Subgroups — #31912724

Order 18176.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986112	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985813	C₂	1	1	yes	maximal, characteristic
3	#22590057	C₂	1	1	yes	maximal, characteristic
4	#22588224	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586446	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586439	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#347×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₈₄	—	2	1	no	—
10	#17986112	—	2	2	no	—
11	#17986112	C₄	1	4	yes	complemented
12	C₂×#14556064	C₂²	1	1	yes	characteristic
13	#17986037	C₂²	1	1	yes	characteristic
14	C₂×#14556060	C₂²	1	1	yes	characteristic
15	C₂×#14556050	—	2	1	no	—
16	#17985954	—	2	1	no	—
17	#17985813	C₄	1	4	yes	complemented
18	#17985801	C₄	1	4	yes	complemented
19	#17985759	—	2	1	no	—
20	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
23	#347×C₇₁	—	2	2	no	—
24	C₂²×D₂₈₄	—	2	1	no	—
25	C₂²×C₄×D₇₁	—	4	1	no	—
26	C₂×D₅₆₈	—	2	1	no	—
27	C₂×D₅₆₈	—	2	1	no	—
28	C₂×D₅₆₈	—	2	1	no	—
29	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
30	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
31	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
32	C₂×#11765849	—	2	1	no	—
33	C₄×D₂₈₄	—	2	4	no	—
34	C₄×D₂₈₄	—	4	2	no	—
35	#14556087	—	2	1	no	—
36	C₂×C₄⋊Dic₇₁	—	2	1	no	—
37	C₂×(C₂×D₇₁)⋊C₄	—	4	1	no	—
38	#14556064	—	2	1	no	—
39	#14556064	C₂×C₄	1	2	yes	complemented
40	#14556062	C₂×C₄	1	2	yes	complemented
41	#14556060	—	2	1	no	—
42	#14556060	—	2	1	no	—
43	#14556060	C₂×C₄	1	2	yes	—
44	#14556050	—	2	4	no	—
45	#14556038	—	4	1	no	—
46	C₈×Dic₇₁	—	4	1	no	—
47	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
50	M₃₂×C₇₁	—	2	1	no	—
51	M₃₂×C₇₁	—	2	1	no	—
52	M₃₂×C₇₁	—	2	1	no	—
53	M₃₂×C₇₁	C₂×C₄	1	2	yes	complemented
54	C₂×C₁₆×C₇₁	—	2	1	no	—
55	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
56	#97×C₇₁	C₂×C₄	1	2	yes	—
57	#97×C₇₁	D₄	1	2	yes	—
58	C₂³×D₇₁	—	4	1	no	—
59	C₂×D₂₈₄	—	2	2	no	—
60	C₂×D₂₈₄	—	2	2	no	—
61	C₂×D₂₈₄	—	2	2	no	—
62	C₂×D₂₈₄	—	4	1	no	—
63	C₂×D₂₈₄	—	4	1	no	—
64	C₂×D₂₈₄	—	4	1	no	—
65	C₂²×Dic₇₁	—	4	1	no	—
66	C₂×C₄×D₇₁	—	4	4	no	—
67	C₂×C₄×D₇₁	—	8	4	no	—
68	D₅₆₈	—	2	2	no	—
69	D₅₆₈	—	2	2	no	—
70	D₅₆₈	—	2	2	no	—
71	D₅₆₈	C₄²	1	4	yes	—
72	#11765849	—	2	2	no	—
73	#11765849	—	4	1	no	—
74	C₂×C₇₁⋊C₈	—	4	1	no	—
75	C₄⋊Dic₇₁	—	2	2	no	—
76	C₄⋊Dic₇₁	—	2	2	no	—
77	C₄⋊Dic₇₁	—	4	1	no	—
78	C₂²⋊Dic₇₁	—	4	1	no	—
79	(C₂×D₇₁)⋊C₄	—	4	4	no	—
80	C₄×Dic₇₁	—	4	1	no	—
81	C₂²×C₄×C₇₁	—	2	1	no	—
82	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
92	C₄²×C₇₁	—	2	1	no	—
93	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
94	C₁₆×C₇₁	—	2	2	no	—
95	C₁₆×C₇₁	—	2	2	no	—
96	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂²×D₇₁	—	4	2	no	—
98	C₂²×D₇₁	—	4	2	no	—
99	C₂²×D₇₁	—	4	2	no	—
100	C₂²×D₇₁	—	8	2	no	—
101	C₂²×D₇₁	—	8	2	no	—
102	C₂×C₄×C₇₁	—	2	2	no	—
103	C₂×C₄×C₇₁	—	4	2	no	—
104	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
110	C₂×Dic₇₁	—	4	2	no	—
111	C₂×Dic₇₁	—	4	2	no	—
112	C₂×Dic₇₁	—	8	2	no	—
113	D₂₈₄	—	2	4	no	—
114	D₂₈₄	—	4	2	no	—
115	D₂₈₄	—	4	2	no	—
116	D₂₈₄	—	4	2	no	—
117	C₄×D₇₁	—	8	8	no	—
118	C₈×C₇₁	—	2	2	no	—
119	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
120	C₈×C₇₁	#117	1	1	yes	characteristic
121	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
123	C₇₁⋊C₈	—	4	2	no	—
124	C₂²×C₇₁	—	2	2	no	—
125	C₂²×C₇₁	#418	1	1	yes	characteristic
126	C₂²×C₇₁	#440	1	1	yes	characteristic
127	C₂²×C₇₁	#442	1	1	yes	characteristic
128	C₂×D₇₁	—	4	4	no	—
129	C₂×D₇₁	—	8	2	no	—
130	C₂×D₇₁	—	8	2	no	—
131	C₂×D₇₁	—	8	4	no	—
132	C₂×D₇₁	—	8	4	no	—
133	C₄×C₇₁	—	4	2	no	—
134	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₇₁	#417	1	1	yes	characteristic
136	C₄×C₇₁	#440	1	1	yes	characteristic
137	C₄×C₇₁	#442	1	1	yes	characteristic
138	Dic₇₁	—	8	2	no	—
139	Dic₇₁	—	8	2	no	—
140	#12647	—	71	1	no	maximal, Sylow
141	C₂×C₇₁	—	2	1	no	—
142	C₂×C₇₁	—	2	1	no	—
143	C₂×C₇₁	#1762	1	1	yes	characteristic
144	C₂×C₇₁	AGL(1,ℤ/16)	1	2	yes	—
145	D₇₁	—	8	4	no	—
146	C₂×#442	—	71	1	no	—
147	C₂×#361	—	71	1	no	—
148	#2140	—	71	1	no	—
149	C₂×#347	D₇₁	1	1	yes	characteristic, complemented
150	#1773	—	71	1	no	—
151	#1386	—	71	1	no	—
152	#1383	—	71	1	no	—
153	C₇₁	#12647	1	1	yes	characteristic, complemented, Sylow
154	C₂²×D₈	—	71	1	no	—
155	C₂×C₄×D₄	—	142	1	no	—
156	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
157	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
158	#442	—	71	4	no	—
159	#442	—	142	2	no	—
160	C₂×#108	—	71	1	no	—
161	#427	—	71	1	no	—
162	C₂×#106	—	71	1	no	—
163	C₂×#102	—	142	1	no	—
164	#405	—	142	1	no	—
165	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
166	#361	—	71	4	no	—
167	#357	—	71	4	no	—
168	#347	—	2	2	no	—
169	#339	—	142	1	no	—
170	C₂²×D₄	—	142	1	no	—
171	C₂³×C₄	—	284	1	no	—
172	C₂×D₈	—	71	2	no	—
173	C₂×D₈	—	71	2	no	—
174	C₂×D₈	—	71	2	no	—
175	C₂×D₈	—	142	1	no	—
176	C₂×D₈	—	142	1	no	—
177	C₂×D₈	—	142	1	no	—
178	C₂×M₁₆	—	142	1	no	—
179	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
181	C₄×D₄	—	284	2	no	—
182	C₄×D₄	—	142	4	no	—
183	#117	—	142	1	no	—
184	C₂×C₄⋊C₄	—	142	1	no	—
185	C₂×C₂²⋊C₄	—	284	1	no	—
186	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	C₄×D₇₁	1	2	yes	complemented
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	C₄×D₇₁	1	2	yes	complemented
193	#108	—	71	2	no	—
194	#108	—	142	1	no	—
195	#107	—	71	2	no	—
196	#106	—	71	2	no	—
197	#106	—	142	1	no	—
198	#106	—	142	1	no	—
199	#102	—	142	4	no	—
200	#97	—	284	1	no	—
201	#97	C₄×D₇₁	1	2	yes	—
202	#97	C₂²⋊D₇₁	1	2	yes	—
203	C₄×C₈	—	284	1	no	—
204	C₂⁴	—	284	1	no	—
205	C₂×D₄	—	284	1	no	—
206	C₂×D₄	—	284	1	no	—
207	C₂×D₄	—	284	1	no	—
208	C₂×D₄	—	142	2	no	—
209	C₂×D₄	—	142	2	no	—
210	C₂×D₄	—	142	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	284	4	no	—
213	C₂²×C₄	—	284	1	no	—
214	C₂²×C₄	—	568	4	no	—
215	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
216	D₈	—	71	4	no	—
217	D₈	—	142	2	no	—
218	D₈	—	142	2	no	—
219	D₈	—	142	2	no	—
220	M₁₆	—	284	1	no	—
221	M₁₆	—	142	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	284	1	no	—
225	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
229	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
230	C₂×C₈	#11765865	1	1	yes	characteristic
231	C₂×C₈	#11765865	1	1	yes	characteristic
232	C₄⋊C₄	—	284	1	no	—
233	C₄⋊C₄	—	142	2	no	—
234	C₄⋊C₄	—	142	2	no	—
235	C₂²⋊C₄	—	284	4	no	—
236	C₂²⋊C₄	—	284	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	284	1	no	—
239	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₂³	—	284	2	no	—
243	C₂³	—	284	2	no	—
244	C₂³	—	284	2	no	—
245	C₂³	—	568	2	no	—
246	C₂³	—	568	2	no	—
247	C₂³	#14556156	1	1	yes	characteristic
248	D₄	—	284	2	no	—
249	D₄	—	284	2	no	—
250	D₄	—	284	2	no	—
251	D₄	—	142	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	284	2	no	—
255	C₂×C₄	—	284	2	no	—
256	C₂×C₄	—	568	8	no	—
257	C₂×C₄	—	568	2	no	—
258	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
261	C₂×C₄	#14556156	1	1	yes	characteristic
262	C₂×C₄	#14556156	1	1	yes	characteristic
263	C₂×C₄	#14556161	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	284	2	no	—
266	C₈	C₄²×D₇₁	1	1	yes	characteristic
267	C₈	#14556085	1	1	yes	characteristic
268	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	284	4	no	—
272	C₂²	—	568	2	no	—
273	C₂²	—	568	2	no	—
274	C₂²	—	568	4	no	—
275	C₂²	—	568	4	no	—
276	C₂²	#17986005	1	1	yes	characteristic
277	C₂²	#17986101	1	1	yes	characteristic
278	C₂²	#17986112	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	568	2	no	—
281	C₄	—	568	2	no	—
282	C₄	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
283	C₄	#17986001	1	1	yes	characteristic
284	C₄	#17986101	1	1	yes	characteristic
285	C₄	#17986112	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	568	4	no	—
289	C₂	#22588166	1	1	yes	characteristic
290	C₂	#22590240	1	2	yes	—

	subgroup	G/H
center	C₂²	#17986112
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#347×C₇₁	C₂
socle	C₂²×C₇₁	#442
prime radical	C₂²×D₅₆₈	C₂²
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#347	D₇₁
hypocenter	C₇₁	#12647
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #22588224	C₂	semidirect
5. #22586446	C₂	semidirect
6. #22586439	C₂	semidirect
7. C₂×#347×C₇₁	C₂	semidirect
11. #17986112	C₄	semidirect
17. #17985813	C₄	semidirect
18. #17985801	C₄	semidirect
39. #14556064	C₂×C₄	semidirect
40. #14556062	C₂×C₄	semidirect
53. M₃₂×C₇₁	C₂×C₄	semidirect
55. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
149. C₂×#347	D₇₁	semidirect
153. C₇₁	#12647	semidirect
190. M₃₂	C₄×D₇₁	semidirect
192. C₂×C₁₆	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986112	2
C₂×#17985813	2
#22590057	2
#22588224	2
#22586446	2
#22586439	2
C₂×#347×C₇₁	2
#12647	71