Subgroups — #32152961

Order 18176.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598879	C₂	1	2	yes	maximal
2	#22590626	C₂	1	4	yes	maximal, complemented
3	#22590619	C₂	1	2	yes	maximal, complemented
4	#22590615	C₂	1	2	yes	maximal, complemented
5	#22590128	C₂	1	2	yes	maximal
6	#22590124	C₂	1	1	yes	maximal, characteristic
7	C₂×#17985825	C₂	1	1	yes	maximal, characteristic
8	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
9	#17986627	—	2	2	no	—
10	#17986408	—	2	4	no	—
11	#17986404	C₂²	1	4	yes	—
12	#17986401	C₂²	1	4	yes	—
13	#17986041	—	2	2	no	—
14	#17986037	C₂²	1	2	yes	—
15	C₂×#14556062	C₂²	1	2	yes	—
16	#17985827	C₂²	1	4	yes	—
17	#17985827	C₂²	1	2	yes	—
18	#17985825	C₂²	1	1	yes	characteristic
19	#17985825	C₂²	1	2	yes	—
20	#17985825	C₂²	1	4	yes	—
21	#17985825	C₂²	1	1	yes	characteristic
22	#17985825	C₂²	1	2	yes	—
23	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	#405×C₇₁	C₂²	1	1	yes	characteristic
26	#346×C₇₁	C₂²	1	2	yes	—
27	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
28	#14556128	—	2	4	no	—
29	#14556127	—	4	2	no	—
30	#14556120	—	2	4	no	—
31	#14556119	—	4	2	no	—
32	#14556105	—	4	4	no	—
33	#14556087	—	2	2	no	—
34	C₂×C₄⋊Dic₇₁	—	2	2	no	—
35	#14556062	—	2	4	no	—
36	#14556062	—	2	2	no	—
37	#14556062	C₂³	1	4	yes	—
38	#14556062	C₂³	1	2	yes	—
39	#14556062	C₂³	1	2	yes	—
40	#14556060	—	2	4	no	—
41	#14556060	—	2	2	no	—
42	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	#117×C₇₁	D₄	1	1	yes	characteristic
45	M₃₂×C₇₁	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₇₁	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
50	#97×C₇₁	D₄	1	2	yes	—
51	C₄×C₈×C₇₁	C₂³	1	2	yes	—
52	C₂×Dic₁₄₂	—	4	4	no	—
53	C₂²×Dic₇₁	—	4	2	no	—
54	C₄⋊Dic₇₁	—	2	2	no	—
55	C₄⋊Dic₇₁	—	2	4	no	—
56	C₄⋊Dic₇₁	—	2	2	no	—
57	C₄⋊Dic₇₁	—	4	2	no	—
58	C₄⋊Dic₇₁	—	4	4	no	—
59	Dic₇₁⋊C₄	—	4	4	no	—
60	Dic₇₁⋊C₄	—	4	4	no	—
61	C₂²⋊Dic₇₁	—	4	2	no	—
62	C₄×Dic₇₁	—	4	2	no	—
63	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
64	M₁₆×C₇₁	Q₁₆	1	4	yes	—
65	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
71	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₇₁	D₈	1	2	yes	—
73	C₄²×C₇₁	C₂×D₄	1	2	yes	—
74	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
75	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
76	C₁₆×C₇₁	C₂×Q₈	1	4	yes	—
77	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₇₁	—	4	2	no	—
85	C₂×Dic₇₁	—	4	2	no	—
86	C₂×Dic₇₁	—	4	4	no	—
87	C₂×Dic₇₁	—	16	2	no	—
88	Dic₁₄₂	—	16	4	no	—
89	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
90	C₈×C₇₁	C₂×Q₁₆	1	4	yes	—
91	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₇₁	—	2	1	no	—
94	C₂²×C₇₁	—	2	1	no	—
95	C₂²×C₇₁	#500	1	2	yes	—
96	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₇₁	—	2	2	no	—
98	C₄×C₇₁	—	2	2	no	—
99	C₄×C₇₁	#500	1	2	yes	—
100	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₇₁	—	8	2	no	—
103	Dic₇₁	—	8	4	no	—
104	Dic₇₁	—	8	2	no	—
105	#33729	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	—	2	2	no	—
107	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
108	C₂×C₇₁	#3415	1	1	yes	characteristic
109	C₂×C₇₁	#3417	1	1	yes	characteristic
110	#3164	—	71	2	no	—
111	#2255	—	71	4	no	—
112	#2253	—	71	2	no	—
113	#2252	—	71	2	no	—
114	#2159	—	71	2	no	—
115	#2158	—	71	1	no	—
116	C₂×#366	—	71	1	no	—
117	#2108	D₇₁	1	1	yes	characteristic
118	C₇₁	#33729	1	1	yes	characteristic, complemented, Sylow
119	#544	—	142	2	no	—
120	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
121	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
122	#501	—	142	4	no	—
123	#500	—	71	4	no	—
124	#499	—	71	4	no	—
125	#428	—	142	2	no	—
126	#427	—	71	2	no	—
127	C₂×#107	—	71	2	no	—
128	#405	C₂×D₇₁	1	1	yes	characteristic
129	#367	—	71	2	no	—
130	#367	—	71	4	no	—
131	#366	—	71	2	no	—
132	#366	—	71	2	no	—
133	#366	—	71	1	no	—
134	#366	—	71	1	no	—
135	#366	—	71	4	no	—
136	#346	C₂×D₇₁	1	2	yes	—
137	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
138	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	284	2	no	—
141	C₄⋊Q₈	—	142	4	no	—
142	#125	—	284	2	no	—
143	#125	—	142	4	no	—
144	C₂²⋊Q₈	—	284	4	no	—
145	#117	—	142	2	no	—
146	#117	D₂₈₄	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	142	2	no	—
148	M₃₂	Dic₁₄₂	1	4	yes	complemented
149	C₂×C₁₆	Dic₁₄₂	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
152	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
153	#107	—	71	4	no	—
154	#107	—	71	2	no	—
155	#107	—	71	2	no	—
156	#107	—	142	2	no	—
157	#107	—	142	4	no	—
158	#106	—	142	2	no	—
159	#106	—	142	4	no	—
160	#97	D₂₈₄	1	2	yes	—
161	C₄×C₈	C₂²×D₇₁	1	2	yes	—
162	C₂×Q₈	—	284	4	no	—
163	C₂²×C₄	—	284	2	no	—
164	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
165	M₁₆	Dic₂₈₄	1	4	yes	—
166	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
168	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
169	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
170	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
171	C₄⋊C₄	—	284	4	no	—
172	C₄⋊C₄	—	284	4	no	—
173	C₄⋊C₄	—	284	2	no	—
174	C₄⋊C₄	—	284	4	no	—
175	C₄⋊C₄	—	142	2	no	—
176	C₄⋊C₄	—	142	4	no	—
177	C₄⋊C₄	—	142	2	no	—
178	C₄⋊C₄	D₅₆₈	1	1	yes	characteristic
179	C₄⋊C₄	D₅₆₈	1	1	yes	characteristic
180	C₂²⋊C₄	—	284	2	no	—
181	C₂²⋊C₄	D₅₆₈	1	2	yes	—
182	C₄²	—	284	2	no	—
183	C₄²	C₂×D₂₈₄	1	2	yes	—
184	C₁₆	C₂×Dic₁₄₂	1	2	yes	—
185	C₁₆	C₂×Dic₁₄₂	1	2	yes	—
186	C₁₆	C₂×Dic₁₄₂	1	4	yes	—
187	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
188	Q₈	—	1136	4	no	—
189	C₂×C₄	—	284	2	no	—
190	C₂×C₄	—	284	2	no	—
191	C₂×C₄	—	284	4	no	—
192	C₂×C₄	—	1136	2	no	—
193	C₂×C₄	#14556128	1	2	yes	—
194	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
195	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
196	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
198	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
199	C₈	#14556128	1	2	yes	—
200	C₈	C₂×Dic₂₈₄	1	4	yes	—
201	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#17986404	1	2	yes	—
206	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	568	4	no	—
210	C₄	—	568	2	no	—
211	C₄	—	568	2	no	—
212	C₄	#17986404	1	2	yes	—
213	C₄	C₂×#14556128	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#17986404	1	1	yes	characteristic
217	C₂	#22601481	1	1	yes	characteristic
218	C₂	#22601493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#33729
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22590626	C₂	semidirect
3. #22590619	C₂	semidirect
4. #22590615	C₂	semidirect
45. M₃₂×C₇₁	Q₈	semidirect
46. C₂×C₁₆×C₇₁	Q₈	semidirect
118. C₇₁	#33729	semidirect
148. M₃₂	Dic₁₄₂	semidirect
149. C₂×C₁₆	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22598879	2
#22590626	2
#22590619	2
#22590615	2
#22590128	2
#22590124	2
C₂×#17985825	2
C₇₁×#2108	2
#33729	71