Subgroups — #32152944

Order 18176.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	2	yes	maximal
2	#22590622	C₂	1	2	yes	maximal, complemented
3	#22590619	C₂	1	4	yes	maximal, complemented
4	#22590615	C₂	1	2	yes	maximal, complemented
5	#22590124	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
7	#17986556	—	2	2	no	—
8	#17986404	C₂²	1	4	yes	—
9	#17986401	—	2	2	no	—
10	#17986401	C₂²	1	4	yes	—
11	#17986397	—	2	2	no	—
12	#17986037	—	2	2	no	—
13	#17986037	C₂²	1	4	yes	—
14	#17985827	C₂²	1	8	yes	—
15	#17985825	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
20	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	#14556128	—	2	4	no	—
22	#14556120	—	2	4	no	—
23	#14556107	—	4	4	no	—
24	C₄×Dic₁₄₂	—	4	4	no	—
25	#14556087	—	2	4	no	—
26	#14556062	—	2	4	no	—
27	#14556062	C₂³	1	8	yes	—
28	#14556060	—	2	4	no	—
29	#14556060	—	2	4	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₁	Q₈	1	8	yes	complemented
34	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
35	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
36	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₄₂	—	4	4	no	—
41	C₄⋊Dic₇₁	—	2	8	no	—
42	C₄⋊Dic₇₁	—	4	4	no	—
43	Dic₇₁⋊C₄	—	4	8	no	—
44	C₂²⋊Dic₇₁	—	4	4	no	—
45	C₄×Dic₇₁	—	4	4	no	—
46	C₂²×C₄×C₇₁	D₈	1	2	yes	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
49	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
53	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
54	C₄²×C₇₁	D₈	1	2	yes	—
55	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₇₁	C₂×Q₈	1	4	yes	—
58	C₁₆×C₇₁	C₂×Q₈	1	4	yes	—
59	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	4	no	—
61	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₇₁	—	4	8	no	—
68	Dic₁₄₂	—	8	8	no	—
69	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
70	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
71	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
72	C₈×C₇₁	C₂²×Q₈	1	2	yes	—
73	C₂²×C₇₁	—	2	2	no	—
74	C₂²×C₇₁	#500	1	2	yes	—
75	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₇₁	—	2	2	no	—
77	C₄×C₇₁	#500	1	2	yes	—
78	C₄×C₇₁	#508	1	4	yes	—
79	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
81	Dic₇₁	—	8	8	no	—
82	#33726	—	71	1	no	maximal, Sylow
83	C₂×C₇₁	—	2	2	no	—
84	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
85	C₂×C₇₁	C₂×#508	1	2	yes	—
86	#3160	—	71	2	no	—
87	#2254	—	71	2	no	—
88	#2253	—	71	4	no	—
89	#2252	—	71	2	no	—
90	#2158	—	71	4	no	—
91	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
92	C₇₁	#33726	1	1	yes	characteristic, complemented, Sylow
93	#533	—	142	2	no	—
94	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
95	#500	—	71	4	no	—
96	#499	—	71	4	no	—
97	#499	—	142	2	no	—
98	#498	—	142	2	no	—
99	#427	—	71	4	no	—
100	#427	—	142	2	no	—
101	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
102	#367	—	71	8	no	—
103	#366	—	71	8	no	—
104	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
105	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
106	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
107	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	C₄⋊Q₈	—	142	4	no	—
110	#125	—	142	4	no	—
111	C₂²⋊Q₈	—	284	4	no	—
112	C₄×Q₈	—	284	4	no	—
113	#117	—	142	4	no	—
114	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
115	C₂×C₁₆	Dic₁₄₂	1	8	yes	complemented
116	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
117	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
118	#107	—	71	8	no	—
119	#107	—	142	4	no	—
120	#106	—	142	4	no	—
121	#106	—	142	4	no	—
122	C₄×C₈	D₂₈₄	1	1	yes	characteristic
123	C₄×C₈	D₂₈₄	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
126	C₂×Q₈	—	284	4	no	—
127	C₂²×C₄	D₅₆₈	1	2	yes	—
128	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
129	C₂×C₈	Dic₂₈₄	1	4	yes	—
130	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
131	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
132	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
133	C₂×C₈	C₂×Dic₁₄₂	1	4	yes	—
134	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
135	C₄⋊C₄	—	284	8	no	—
136	C₄⋊C₄	—	284	4	no	—
137	C₄⋊C₄	—	142	8	no	—
138	C₂²⋊C₄	—	284	4	no	—
139	C₄²	—	284	4	no	—
140	C₄²	D₅₆₈	1	2	yes	—
141	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
142	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
143	C₁₆	C₂×Dic₁₄₂	1	4	yes	—
144	C₁₆	C₂×Dic₁₄₂	1	4	yes	—
145	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
146	Q₈	—	568	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	284	8	no	—
149	C₂×C₄	#14556128	1	2	yes	—
150	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
151	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
152	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
154	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
155	C₈	#14556128	1	2	yes	—
156	C₈	C₂×Dic₂₈₄	1	2	yes	—
157	C₈	C₂×Dic₂₈₄	1	2	yes	—
158	C₈	C₂²×Dic₁₄₂	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#17986404	1	2	yes	—
161	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	568	8	no	—
164	C₄	#17986404	1	2	yes	—
165	C₄	#17986431	1	4	yes	—
166	C₄	C₂×#14556128	1	1	yes	characteristic
167	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#17986404	1	1	yes	characteristic
170	C₂	C₂×#17986431	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#33726
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22590622	C₂	semidirect
3. #22590619	C₂	semidirect
4. #22590615	C₂	semidirect
33. C₂×C₁₆×C₇₁	Q₈	semidirect
92. C₇₁	#33726	semidirect
115. 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
#22598854	2
#22590622	2
#22590619	2
#22590615	2
#22590124	2
C₂×C₄×C₁₆×C₇₁	2
#33726	71