Subgroups — C₂²×#17985754 #31994803

Order 18176.

3086 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×#17985754	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄²×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₇₁⋊C₈	—	2	8	no	—
5	C₂³×C₇₁⋊C₈	—	2	8	no	—
6	C₂³×C₇₁⋊C₈	C₄	1	16	yes	complemented
7	C₂³×C₇₁⋊C₈	C₂²	1	12	yes	—
8	#17985754	C₂²	1	16	yes	direct product factor
9	C₂⁴×C₄×C₇₁	C₄	1	2	yes	—
10	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₇₁	—	2	8	no	—
12	C₂²×C₄²×C₇₁	C₄	1	6	yes	—
13	C₂²×C₄²×C₇₁	C₂²	1	6	yes	—
14	C₂²×C₇₁⋊C₈	—	2	64	no	—
15	C₂²×C₇₁⋊C₈	—	2	24	no	—
16	C₂²×C₇₁⋊C₈	—	2	24	no	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	48	yes	complemented
18	C₂²×C₇₁⋊C₈	C₂³	1	8	yes	—
19	C₂⁵×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₇₁	—	2	8	no	—
21	C₂³×C₄×C₇₁	—	2	8	no	—
22	C₂³×C₄×C₇₁	C₈	1	16	yes	complemented
23	C₂³×C₄×C₇₁	C₂×C₄	1	12	yes	—
24	C₂³×C₄×C₇₁	C₂×C₄	1	3	yes	—
25	C₂³×C₄×C₇₁	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₇₁	D₄	1	4	yes	—
27	C₂³×C₄×C₇₁	D₄	1	8	yes	—
28	C₂³×C₄×C₇₁	Q₈	1	4	yes	—
29	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
30	C₂×C₄²×C₇₁	—	2	24	no	—
31	C₂×C₄²×C₇₁	—	2	24	no	—
32	C₂×C₄²×C₇₁	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₇₁	C₂³	1	4	yes	—
34	C₂×C₇₁⋊C₈	—	2	96	no	—
35	C₂×C₇₁⋊C₈	—	2	8	no	—
36	C₂×C₇₁⋊C₈	—	2	8	no	—
37	C₂×C₇₁⋊C₈	C₂²×C₄	1	16	yes	complemented
38	C₂⁴×C₇₁	C₄²	1	4	yes	—
39	C₂⁴×C₇₁	C₂²⋊C₄	1	4	yes	—
40	C₂⁴×C₇₁	C₄⋊C₄	1	4	yes	—
41	C₂⁴×C₇₁	C₂×C₈	1	8	yes	—
42	C₂⁴×C₇₁	M₁₆	1	8	yes	—
43	C₂⁴×C₇₁	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₇₁	—	2	64	no	—
45	C₂²×C₄×C₇₁	—	2	24	no	—
46	C₂²×C₄×C₇₁	—	2	24	no	—
47	C₂²×C₄×C₇₁	C₄²	1	12	yes	—
48	C₂²×C₄×C₇₁	C₂²⋊C₄	1	32	yes	—
49	C₂²×C₄×C₇₁	C₂²⋊C₄	1	12	yes	—
50	C₂²×C₄×C₇₁	C₄⋊C₄	1	32	yes	—
51	C₂²×C₄×C₇₁	C₄⋊C₄	1	12	yes	—
52	C₂²×C₄×C₇₁	C₂×C₈	1	48	yes	complemented
53	C₂²×C₄×C₇₁	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₇₁	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₇₁	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₇₁	C₂×D₄	1	6	yes	—
58	C₂²×C₄×C₇₁	C₂×Q₈	1	6	yes	—
59	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
60	C₄²×C₇₁	—	2	24	no	—
61	C₄²×C₇₁	—	2	8	no	—
62	C₂³×C₇₁	#95	1	16	yes	—
63	C₂³×C₇₁	C₄×C₈	1	16	yes	—
64	C₂³×C₇₁	#97	1	16	yes	—
65	C₂³×C₇₁	C₂²⋊C₈	1	32	yes	—
66	C₂³×C₇₁	C₄⋊C₈	1	32	yes	—
67	C₂³×C₇₁	C₂×C₄²	1	6	yes	—
68	C₂³×C₇₁	C₂×C₂²⋊C₄	1	6	yes	—
69	C₂³×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
70	C₂³×C₇₁	C₂²×C₈	1	12	yes	—
71	C₂³×C₇₁	C₂×M₁₆	1	12	yes	—
72	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	—	2	8	no	—
74	C₂×C₄×C₇₁	—	2	8	no	—
75	C₂×C₄×C₇₁	—	2	96	no	—
76	C₂×C₄×C₇₁	#95	1	48	yes	—
77	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	24	yes	—
79	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	24	yes	—
81	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₇₁	C₂²×C₈	1	16	yes	complemented
83	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
85	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
87	C₇₁⋊C₈	—	2	16	no	—
88	C₂²×C₇₁	#336	1	64	yes	complemented
89	C₂²×C₇₁	C₂×#95	1	12	yes	—
90	C₂²×C₇₁	C₂×C₄×C₈	1	12	yes	—
91	C₂²×C₇₁	C₂×#97	1	12	yes	—
92	C₂²×C₇₁	C₂×C₂²⋊C₈	1	24	yes	—
93	C₂²×C₇₁	C₂×C₄⋊C₈	1	24	yes	—
94	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
95	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₇₁	C₂³×C₈	1	2	yes	—
98	C₂²×C₇₁	C₂²×M₁₆	1	2	yes	—
99	C₄×C₇₁	—	2	16	no	—
100	C₄×C₇₁	C₂×#95	1	12	yes	—
101	C₄×C₇₁	C₂²×C₂²⋊C₄	1	2	yes	—
102	C₄×C₇₁	C₂²×C₄⋊C₄	1	2	yes	—
103	C₂²×#336	—	71	1	no	maximal, Sylow
104	C₂×C₇₁	C₂×#336	1	24	yes	complemented
105	C₂×C₇₁	C₂²×#95	1	1	yes	characteristic
106	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
107	C₂×C₇₁	C₂²×#97	1	1	yes	characteristic
108	C₂×C₇₁	C₂²×C₂²⋊C₈	1	2	yes	—
109	C₂×C₇₁	C₂²×C₄⋊C₈	1	2	yes	—
110	C₂⁴×C₈	—	71	2	no	—
111	C₂³×C₄²	D₇₁	1	1	yes	characteristic
112	C₂×#336	—	71	12	no	—
113	C₇₁	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴×C₄	Dic₇₁	1	2	yes	—
115	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
116	C₂³×C₈	—	71	16	no	—
117	C₂³×C₈	—	71	12	no	—
118	C₂³×C₈	—	142	8	no	—
119	C₂³×C₈	—	142	8	no	—
120	C₂²×C₄²	—	2	8	no	—
121	C₂²×C₄²	Dic₇₁	1	6	yes	—
122	C₂²×C₄²	C₂×D₇₁	1	6	yes	—
123	#336	—	71	16	no	—
124	C₂⁵	C₂×Dic₇₁	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	—	2	8	no	—
127	C₂³×C₄	C₇₁⋊C₈	1	16	yes	complemented
128	C₂³×C₄	Dic₁₄₂	1	4	yes	—
129	C₂³×C₄	C₄×D₇₁	1	8	yes	—
130	C₂³×C₄	D₂₈₄	1	4	yes	—
131	C₂³×C₄	C₂×Dic₇₁	1	12	yes	—
132	C₂³×C₄	C₂×Dic₇₁	1	3	yes	—
133	C₂³×C₄	C₂²⋊D₇₁	1	8	yes	—
134	C₂³×C₄	C₂²×D₇₁	1	3	yes	—
135	C₂²×C₈	—	71	48	no	—
136	C₂²×C₈	—	71	8	no	—
137	C₂²×C₈	—	142	64	no	—
138	C₂²×C₈	—	142	24	no	—
139	C₂²×C₈	—	142	24	no	—
140	C₂×C₄²	—	2	24	no	—
141	C₂×C₄²	—	2	24	no	—
142	C₂×C₄²	C₂×Dic₇₁	1	12	yes	—
143	C₂×C₄²	C₂²×D₇₁	1	4	yes	—
144	C₂⁴	C₄×Dic₇₁	1	4	yes	—
145	C₂⁴	C₂²⋊Dic₇₁	1	4	yes	—
146	C₂⁴	C₄⋊Dic₇₁	1	4	yes	—
147	C₂⁴	C₂×C₇₁⋊C₈	1	8	yes	—
148	C₂⁴	#11765849	1	8	yes	—
149	C₂⁴	C₂²×Dic₇₁	1	3	yes	—
150	C₂²×C₄	—	2	64	no	—
151	C₂²×C₄	—	2	24	no	—
152	C₂²×C₄	—	2	24	no	—
153	C₂²×C₄	C₄×Dic₇₁	1	12	yes	—
154	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	32	yes	—
155	C₂²×C₄	C₂²⋊Dic₇₁	1	12	yes	—
156	C₂²×C₄	Dic₇₁⋊C₄	1	32	yes	—
157	C₂²×C₄	C₄⋊Dic₇₁	1	12	yes	—
158	C₂²×C₄	C₂×C₇₁⋊C₈	1	48	yes	complemented
159	C₂²×C₄	C₂×C₄×D₇₁	1	12	yes	—
160	C₂²×C₄	C₂²×Dic₇₁	1	8	yes	—
161	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₇₁	1	12	yes	—
163	C₂²×C₄	C₂×D₂₈₄	1	6	yes	—
164	C₂²×C₄	C₂×Dic₁₄₂	1	6	yes	—
165	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
166	C₂×C₈	—	71	16	no	—
167	C₂×C₈	—	142	8	no	—
168	C₂×C₈	—	142	8	no	—
169	C₂×C₈	—	142	96	no	—
170	C₄²	—	2	8	no	—
171	C₄²	—	2	24	no	—
172	C₂³	C₇₁⋊#95	1	16	yes	—
173	C₂³	C₄×C₇₁⋊C₈	1	16	yes	—
174	C₂³	#14556039	1	16	yes	—
175	C₂³	C₂²⋊(C₇₁⋊C₈)	1	32	yes	—
176	C₂³	C₄⋊(C₇₁⋊C₈)	1	32	yes	—
177	C₂³	C₂×C₄×Dic₇₁	1	6	yes	—
178	C₂³	C₂×C₂²⋊Dic₇₁	1	6	yes	—
179	C₂³	C₂×C₄⋊Dic₇₁	1	6	yes	—
180	C₂³	C₂²×C₇₁⋊C₈	1	12	yes	—
181	C₂³	C₂×#11765849	1	12	yes	—
182	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	96	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	C₇₁⋊#95	1	48	yes	—
187	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
188	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	24	yes	—
189	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	6	yes	—
190	C₂×C₄	C₂×Dic₇₁⋊C₄	1	24	yes	—
191	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
192	C₂×C₄	C₂²×C₇₁⋊C₈	1	16	yes	complemented
193	C₂×C₄	C₂²×C₄×D₇₁	1	2	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₇₁	1	2	yes	—
195	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
197	C₈	—	142	16	no	—
198	C₂²	#17985754	1	64	yes	direct product factor
199	C₂²	C₂×C₇₁⋊#95	1	12	yes	—
200	C₂²	C₂×C₄×C₇₁⋊C₈	1	12	yes	—
201	C₂²	C₂×#14556039	1	12	yes	—
202	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	24	yes	—
203	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	24	yes	—
204	C₂²	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
205	C₂²	C₂²×C₂²⋊Dic₇₁	1	1	yes	characteristic
206	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
207	C₂²	C₂³×C₇₁⋊C₈	1	2	yes	—
208	C₂²	C₂²×#11765849	1	2	yes	—
209	C₄	—	2	16	no	—
210	C₄	C₂×C₇₁⋊#95	1	12	yes	—
211	C₄	C₂²×(C₂×D₇₁)⋊C₄	1	2	yes	—
212	C₄	C₂²×Dic₇₁⋊C₄	1	2	yes	—
213	C₂	C₂×#17985754	1	24	yes	direct product factor
214	C₂	C₂²×C₇₁⋊#95	1	1	yes	characteristic
215	C₂	C₂²×C₄×C₇₁⋊C₈	1	1	yes	characteristic
216	C₂	C₂²×#14556039	1	1	yes	characteristic
217	C₂	C₂²×C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
218	C₂	C₂²×C₄⋊(C₇₁⋊C₈)	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₇₁
commutator	C₂×C₇₁	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂³×C₄²×C₇₁	C₂
socle, prime radical	C₂⁵×C₇₁	C₂×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂³×C₄²	D₇₁
hypocenter	C₇₁	C₂²×#336
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#17985754	C₂	direct (N × H)
6. C₂³×C₇₁⋊C₈	C₄	semidirect
8. #17985754	C₂²	direct (N × H)
17. C₂²×C₇₁⋊C₈	C₂×C₄	semidirect
22. C₂³×C₄×C₇₁	C₈	semidirect
37. C₂×C₇₁⋊C₈	C₂²×C₄	semidirect
52. C₂²×C₄×C₇₁	C₂×C₈	semidirect
82. C₂×C₄×C₇₁	C₂²×C₈	semidirect
88. C₂²×C₇₁	#336	semidirect
104. C₂×C₇₁	C₂×#336	semidirect
113. C₇₁	C₂²×#336	semidirect
127. C₂³×C₄	C₇₁⋊C₈	semidirect
158. C₂²×C₄	C₂×C₇₁⋊C₈	semidirect
192. C₂×C₄	C₂²×C₇₁⋊C₈	semidirect
198. C₂²	#17985754	direct (N × H)
213. C₂	C₂×#17985754	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇₁⋊C₈	2
C₂×#17985754	2
C₂³×C₄²×C₇₁	2
C₂²×#336	71

Subgroups — C22×#17985754 #31994803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17985754 #31994803