Subgroups — C₂²×#17985721 #31963021

Order 18176.

1182 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985721	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇₁⋊C₈	C₄	1	2	yes	—
6	C₂³×C₇₁⋊C₈	C₄	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	6	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₄	1	6	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
11	#17985721	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₇₁	C₄	1	2	yes	—
13	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₄	1	6	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	6	yes	—
16	C₂²×C₇₁⋊C₈	—	2	8	no	—
17	C₂²×C₇₁⋊C₈	C₈	1	16	yes	complemented
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
20	C₄×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
21	C₄×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
22	C₄×C₇₁⋊C₈	C₂³	1	4	yes	—
23	C₄×C₇₁⋊C₈	C₂³	1	4	yes	—
24	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	—	2	8	no	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	12	yes	—
29	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
33	C₄×C₈×C₇₁	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₇₁	C₂³	1	4	yes	—
35	C₂×C₇₁⋊C₈	—	2	24	no	—
36	C₂×C₇₁⋊C₈	C₂×C₈	1	48	yes	complemented
37	C₂×C₇₁⋊C₈	C₂²×C₄	1	8	yes	—
38	C₂×C₇₁⋊C₈	C₂²×C₄	1	8	yes	—
39	C₂⁴×C₇₁	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₄²	1	3	yes	—
41	C₂²×C₄×C₇₁	C₄²	1	3	yes	—
42	C₂²×C₄×C₇₁	C₄²	1	3	yes	—
43	C₂²×C₄×C₇₁	C₂×C₈	1	8	yes	—
44	C₂²×C₄×C₇₁	M₁₆	1	8	yes	—
45	C₂²×C₄×C₇₁	M₁₆	1	8	yes	—
46	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₇₁	—	2	24	no	—
50	C₂×C₈×C₇₁	C₂²×C₄	1	8	yes	—
51	C₄²×C₇₁	C₄²	1	6	yes	—
52	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
53	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
54	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
55	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
56	C₂³×C₇₁	C₄×C₈	1	4	yes	—
57	C₂³×C₇₁	#97	1	4	yes	—
58	C₂³×C₇₁	#97	1	4	yes	—
59	C₂³×C₇₁	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₇₁	C₄×C₈	1	12	yes	—
61	C₂×C₄×C₇₁	#97	1	12	yes	—
62	C₂×C₄×C₇₁	#97	1	12	yes	—
63	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
64	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
65	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
66	C₂×C₄×C₇₁	C₂²×C₈	1	12	yes	—
67	C₂×C₄×C₇₁	C₂×M₁₆	1	12	yes	—
68	C₂×C₄×C₇₁	C₂×M₁₆	1	12	yes	—
69	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₇₁	—	2	8	no	—
73	C₇₁⋊C₈	—	2	8	no	—
74	C₇₁⋊C₈	C₂²×C₈	1	16	yes	complemented
75	C₂²×C₇₁	#322	1	16	yes	complemented
76	C₂²×C₇₁	C₂×C₄×C₈	1	6	yes	—
77	C₂²×C₇₁	C₂×#97	1	6	yes	—
78	C₂²×C₇₁	C₂×#97	1	6	yes	—
79	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
80	C₄×C₇₁	C₂×C₄×C₈	1	6	yes	—
81	C₄×C₇₁	C₂×#97	1	6	yes	—
82	C₄×C₇₁	C₂×#97	1	6	yes	—
83	C₄×C₇₁	C₂³×C₈	1	2	yes	—
84	C₄×C₇₁	C₂²×M₁₆	1	2	yes	—
85	C₄×C₇₁	C₂²×M₁₆	1	2	yes	—
86	C₂²×#322	—	71	1	no	maximal, Sylow
87	C₂×C₇₁	C₂×#322	1	12	yes	complemented
88	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
89	C₂×C₇₁	C₂²×#97	1	1	yes	characteristic
90	C₂×C₇₁	C₂²×#97	1	1	yes	characteristic
91	C₂²×C₄×C₈	—	71	1	no	—
92	C₂²×C₄×C₈	—	71	1	no	—
93	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
94	C₂×#322	—	71	12	no	—
95	C₇₁	C₂²×#322	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₈	—	71	2	no	—
97	C₂³×C₈	—	71	2	no	—
98	C₂³×C₈	Dic₇₁	1	2	yes	—
99	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
100	C₂×C₄×C₈	—	71	6	no	—
101	C₂×C₄×C₈	—	71	6	no	—
102	C₂×C₄×C₈	—	71	6	no	—
103	C₂×C₄×C₈	—	71	6	no	—
104	C₂×C₄×C₈	Dic₇₁	1	6	yes	—
105	C₂×C₄×C₈	C₂×D₇₁	1	6	yes	—
106	#322	—	71	16	no	—
107	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
108	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
109	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
110	C₂²×C₈	—	2	8	no	—
111	C₂²×C₈	—	71	16	no	—
112	C₂²×C₈	—	71	12	no	—
113	C₂²×C₈	—	71	12	no	—
114	C₂²×C₈	—	142	8	no	—
115	C₂²×C₈	C₂×Dic₇₁	1	12	yes	—
116	C₂×C₄²	C₄×D₇₁	1	3	yes	—
117	C₂×C₄²	C₄×D₇₁	1	3	yes	—
118	C₂×C₄²	C₂×Dic₇₁	1	3	yes	—
119	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
120	C₄×C₈	—	71	12	no	—
121	C₄×C₈	—	71	12	no	—
122	C₄×C₈	—	71	4	no	—
123	C₄×C₈	—	71	4	no	—
124	C₄×C₈	C₂×Dic₇₁	1	12	yes	—
125	C₄×C₈	C₂²×D₇₁	1	4	yes	—
126	C₂⁴	C₄×Dic₇₁	1	1	yes	characteristic
127	C₂²×C₄	C₄×Dic₇₁	1	3	yes	—
128	C₂²×C₄	C₄×Dic₇₁	1	3	yes	—
129	C₂²×C₄	C₄×Dic₇₁	1	3	yes	—
130	C₂²×C₄	C₈×D₇₁	1	8	yes	—
131	C₂²×C₄	#11765848	1	8	yes	—
132	C₂²×C₄	#11765849	1	8	yes	—
133	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
134	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
135	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
136	C₂×C₈	—	2	24	no	—
137	C₂×C₈	—	71	8	no	—
138	C₂×C₈	—	71	48	no	—
139	C₂×C₈	—	71	8	no	—
140	C₂×C₈	—	142	24	no	—
141	C₂×C₈	C₂²×Dic₇₁	1	8	yes	—
142	C₄²	C₄×Dic₇₁	1	6	yes	—
143	C₄²	C₂×C₄×D₇₁	1	3	yes	—
144	C₄²	C₂×C₄×D₇₁	1	3	yes	—
145	C₄²	C₂²×Dic₇₁	1	3	yes	—
146	C₄²	C₂³×D₇₁	1	1	yes	characteristic
147	C₂³	C₈×Dic₇₁	1	4	yes	—
148	C₂³	#14556038	1	4	yes	—
149	C₂³	#14556039	1	4	yes	—
150	C₂³	C₂×C₄×Dic₇₁	1	3	yes	—
151	C₂×C₄	C₈×Dic₇₁	1	12	yes	—
152	C₂×C₄	#14556038	1	12	yes	—
153	C₂×C₄	#14556039	1	12	yes	—
154	C₂×C₄	C₂×C₄×Dic₇₁	1	3	yes	—
155	C₂×C₄	C₂×C₄×Dic₇₁	1	3	yes	—
156	C₂×C₄	C₂×C₄×Dic₇₁	1	3	yes	—
157	C₂×C₄	C₂×C₈×D₇₁	1	12	yes	—
158	C₂×C₄	C₂×#11765848	1	12	yes	—
159	C₂×C₄	C₂×#11765849	1	12	yes	—
160	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
162	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
163	C₈	—	2	8	no	—
164	C₈	—	71	16	no	—
165	C₈	—	142	8	no	—
166	C₂²	#17985721	1	16	yes	direct product factor
167	C₂²	C₂×C₈×Dic₇₁	1	6	yes	—
168	C₂²	C₂×#14556038	1	6	yes	—
169	C₂²	C₂×#14556039	1	6	yes	—
170	C₂²	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
171	C₄	C₂×C₈×Dic₇₁	1	6	yes	—
172	C₄	C₂×#14556038	1	6	yes	—
173	C₄	C₂×#14556039	1	6	yes	—
174	C₄	C₂²×C₈×D₇₁	1	2	yes	—
175	C₄	C₂²×#11765848	1	2	yes	—
176	C₄	C₂²×#11765849	1	2	yes	—
177	C₂	C₂×#17985721	1	12	yes	direct product factor
178	C₂	C₂²×C₈×Dic₇₁	1	1	yes	characteristic
179	C₂	C₂²×#14556038	1	1	yes	characteristic
180	C₂	C₂²×#14556039	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₇₁
commutator	C₂×C₇₁	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₇₁
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₄²
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₁
hypocenter	C₇₁	C₂²×#322
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#17985721	C₂	direct (N × H)
11. #17985721	C₂²	direct (N × H)
17. C₂²×C₇₁⋊C₈	C₈	semidirect
36. C₂×C₇₁⋊C₈	C₂×C₈	semidirect
74. C₇₁⋊C₈	C₂²×C₈	semidirect
75. C₂²×C₇₁	#322	semidirect
87. C₂×C₇₁	C₂×#322	semidirect
95. C₇₁	C₂²×#322	semidirect
166. C₂²	#17985721	direct (N × H)
177. C₂	C₂×#17985721	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇₁⋊C₈	2
C₂×#17985721	2
C₂²×C₄×C₈×C₇₁	2
C₂²×#322	71

Subgroups — C22×#17985721 #31963021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17985721 #31963021