Subgroups — C₂×#22590099 #32150435

Order 18176.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22590099	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	4	yes	maximal, complemented
5	C₂×#405×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	—	2	4	no	—
7	C₂×#14556068	C₂²	1	4	yes	—
8	C₂²×C₇₁⋊C₁₆	—	2	4	no	—
9	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
12	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	16	yes	complemented
13	C₂²×M₁₆×C₇₁	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₁	C₄	1	1	yes	characteristic
16	#405×C₇₁	C₄	1	4	yes	—
17	#405×C₇₁	C₂²	1	4	yes	—
18	C₂×#97×C₇₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
20	#14556068	—	2	8	no	—
21	C₂×C₇₁⋊C₁₆	—	2	4	no	—
22	C₂×C₇₁⋊C₁₆	—	2	8	no	—
23	C₂×C₇₁⋊C₁₆	—	2	4	no	—
24	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
25	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
26	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₇₁	—	2	4	no	—
28	C₂×M₁₆×C₇₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	D₄	1	4	yes	—
33	C₂²×C₈×C₇₁	Q₈	1	4	yes	—
34	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	#117×C₇₁	C₈	1	4	yes	—
37	#117×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₇₁	C₈	1	2	yes	—
39	C₂×C₂²⋊C₄×C₇₁	C₈	1	2	yes	—
40	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
41	#97×C₇₁	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₇₁	C₂³	1	4	yes	—
44	C₇₁⋊C₁₆	—	2	4	no	—
45	C₇₁⋊C₁₆	—	2	4	no	—
46	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₇₁	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇₁	M₁₆	1	4	yes	—
52	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
54	M₁₆×C₇₁	—	2	8	no	—
55	C₂×C₈×C₇₁	—	2	4	no	—
56	C₂×C₈×C₇₁	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
64	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
67	C₂²⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
68	C₄²×C₇₁	C₂×C₈	1	4	yes	—
69	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
70	C₂³×C₇₁	—	2	4	no	—
71	C₂³×C₇₁	C₄⋊C₈	1	4	yes	—
72	C₂³×C₇₁	C₂²×C₈	1	2	yes	—
73	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	—	2	8	no	—
75	C₂×C₄×C₇₁	C₄⋊C₈	1	8	yes	—
76	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₇₁	C₂²×C₈	1	8	yes	—
80	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
84	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
85	C₈×C₇₁	—	2	4	no	—
86	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
87	C₈×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₇₁	—	2	4	no	—
91	C₂²×C₇₁	—	2	8	no	—
92	C₂²×C₇₁	C₂×C₄⋊C₈	1	4	yes	—
93	C₂²×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
94	C₂²×C₇₁	#504	1	4	yes	—
95	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₇₁	—	2	8	no	—
97	C₄×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
98	C₄×C₇₁	C₂×C₄⋊C₈	1	4	yes	—
99	C₄×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
101	C₂×#2151	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	—	2	4	no	—
103	C₂×C₇₁	#2151	1	4	yes	complemented
104	C₂×C₇₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₇₁	C₂×#504	1	2	yes	—
106	C₂²×M₃₂	—	71	1	no	—
107	C₂³×C₁₆	—	71	1	no	—
108	C₂×#405	D₇₁	1	1	yes	characteristic
109	#2151	—	71	8	no	—
110	C₂×C₄⋊C₁₆	—	71	4	no	—
111	C₇₁	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
112	C₂²×M₁₆	Dic₇₁	1	1	yes	characteristic
113	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
114	C₂×#117	Dic₇₁	1	1	yes	characteristic
115	C₂×M₃₂	—	71	4	no	—
116	C₂×M₃₂	—	142	4	no	—
117	C₂²×C₁₆	—	71	4	no	—
118	C₂²×C₁₆	—	71	2	no	—
119	C₂²×C₁₆	—	71	2	no	—
120	C₂²×C₁₆	—	142	4	no	—
121	#405	Dic₇₁	1	4	yes	—
122	#405	C₂×D₇₁	1	4	yes	—
123	C₂×#97	Dic₇₁	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
125	C₄⋊C₁₆	—	71	16	no	—
126	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂×Dic₇₁	1	4	yes	—
129	C₂²×C₈	Dic₁₄₂	1	4	yes	—
130	C₂²×C₈	D₂₈₄	1	4	yes	—
131	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
132	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
133	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
134	C₂²×C₈	C₂²×D₇₁	1	2	yes	—
135	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
136	#117	C₇₁⋊C₈	1	4	yes	—
137	#117	C₂×Dic₇₁	1	4	yes	—
138	C₂×C₄⋊C₄	C₇₁⋊C₈	1	2	yes	—
139	C₂×C₂²⋊C₄	C₇₁⋊C₈	1	2	yes	—
140	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
141	M₃₂	—	142	8	no	—
142	C₂×C₁₆	—	71	4	no	—
143	C₂×C₁₆	—	71	4	no	—
144	C₂×C₁₆	—	142	4	no	—
145	C₂×C₁₆	—	142	8	no	—
146	C₂×C₁₆	—	142	4	no	—
147	#97	C₂×Dic₇₁	1	8	yes	—
148	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
149	C₄×C₈	C₂²×D₇₁	1	4	yes	—
150	C₂⁴	C₂×C₇₁⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊Dic₇₁	1	4	yes	—
152	C₂²×C₄	C₂×C₇₁⋊C₈	1	4	yes	—
153	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
155	C₂²×C₄	#11765849	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
158	M₁₆	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
161	C₂×C₈	C₄⋊Dic₇₁	1	8	yes	—
162	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
163	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
164	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
165	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
166	C₂×C₈	C₂×D₂₈₄	1	4	yes	—
167	C₂×C₈	C₂×Dic₁₄₂	1	4	yes	—
168	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
169	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
170	C₄⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
171	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
172	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
173	C₄²	C₂²×Dic₇₁	1	4	yes	—
174	C₁₆	—	142	4	no	—
175	C₁₆	—	142	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
178	C₂³	C₂²×C₇₁⋊C₈	1	2	yes	—
179	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	8	yes	—
182	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₇₁	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
185	C₂×C₄	C₂²×C₇₁⋊C₈	1	8	yes	—
186	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
188	C₂×C₄	C₂×#11765849	1	2	yes	—
189	C₂×C₄	C₂×#11765849	1	4	yes	—
190	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	C₂×C₄⋊Dic₇₁	1	2	yes	—
193	C₈	C₂×C₄⋊Dic₇₁	1	4	yes	—
194	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	4	yes	—
199	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	2	yes	—
200	C₂²	#17986416	1	4	yes	—
201	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
202	C₄	—	2	8	no	—
203	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	2	yes	—
204	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	4	yes	—
205	C₄	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
206	C₄	C₂²×#11765849	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#22590099	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
210	C₂	C₂×#17986416	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#405×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₈
prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×#405	D₇₁
hypocenter	C₇₁	C₂×#2151
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #22590099	C₂	direct (N × H)
4. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
12. C₄⋊(C₇₁⋊C₁₆)	C₂²	semidirect
103. C₂×C₇₁	#2151	semidirect
111. C₇₁	C₂×#2151	semidirect
208. C₂	#22590099	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×#14556068	2
C₂³×C₇₁⋊C₁₆	2
#22590099	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂×#405×C₇₁	2
C₂×#2151	71

Subgroups — C2×#22590099 #32150435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590099 #32150435