Subgroups — C₂²×#17986766 #32911217

Order 18176.

11118 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986766	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14556135	C₂	1	8	yes	maximal, complemented
3	C₂³×#11765849	C₂	1	6	yes	maximal, complemented
4	C₂³×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986766	C₂²	1	256	yes	direct product factor
6	C₂×#14556135	C₂²	1	192	yes	complemented
7	C₂²×#11765849	C₂²	1	4	yes	—
8	C₂²×#11765849	C₂²	1	12	yes	complemented
9	C₂²×#11765849	C₂²	1	144	yes	complemented
10	C₂³×C₇₁⋊C₈	C₂²	1	12	yes	complemented
11	C₂²×#41×C₇₁	C₄	1	4	yes	—
12	C₂²×#41×C₇₁	C₄	1	24	yes	—
13	C₂²×#41×C₇₁	C₂²	1	4	yes	—
14	C₂²×#41×C₇₁	C₂²	1	24	yes	—
15	C₂³×Q₈×C₇₁	C₄	1	1	yes	characteristic
16	C₂³×D₄×C₇₁	C₄	1	3	yes	—
17	C₂⁴×C₄×C₇₁	C₂²	1	3	yes	—
18	#14556135	C₂³	1	512	yes	complemented
19	C₂×#11765849	C₂³	1	48	yes	—
20	C₂×#11765849	C₂³	1	144	yes	complemented
21	C₂×#11765849	C₂³	1	384	yes	complemented
22	C₂²×C₇₁⋊C₈	C₂³	1	8	yes	—
23	C₂²×C₇₁⋊C₈	C₂³	1	144	yes	complemented
24	C₂⁵×C₇₁	C₂×C₄	1	3	yes	—
25	C₂×#41×C₇₁	C₂×C₄	1	48	yes	—
26	C₂×#41×C₇₁	C₂×C₄	1	96	yes	—
27	C₂×#41×C₇₁	C₂×C₄	1	192	yes	—
28	C₂×#41×C₇₁	C₂³	1	48	yes	—
29	C₂×#41×C₇₁	C₂³	1	64	yes	—
30	C₂²×Q₈×C₇₁	C₂×C₄	1	4	yes	—
31	C₂²×Q₈×C₇₁	C₂×C₄	1	24	yes	—
32	C₂²×D₄×C₇₁	C₂×C₄	1	12	yes	—
33	C₂²×D₄×C₇₁	C₂×C₄	1	72	yes	—
34	C₂³×C₄×C₇₁	C₂×C₄	1	3	yes	—
35	C₂³×C₄×C₇₁	C₂×C₄	1	6	yes	—
36	C₂³×C₄×C₇₁	C₂×C₄	1	36	yes	—
37	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₇₁	C₂³	1	6	yes	—
39	C₂³×C₄×C₇₁	C₂³	1	36	yes	—
40	#11765849	C₂⁴	1	64	yes	—
41	#11765849	C₂⁴	1	192	yes	complemented
42	C₂×C₇₁⋊C₈	C₂⁴	1	48	yes	—
43	C₂×C₇₁⋊C₈	C₂⁴	1	192	yes	complemented
44	C₂⁴×C₇₁	—	4	12	no	—
45	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂⁴×C₇₁	C₂²×C₄	1	6	yes	—
47	C₂⁴×C₇₁	C₂²×C₄	1	36	yes	—
48	#41×C₇₁	C₂²×C₄	1	192	yes	—
49	#41×C₇₁	C₂²×C₄	1	256	yes	—
50	#41×C₇₁	C₂⁴	1	64	yes	—
51	C₂×Q₈×C₇₁	C₂²×C₄	1	48	yes	—
52	C₂×Q₈×C₇₁	C₂²×C₄	1	64	yes	—
53	C₂×D₄×C₇₁	C₂²×C₄	1	144	yes	—
54	C₂×D₄×C₇₁	C₂²×C₄	1	192	yes	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₂²×C₄	1	36	yes	—
58	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₇₁	C₂²×C₄	1	36	yes	—
60	C₂²×C₄×C₇₁	C₂²×C₄	1	72	yes	—
61	C₂²×C₄×C₇₁	C₂²×C₄	1	144	yes	—
62	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
63	C₂²×C₄×C₇₁	C₂⁴	1	36	yes	—
64	C₂²×C₄×C₇₁	C₂⁴	1	48	yes	—
65	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
66	C₂³×C₇₁	—	2	4	no	—
67	C₂³×C₇₁	—	2	24	no	—
68	C₂³×C₇₁	—	4	72	no	—
69	C₂³×C₇₁	C₂³×C₄	1	6	yes	—
70	C₂³×C₇₁	C₂³×C₄	1	36	yes	—
71	C₂³×C₇₁	C₂³×C₄	1	48	yes	—
72	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
73	Q₈×C₇₁	C₂³×C₄	1	64	yes	—
74	D₄×C₇₁	C₂³×C₄	1	192	yes	—
75	C₂×C₄×C₇₁	C₂³×C₄	1	12	yes	—
76	C₂×C₄×C₇₁	C₂³×C₄	1	6	yes	—
77	C₂×C₄×C₇₁	C₂³×C₄	1	3	yes	—
78	C₂×C₄×C₇₁	C₂³×C₄	1	48	yes	—
79	C₂×C₄×C₇₁	C₂³×C₄	1	36	yes	—
80	C₂×C₄×C₇₁	C₂³×C₄	1	72	yes	—
81	C₂×C₄×C₇₁	C₂³×C₄	1	96	yes	—
82	C₂×C₄×C₇₁	C₂⁵	1	3	yes	—
83	C₂×C₄×C₇₁	C₂⁵	1	4	yes	—
84	C₂×C₄×C₇₁	C₂⁵	1	24	yes	—
85	C₇₁⋊C₈	C₂⁵	1	32	yes	—
86	C₂²×C₇₁	—	2	12	no	—
87	C₂²×C₇₁	—	2	72	no	—
88	C₂²×C₇₁	—	4	48	no	—
89	C₂²×C₇₁	#568	1	4	yes	complemented
90	C₂²×C₇₁	C₂⁴×C₄	1	4	yes	—
91	C₂²×C₇₁	C₂⁴×C₄	1	24	yes	—
92	C₂²×C₇₁	C₂⁴×C₄	1	3	yes	—
93	C₄×C₇₁	C₂⁴×C₄	1	3	yes	—
94	C₄×C₇₁	C₂⁴×C₄	1	4	yes	—
95	C₄×C₇₁	C₂⁴×C₄	1	24	yes	—
96	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
97	C₂²×#568	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	—	2	4	no	—
99	C₂×C₇₁	—	2	24	no	—
100	C₂×C₇₁	C₂×#568	1	6	yes	complemented
101	C₂×C₇₁	C₂⁵×C₄	1	1	yes	characteristic
102	C₂³×#41	D₇₁	1	1	yes	characteristic
103	C₂×#568	—	71	48	no	—
104	C₂²×#131	—	71	8	no	—
105	C₂³×M₁₆	—	71	6	no	—
106	C₇₁	C₂²×#568	1	1	yes	characteristic, complemented, Sylow
107	C₂²×#41	Dic₇₁	1	4	yes	—
108	C₂²×#41	Dic₇₁	1	24	yes	—
109	C₂²×#41	C₂×D₇₁	1	4	yes	—
110	C₂²×#41	C₂×D₇₁	1	24	yes	—
111	C₂³×Q₈	Dic₇₁	1	1	yes	characteristic
112	C₂³×D₄	Dic₇₁	1	3	yes	—
113	C₂⁴×C₄	C₂×D₇₁	1	3	yes	—
114	#568	—	71	256	no	—
115	C₂×#131	—	71	192	no	—
116	C₂²×M₁₆	—	71	4	no	—
117	C₂²×M₁₆	—	71	12	no	—
118	C₂²×M₁₆	—	71	144	no	—
119	C₂³×C₈	—	71	12	no	—
120	C₂⁵	C₂×Dic₇₁	1	3	yes	—
121	C₂×#41	C₂×Dic₇₁	1	48	yes	—
122	C₂×#41	C₂×Dic₇₁	1	96	yes	—
123	C₂×#41	C₂×Dic₇₁	1	192	yes	—
124	C₂×#41	C₂²×D₇₁	1	48	yes	—
125	C₂×#41	C₂²×D₇₁	1	64	yes	—
126	C₂²×Q₈	C₂×Dic₇₁	1	4	yes	—
127	C₂²×Q₈	C₂×Dic₇₁	1	24	yes	—
128	C₂²×D₄	C₂×Dic₇₁	1	12	yes	—
129	C₂²×D₄	C₂×Dic₇₁	1	72	yes	—
130	C₂³×C₄	C₂×Dic₇₁	1	3	yes	—
131	C₂³×C₄	C₂×Dic₇₁	1	6	yes	—
132	C₂³×C₄	C₂×Dic₇₁	1	36	yes	—
133	C₂³×C₄	C₂²×D₇₁	1	36	yes	—
134	C₂³×C₄	C₂²×D₇₁	1	6	yes	—
135	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
136	#131	—	71	512	no	—
137	C₂×M₁₆	—	71	48	no	—
138	C₂×M₁₆	—	71	144	no	—
139	C₂×M₁₆	—	71	384	no	—
140	C₂²×C₈	—	71	144	no	—
141	C₂²×C₈	—	71	8	no	—
142	C₂⁴	—	4	12	no	—
143	C₂⁴	C₂²×Dic₇₁	1	1	yes	characteristic
144	C₂⁴	C₂²×Dic₇₁	1	6	yes	—
145	C₂⁴	C₂²×Dic₇₁	1	36	yes	—
146	#41	C₂²×Dic₇₁	1	192	yes	—
147	#41	C₂²×Dic₇₁	1	256	yes	—
148	#41	C₂³×D₇₁	1	64	yes	—
149	C₂×Q₈	C₂²×Dic₇₁	1	48	yes	—
150	C₂×Q₈	C₂²×Dic₇₁	1	64	yes	—
151	C₂×D₄	C₂²×Dic₇₁	1	144	yes	—
152	C₂×D₄	C₂²×Dic₇₁	1	192	yes	—
153	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
154	C₂²×C₄	C₂²×Dic₇₁	1	6	yes	—
155	C₂²×C₄	C₂²×Dic₇₁	1	36	yes	—
156	C₂²×C₄	C₂²×Dic₇₁	1	6	yes	—
157	C₂²×C₄	C₂²×Dic₇₁	1	72	yes	—
158	C₂²×C₄	C₂²×Dic₇₁	1	144	yes	—
159	C₂²×C₄	C₂²×Dic₇₁	1	36	yes	—
160	C₂²×C₄	C₂³×D₇₁	1	36	yes	—
161	C₂²×C₄	C₂³×D₇₁	1	6	yes	—
162	C₂²×C₄	C₂³×D₇₁	1	48	yes	—
163	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
164	M₁₆	—	71	64	no	—
165	M₁₆	—	71	192	no	—
166	C₂×C₈	—	71	48	no	—
167	C₂×C₈	—	71	192	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	24	no	—
170	C₂³	—	4	72	no	—
171	C₂³	C₂³×Dic₇₁	1	6	yes	—
172	C₂³	C₂³×Dic₇₁	1	36	yes	—
173	C₂³	C₂³×Dic₇₁	1	48	yes	—
174	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
175	Q₈	C₂³×Dic₇₁	1	64	yes	—
176	D₄	C₂³×Dic₇₁	1	192	yes	—
177	C₂×C₄	C₂³×Dic₇₁	1	12	yes	—
178	C₂×C₄	C₂³×Dic₇₁	1	6	yes	—
179	C₂×C₄	C₂³×Dic₇₁	1	3	yes	—
180	C₂×C₄	C₂³×Dic₇₁	1	48	yes	—
181	C₂×C₄	C₂³×Dic₇₁	1	36	yes	—
182	C₂×C₄	C₂³×Dic₇₁	1	72	yes	—
183	C₂×C₄	C₂³×Dic₇₁	1	96	yes	—
184	C₂×C₄	C₂⁴×D₇₁	1	3	yes	—
185	C₂×C₄	C₂⁴×D₇₁	1	24	yes	—
186	C₂×C₄	C₂⁴×D₇₁	1	4	yes	—
187	C₈	—	71	32	no	—
188	C₂²	—	2	12	no	—
189	C₂²	—	2	72	no	—
190	C₂²	—	4	48	no	—
191	C₂²	#17986766	1	4	yes	direct product factor
192	C₂²	C₂⁴×Dic₇₁	1	4	yes	—
193	C₂²	C₂⁴×Dic₇₁	1	24	yes	—
194	C₂²	C₂⁴×Dic₇₁	1	3	yes	—
195	C₄	C₂⁴×Dic₇₁	1	4	yes	—
196	C₄	C₂⁴×Dic₇₁	1	3	yes	—
197	C₄	C₂⁴×Dic₇₁	1	24	yes	—
198	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
199	C₂	—	2	4	no	—
200	C₂	—	2	24	no	—
201	C₂	C₂×#17986766	1	6	yes	direct product factor
202	C₂	C₂⁵×Dic₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₇₁
commutator	C₂×C₇₁	C₂⁵×C₄
Frattini	C₄	C₂⁵×D₇₁
Fitting, prime radical	C₂³×#41×C₇₁	C₂
socle	C₂³×C₇₁	C₂³×C₄
cosocle	C₄×C₇₁	C₂⁶
hypercenter	C₂³×#41	D₇₁
hypocenter	C₇₁	C₂²×#568
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#17986766	C₂	direct (N × H)
2. C₂²×#14556135	C₂	semidirect
3. C₂³×#11765849	C₂	semidirect
5. #17986766	C₂²	direct (N × H)
6. C₂×#14556135	C₂²	semidirect
8. C₂²×#11765849	C₂²	semidirect
9. C₂²×#11765849	C₂²	semidirect
10. C₂³×C₇₁⋊C₈	C₂²	semidirect
18. #14556135	C₂³	semidirect
20. C₂×#11765849	C₂³	semidirect
21. C₂×#11765849	C₂³	semidirect
23. C₂²×C₇₁⋊C₈	C₂³	semidirect
41. #11765849	C₂⁴	semidirect
43. C₂×C₇₁⋊C₈	C₂⁴	semidirect
89. C₂²×C₇₁	#568	semidirect
100. C₂×C₇₁	C₂×#568	semidirect
106. C₇₁	C₂²×#568	semidirect
191. C₂²	#17986766	direct (N × H)
201. C₂	C₂×#17986766	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17986766	2
C₂²×#14556135	2
C₂³×#11765849	2
C₂³×#41×C₇₁	2
C₂²×#568	71

Subgroups — C22×#17986766 #32911217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17986766 #32911217