	no	—
221	C₂²⋊Q₈	—	268	1	no	—
222	C₂²⋊Q₈	—	268	2	no	—
223	C₄×Q₈	—	67	1	no	—
224	C₄×Q₈	—	134	2	no	—
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	134	2	no	—
227	C₄×D₄	—	134	1	no	—
228	#117	—	268	1	no	—
229	C₂×C₄⋊C₄	—	268	1	no	—
230	SD₃₂	—	134	4	no	—
231	D₁₆	—	134	2	no	—
232	C₂×C₁₆	—	67	1	no	—
233	C₂×C₁₆	—	67	1	no	—
234	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	67	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	—	67	2	no	—
241	C₄⋊C₈	—	67	1	no	—
242	#103	—	134	2	no	—
243	#102	—	2	1	no	—
244	#102	—	134	2	no	—
245	#102	—	134	1	no	—
246	C₄×C₈	—	67	2	no	—
247	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	67	1	no	—
250	C₂×Q₈	—	134	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	134	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	268	2	no	—
255	C₂²×C₄	—	268	1	no	—
256	Q₁₆	—	536	1	no	—
257	Q₁₆	—	134	2	no	—
258	Q₁₆	—	134	2	no	—
259	SD₁₆	—	536	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	67	2	no	—
263	D₈	—	134	1	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	268	1	no	—
274	C₄⋊C₄	—	268	1	no	—
275	C₄⋊C₄	—	67	1	no	—
276	C₄⋊C₄	—	67	2	no	—
277	C₄⋊C₄	—	134	1	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	268	2	no	—
280	C₂²⋊C₄	—	268	2	no	—
281	C₂²⋊C₄	—	268	2	no	—
282	C₂²⋊C₄	—	268	1	no	—
283	C₄²	—	268	2	no	—
284	C₄²	—	67	2	no	—
285	C₄²	D₄×D₆₇	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	134	1	no	—
288	C₁₆	—	134	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	268	1	no	—
291	Q₈	—	268	2	no	—
292	Q₈	—	268	2	no	—
293	Q₈	—	16	1	no	—
294	Q₈	—	134	2	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	268	1	no	—
298	D₄	—	134	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	268	1	no	—
304	C₂×C₄	—	536	2	no	—
305	C₂×C₄	—	1072	2	no	—
306	C₂×C₄	—	67	2	no	—
307	C₂×C₄	—	67	2	no	—
308	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
309	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	134	2	no	—
313	C₈	#41×D₆₇	1	1	yes	characteristic
314	C₈	#14524091	1	1	yes	characteristic
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	268	2	no	—
319	C₂²	—	536	1	no	—
320	C₂²	—	536	1	no	—
321	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	536	1	no	—
326	C₄	—	536	2	no	—
327	C₄	—	134	2	no	—
328	C₄	—	134	2	no	—
329	C₄	#16863827	1	2	yes	—
330	C₄	#16864056	1	1	yes	characteristic
331	C₄	#16864205	1	1	yes	characteristic
332	C₂	—	16	1	no	—
333	C₂	—	536	2	no	—
334	C₂	#22572843	1	1	yes	characteristic
335	C₂	C₂×#16863827	1	1	yes	characteristic
336	C₂	#22573355	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×#2226	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#14523984	C₂²
hypercenter	#2226	D₆₇
hypocenter	C₆₇	#33927
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
1. #22573195	C₂	semidirect
2. #22573078	C₂	semidirect
3. #22562439	C₂	semidirect
4. #22562438	C₂	semidirect
5. #22562376	C₂	semidirect
7. #22562255	C₂	semidirect
8. #22562111	C₂	semidirect
9. #22562060	C₂	semidirect
11. C₆₇×#2226	C₂	semidirect
174. #2226	D₆₇	semidirect
180. C₆₇	#33927	semidirect

Maximal subgroups

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

subgroup	index
#22573195	2
#22573078	2
#22562439	2
#22562438	2
#22562376	2
#22562263	2
#22562255	2
#22562111	2
#22562060	2
C₄×#14523984	2
C₆₇×#2226	2
#33927	67