Subgroups — C₂²×#17985824 #32150472

Order 18176.

782 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556062	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985824	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₁	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	C₄	1	8	yes	complemented
7	C₂²×C₇₁⋊C₁₆	C₂²	1	6	yes	—
8	C₂×#14556062	—	2	3	no	—
9	C₂×#14556062	C₂²	1	6	yes	—
10	#17985824	C₂²	1	16	yes	direct product factor
11	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₇₁	—	2	1	no	—
13	C₂×#107×C₇₁	—	2	3	no	—
14	C₂×#107×C₇₁	C₂²	1	6	yes	—
15	C₂³×Dic₇₁	—	4	1	no	—
16	C₂×C₄⋊Dic₇₁	—	2	6	no	—
17	C₂×C₄⋊Dic₇₁	—	4	3	no	—
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	24	yes	complemented
19	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
20	#14556062	—	2	6	no	—
21	#14556062	C₂³	1	4	yes	—
22	C₂³×C₄×C₇₁	—	4	1	no	—
23	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₇₁	D₄	1	3	yes	—
26	C₂²×C₈×C₇₁	Q₈	1	4	yes	—
27	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
28	C₂×C₄⋊C₄×C₇₁	—	2	6	no	—
29	C₂×C₄⋊C₄×C₇₁	—	4	3	no	—
30	#107×C₇₁	—	2	6	no	—
31	#107×C₇₁	C₂³	1	4	yes	—
32	C₂²×Dic₇₁	—	4	6	no	—
33	C₂²×Dic₇₁	—	8	4	no	—
34	C₄⋊Dic₇₁	—	2	4	no	—
35	C₄⋊Dic₇₁	—	4	6	no	—
36	C₇₁⋊C₁₆	C₂²×C₄	1	8	yes	complemented
37	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	—	4	6	no	—
39	C₂²×C₄×C₇₁	—	8	4	no	—
40	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₇₁	D₈	1	3	yes	—
42	C₂²×C₄×C₇₁	Q₁₆	1	4	yes	—
43	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₇₁	C₄⋊C₄	1	12	yes	—
45	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
46	C₂×C₈×C₇₁	C₂×D₄	1	3	yes	—
47	C₂×C₈×C₇₁	C₂×Q₈	1	6	yes	—
48	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₇₁	—	2	4	no	—
50	C₄⋊C₄×C₇₁	—	4	6	no	—
51	C₂³×C₇₁	#107	1	4	yes	—
52	C₂³×C₇₁	D₁₆	1	4	yes	—
53	C₂³×C₇₁	Q₃₂	1	4	yes	—
54	C₂³×C₇₁	C₂×D₈	1	3	yes	—
55	C₂×C₄×C₇₁	—	4	4	no	—
56	C₂×C₄×C₇₁	—	8	12	no	—
57	C₂×C₄×C₇₁	#107	1	12	yes	—
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
59	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
60	C₂×C₄×C₇₁	C₂×Q₁₆	1	6	yes	—
61	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
62	C₂×Dic₇₁	—	4	4	no	—
63	C₂×Dic₇₁	—	8	12	no	—
64	C₈×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
65	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₇₁	#366	1	16	yes	complemented
68	C₂²×C₇₁	C₂×#107	1	6	yes	—
69	C₂²×C₇₁	C₂×D₁₆	1	6	yes	—
70	C₂²×C₇₁	C₂×Q₃₂	1	6	yes	—
71	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
72	C₄×C₇₁	—	8	4	no	—
73	C₄×C₇₁	C₂×#107	1	6	yes	—
74	C₄×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
76	Dic₇₁	—	8	4	no	—
77	C₂²×#366	—	71	1	no	maximal, Sylow
78	C₂×C₇₁	C₂×#366	1	12	yes	complemented
79	C₂×C₇₁	C₂²×#107	1	1	yes	characteristic
80	C₂×C₇₁	C₂²×D₁₆	1	1	yes	characteristic
81	C₂×C₇₁	C₂²×Q₃₂	1	1	yes	characteristic
82	C₂³×C₁₆	—	71	1	no	—
83	C₂²×#107	—	71	1	no	—
84	C₂²×#107	D₇₁	1	1	yes	characteristic
85	C₂×#366	—	71	12	no	—
86	C₇₁	C₂²×#366	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	—	2	1	no	—
89	C₂²×C₄⋊C₄	—	142	1	no	—
90	C₂²×C₁₆	—	71	8	no	—
91	C₂²×C₁₆	—	71	6	no	—
92	C₂×#107	—	2	3	no	—
93	C₂×#107	—	71	6	no	—
94	C₂×#107	—	142	3	no	—
95	C₂×#107	C₂×D₇₁	1	6	yes	—
96	#366	—	71	16	no	—
97	C₂³×C₄	—	4	1	no	—
98	C₂³×C₄	—	284	1	no	—
99	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂²×C₈	Dic₁₄₂	1	4	yes	—
101	C₂²×C₈	C₄×D₇₁	1	4	yes	—
102	C₂²×C₈	C₂²⋊D₇₁	1	3	yes	—
103	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
104	C₂×C₄⋊C₄	—	2	6	no	—
105	C₂×C₄⋊C₄	—	4	3	no	—
106	C₂×C₄⋊C₄	—	284	3	no	—
107	C₂×C₄⋊C₄	—	142	6	no	—
108	C₂×C₁₆	—	71	24	no	—
109	C₂×C₁₆	—	71	4	no	—
110	#107	—	2	6	no	—
111	#107	—	71	4	no	—
112	#107	—	142	6	no	—
113	#107	C₂²×D₇₁	1	4	yes	—
114	C₂⁴	#11765850	1	1	yes	characteristic
115	C₂²×C₄	—	4	6	no	—
116	C₂²×C₄	—	8	4	no	—
117	C₂²×C₄	—	284	6	no	—
118	C₂²×C₄	—	568	4	no	—
119	C₂²×C₄	Dic₇₁⋊C₄	1	4	yes	—
120	C₂²×C₄	#11765850	1	3	yes	—
121	C₂²×C₄	#11765855	1	4	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
123	C₂×C₈	Dic₇₁⋊C₄	1	12	yes	—
124	C₂×C₈	C₂×C₄×D₇₁	1	6	yes	—
125	C₂×C₈	C₂×C₂²⋊D₇₁	1	3	yes	—
126	C₂×C₈	C₂×Dic₁₄₂	1	6	yes	—
127	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	6	no	—
130	C₄⋊C₄	—	284	6	no	—
131	C₄⋊C₄	—	142	4	no	—
132	C₁₆	—	71	8	no	—
133	C₂³	#14556061	1	4	yes	—
134	C₂³	#14556069	1	4	yes	—
135	C₂³	#14556074	1	4	yes	—
136	C₂³	C₂×#11765850	1	3	yes	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	12	no	—
139	C₂×C₄	—	284	4	no	—
140	C₂×C₄	—	568	12	no	—
141	C₂×C₄	#14556061	1	12	yes	—
142	C₂×C₄	C₂×Dic₇₁⋊C₄	1	6	yes	—
143	C₂×C₄	C₂×#11765850	1	3	yes	—
144	C₂×C₄	C₂×#11765855	1	6	yes	—
145	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
146	C₈	C₂×Dic₇₁⋊C₄	1	6	yes	—
147	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
148	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
149	C₂²	#17985824	1	16	yes	direct product factor
150	C₂²	C₂×#14556061	1	6	yes	—
151	C₂²	C₂×#14556069	1	6	yes	—
152	C₂²	C₂×#14556074	1	6	yes	—
153	C₂²	C₂²×#11765850	1	1	yes	characteristic
154	C₄	—	8	4	no	—
155	C₄	—	568	4	no	—
156	C₄	C₂×#14556061	1	6	yes	—
157	C₄	C₂²×Dic₇₁⋊C₄	1	1	yes	characteristic
158	C₄	C₂²×#11765855	1	1	yes	characteristic
159	C₂	C₂×#17985824	1	12	yes	direct product factor
160	C₂	C₂²×#14556061	1	1	yes	characteristic
161	C₂	C₂²×#14556069	1	1	yes	characteristic
162	C₂	C₂²×#14556074	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11765850
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂²×#107×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂²×#107	D₇₁
hypocenter	C₇₁	C₂²×#366
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. C₂×#17985824	C₂	direct (N × H)
6. C₂²×C₇₁⋊C₁₆	C₄	semidirect
10. #17985824	C₂²	direct (N × H)
18. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
36. C₇₁⋊C₁₆	C₂²×C₄	semidirect
67. C₂²×C₇₁	#366	semidirect
78. C₂×C₇₁	C₂×#366	semidirect
86. C₇₁	C₂²×#366	semidirect
149. C₂²	#17985824	direct (N × H)
159. C₂	C₂×#17985824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇₁⋊C₁₆	2
C₂²×#14556062	2
C₂×#17985824	2
C₂²×#107×C₇₁	2
C₂²×#366	71

Subgroups — C22×#17985824 #32150472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17985824 #32150472