Subgroups — #32152982

Order 18176.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	1	yes	maximal, characteristic
2	C₂×#17986401	C₂	1	1	yes	maximal, characteristic
3	#22590626	C₂	1	4	yes	maximal, complemented
4	#22590619	C₂	1	4	yes	maximal, complemented
5	#22590128	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
7	#17986556	—	2	1	no	—
8	C₂×#14556120	—	2	1	no	—
9	#17986404	C₂²	1	2	yes	—
10	#17986401	—	2	1	no	—
11	#17986401	—	2	1	no	—
12	#17986401	—	2	1	no	—
13	#17986401	C₂²	1	2	yes	—
14	#17986401	C₂²	1	2	yes	—
15	#17986401	C₂²	1	2	yes	—
16	#17986397	—	2	1	no	—
17	#17986037	—	2	1	no	—
18	#17986037	C₂²	1	2	yes	—
19	C₂×#14556062	C₂²	1	2	yes	—
20	C₂×#14556060	—	2	1	no	—
21	#17985827	C₂²	1	8	yes	—
22	#17985825	C₂²	1	8	yes	—
23	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
25	#346×C₇₁	C₂²	1	2	yes	—
26	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
27	#14556128	—	2	2	no	—
28	#14556120	—	2	2	no	—
29	#14556120	—	2	2	no	—
30	#14556120	—	2	2	no	—
31	#14556120	—	4	2	no	—
32	#14556107	—	4	2	no	—
33	C₄×Dic₁₄₂	—	4	2	no	—
34	#14556087	—	2	2	no	—
35	C₂×C₄⋊Dic₇₁	—	2	2	no	—
36	C₂×Dic₇₁⋊C₄	—	4	2	no	—
37	#14556062	—	2	2	no	—
38	#14556062	—	2	2	no	—
39	#14556062	C₂³	1	4	yes	—
40	#14556062	C₂³	1	4	yes	—
41	#14556060	—	2	2	no	—
42	#14556060	—	2	2	no	—
43	#14556060	—	2	2	no	—
44	#14556060	—	2	2	no	—
45	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
48	M₃₂×C₇₁	Q₈	1	8	yes	complemented
49	C₂×C₁₆×C₇₁	C₂³	1	4	yes	—
50	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₄₂	—	4	2	no	—
55	C₂²×Dic₇₁	—	4	2	no	—
56	C₄⋊Dic₇₁	—	2	4	no	—
57	C₄⋊Dic₇₁	—	2	4	no	—
58	C₄⋊Dic₇₁	—	4	2	no	—
59	C₄⋊Dic₇₁	—	4	2	no	—
60	Dic₇₁⋊C₄	—	4	4	no	—
61	Dic₇₁⋊C₄	—	4	4	no	—
62	Dic₇₁⋊C₄	—	4	4	no	—
63	C₂²⋊Dic₇₁	—	4	2	no	—
64	C₄×Dic₇₁	—	4	2	no	—
65	C₂²×C₄×C₇₁	D₈	1	2	yes	—
66	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
68	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
73	C₄²×C₇₁	D₈	1	2	yes	—
74	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₇₁	C₂×Q₈	1	8	yes	—
77	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	—	4	2	no	—
79	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₇₁	—	4	4	no	—
86	C₂×Dic₇₁	—	4	4	no	—
87	C₂×Dic₇₁	—	8	4	no	—
88	Dic₁₄₂	—	8	4	no	—
89	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
90	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
91	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
92	C₈×C₇₁	C₂²×Q₈	1	2	yes	—
93	C₂²×C₇₁	—	2	2	no	—
94	C₂²×C₇₁	#500	1	2	yes	—
95	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₇₁	—	2	2	no	—
97	C₄×C₇₁	—	2	2	no	—
98	C₄×C₇₁	#500	1	2	yes	—
99	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
101	Dic₇₁	—	8	4	no	—
102	Dic₇₁	—	8	4	no	—
103	#33732	—	71	1	no	maximal, Sylow
104	C₂×C₇₁	—	2	2	no	—
105	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
106	C₂×C₇₁	#3414	1	2	yes	—
107	#3160	—	71	1	no	—
108	C₂×#499	—	71	1	no	—
109	#2255	—	71	4	no	—
110	#2253	—	71	4	no	—
111	#2159	—	71	4	no	—
112	C₄×M₃₂	D₇₁	1	1	yes	characteristic
113	C₇₁	#33732	1	1	yes	characteristic, complemented, Sylow
114	#533	—	142	1	no	—
115	C₂×#125	—	142	1	no	—
116	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
117	#500	—	71	2	no	—
118	#499	—	71	2	no	—
119	#499	—	71	2	no	—
120	#499	—	71	2	no	—
121	#499	—	142	1	no	—
122	#499	—	142	1	no	—
123	#499	—	142	1	no	—
124	#498	—	142	1	no	—
125	#427	—	71	2	no	—
126	#427	—	142	1	no	—
127	C₂×#107	—	71	2	no	—
128	C₂×#106	—	142	1	no	—
129	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
130	#367	—	71	8	no	—
131	#366	—	71	8	no	—
132	#346	C₂×D₇₁	1	2	yes	—
133	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
134	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	142	2	no	—
137	#125	—	284	2	no	—
138	#125	—	142	2	no	—
139	#125	—	142	2	no	—
140	#125	—	142	2	no	—
141	C₂²⋊Q₈	—	284	2	no	—
142	C₄×Q₈	—	284	2	no	—
143	#117	—	142	2	no	—
144	C₂×C₄⋊C₄	—	284	2	no	—
145	C₂×C₄⋊C₄	—	142	2	no	—
146	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
147	M₃₂	Dic₁₄₂	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₇₁	1	4	yes	—
149	#107	—	71	4	no	—
150	#107	—	71	4	no	—
151	#107	—	142	2	no	—
152	#107	—	142	2	no	—
153	#106	—	142	2	no	—
154	#106	—	142	2	no	—
155	#106	—	142	2	no	—
156	#106	—	142	2	no	—
157	C₄×C₈	D₂₈₄	1	1	yes	characteristic
158	C₄×C₈	D₂₈₄	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
161	C₂×Q₈	—	284	2	no	—
162	C₂²×C₄	—	284	2	no	—
163	C₂²×C₄	D₅₆₈	1	2	yes	—
164	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
165	C₂×C₈	Dic₂₈₄	1	4	yes	—
166	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
168	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
169	C₂×C₈	C₂×Dic₁₄₂	1	4	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	4	no	—
174	C₄⋊C₄	—	284	2	no	—
175	C₄⋊C₄	—	284	2	no	—
176	C₄⋊C₄	—	142	4	no	—
177	C₄⋊C₄	—	142	4	no	—
178	C₂²⋊C₄	—	284	2	no	—
179	C₄²	—	284	2	no	—
180	C₄²	D₅₆₈	1	2	yes	—
181	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
182	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
183	C₁₆	C₂×Dic₁₄₂	1	8	yes	—
184	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
185	Q₈	—	568	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	284	4	no	—
188	C₂×C₄	—	284	4	no	—
189	C₂×C₄	—	568	4	no	—
190	C₂×C₄	#14556128	1	2	yes	—
191	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
192	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
193	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
194	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
195	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
196	C₈	#14556128	1	2	yes	—
197	C₈	C₂×Dic₂₈₄	1	2	yes	—
198	C₈	C₂×Dic₂₈₄	1	2	yes	—
199	C₈	C₂²×Dic₁₄₂	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#17986404	1	2	yes	—
202	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	568	4	no	—
206	C₄	—	568	4	no	—
207	C₄	#17986404	1	2	yes	—
208	C₄	C₂×#14556128	1	1	yes	characteristic
209	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#17986404	1	1	yes	characteristic
212	C₂	#22601476	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₄×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	C₄×M₃₂	D₇₁
hypocenter	C₇₁	#33732
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22590626	C₂	semidirect
4. #22590619	C₂	semidirect
48. M₃₂×C₇₁	Q₈	semidirect
113. C₇₁	#33732	semidirect
147. M₃₂	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22598854	2
C₂×#17986401	2
#22590626	2
#22590619	2
#22590128	2
C₄×M₃₂×C₇₁	2
#33732	71