Subgroups — C₂³×#14556136 #32911213

Order 18176.

11422 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11765865	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556136	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11765848	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₈×D₇₁	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11765865	C₄	1	28	yes	—
8	C₂²×#11765865	C₂²	1	28	yes	—
9	C₂³×Dic₁₄₂	C₄	1	1	yes	characteristic
10	C₂³×D₂₈₄	C₄	1	1	yes	characteristic
11	C₂³×C₂²⋊D₇₁	C₄	1	2	yes	—
12	C₂³×C₄×D₇₁	C₂²	1	2	yes	—
13	C₂×#14556136	C₂²	1	448	yes	direct product factor
14	C₂²×#11765848	C₂²	1	56	yes	complemented
15	C₂³×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂³×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂³×C₇₁⋊C₈	C₂²	1	28	yes	complemented
18	C₂²×C₈×D₇₁	C₂²	1	56	yes	complemented
19	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₇₁	C₂²	1	28	yes	complemented
21	C₂³×C₈×C₇₁	C₂²	1	2	yes	—
22	C₂⁴×D₇₁	C₂×C₄	1	2	yes	—
23	C₂×#11765865	C₂×C₄	1	336	yes	—
24	C₂×#11765865	C₂³	1	112	yes	—
25	C₂²×Dic₁₄₂	C₂×C₄	1	28	yes	—
26	C₂²×D₂₈₄	C₂×C₄	1	28	yes	—
27	C₂²×C₂²⋊D₇₁	C₂×C₄	1	56	yes	—
28	C₂³×Dic₇₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×D₇₁	C₂×C₄	1	28	yes	—
30	C₂²×C₄×D₇₁	C₂³	1	28	yes	—
31	#14556136	C₂³	1	512	yes	direct product factor
32	C₂×#11765848	C₂³	1	224	yes	complemented
33	C₂²×C₇₁⋊C₈	C₂³	1	14	yes	—
34	C₂²×C₇₁⋊C₈	C₂³	1	14	yes	—
35	C₂²×C₇₁⋊C₈	C₂³	1	112	yes	complemented
36	C₂×C₈×D₇₁	C₂³	1	224	yes	complemented
37	C₂⁵×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₁	C₂×C₄	1	14	yes	—
40	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
41	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₇₁	C₂³	1	112	yes	complemented
43	C₂²×C₈×C₇₁	C₂³	1	28	yes	—
44	C₂³×D₇₁	—	2	16	no	—
45	C₂³×D₇₁	C₂²×C₄	1	28	yes	—
46	#11765865	C₂²×C₄	1	448	yes	—
47	#11765865	C₂⁴	1	64	yes	—
48	C₂×Dic₁₄₂	C₂²×C₄	1	112	yes	—
49	C₂×D₂₈₄	C₂²×C₄	1	112	yes	—
50	C₂×C₂²⋊D₇₁	C₂²×C₄	1	224	yes	—
51	C₂²×Dic₇₁	C₂²×C₄	1	28	yes	—
52	C₂×C₄×D₇₁	C₂²×C₄	1	168	yes	—
53	C₂×C₄×D₇₁	C₂⁴	1	56	yes	—
54	#11765848	C₂⁴	1	128	yes	complemented
55	C₂×C₇₁⋊C₈	C₂⁴	1	28	yes	—
56	C₂×C₇₁⋊C₈	C₂⁴	1	28	yes	—
57	C₂×C₇₁⋊C₈	C₂⁴	1	64	yes	complemented
58	C₈×D₇₁	C₂⁴	1	128	yes	complemented
59	C₂⁴×C₇₁	—	2	8	no	—
60	C₂⁴×C₇₁	C₂²×C₄	1	14	yes	—
61	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂²×C₄	1	7	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	14	yes	—
64	C₂²×C₄×C₇₁	C₂²×C₄	1	84	yes	—
65	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
66	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
67	M₁₆×C₇₁	C₂⁴	1	64	yes	complemented
68	C₂×C₈×C₇₁	C₂⁴	1	56	yes	—
69	C₂³×C₇₁	—	2	56	no	—
70	C₂³×C₇₁	#131	1	8	yes	complemented
71	C₂³×C₇₁	C₂³×C₄	1	28	yes	—
72	C₂³×C₇₁	C₂³×C₄	1	7	yes	—
73	C₂²×D₇₁	—	2	112	no	—
74	C₂²×D₇₁	C₂³×C₄	1	56	yes	—
75	C₂×C₄×C₇₁	C₂³×C₄	1	21	yes	—
76	C₂×C₄×C₇₁	C₂³×C₄	1	28	yes	—
77	C₂×C₄×C₇₁	C₂³×C₄	1	56	yes	—
78	C₂×C₄×C₇₁	C₂⁵	1	7	yes	—
79	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
80	C₂²⋊D₇₁	C₂³×C₄	1	128	yes	—
81	C₂×Dic₇₁	C₂³×C₄	1	56	yes	—
82	D₂₈₄	C₂³×C₄	1	64	yes	—
83	C₄×D₇₁	C₂³×C₄	1	112	yes	—
84	C₄×D₇₁	C₂⁵	1	16	yes	—
85	Dic₁₄₂	C₂³×C₄	1	64	yes	—
86	C₈×C₇₁	C₂⁵	1	16	yes	—
87	C₇₁⋊C₈	C₂⁵	1	8	yes	—
88	C₇₁⋊C₈	C₂⁵	1	8	yes	—
89	C₂²×C₇₁	—	2	56	no	—
90	C₂²×C₇₁	C₂×#131	1	28	yes	complemented
91	C₂²×C₇₁	C₂⁴×C₄	1	8	yes	—
92	C₂²×C₇₁	C₂⁴×C₄	1	7	yes	—
93	C₂×D₇₁	—	2	112	no	—
94	C₂×D₇₁	C₂⁴×C₄	1	16	yes	—
95	C₄×C₇₁	C₂⁴×C₄	1	8	yes	—
96	C₄×C₇₁	C₂⁴×C₄	1	7	yes	—
97	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
98	Dic₇₁	C₂⁴×C₄	1	16	yes	—
99	C₂³×#131	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	—	2	8	no	—
101	C₂×C₇₁	C₂²×#131	1	14	yes	complemented
102	C₂×C₇₁	C₂⁵×C₄	1	1	yes	characteristic
103	D₇₁	—	2	16	no	—
104	C₂³×#41	—	71	1	no	—
105	C₂²×#131	—	71	56	no	—
106	C₂³×M₁₆	—	71	2	no	—
107	C₂³×M₁₆	D₇₁	1	1	yes	characteristic, complemented
108	C₂⁴×C₈	—	71	1	no	—
109	C₂⁴×C₈	—	71	2	no	—
110	C₇₁	C₂³×#131	1	1	yes	characteristic, complemented, Sylow
111	C₂²×#41	—	71	28	no	—
112	C₂²×#41	—	71	28	no	—
113	C₂³×Q₈	—	71	1	no	—
114	C₂³×D₄	—	71	2	no	—
115	C₂³×D₄	—	71	1	no	—
116	C₂⁴×C₄	—	71	2	no	—
117	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
118	C₂×#131	—	71	448	no	—
119	C₂²×M₁₆	—	71	56	no	—
120	C₂²×M₁₆	C₂×D₇₁	1	28	yes	complemented
121	C₂³×C₈	—	71	28	no	—
122	C₂³×C₈	—	71	1	no	—
123	C₂³×C₈	—	71	1	no	—
124	C₂³×C₈	—	71	56	no	—
125	C₂³×C₈	C₂×D₇₁	1	2	yes	—
126	C₂⁵	—	71	2	no	—
127	C₂⁵	C₄×D₇₁	1	1	yes	characteristic
128	C₂×#41	—	71	336	no	—
129	C₂×#41	—	71	112	no	—
130	C₂²×Q₈	—	71	28	no	—
131	C₂²×D₄	—	71	56	no	—
132	C₂²×D₄	—	71	28	no	—
133	C₂³×C₄	—	71	2	no	—
134	C₂³×C₄	—	71	28	no	—
135	C₂³×C₄	—	71	28	no	—
136	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
137	C₂³×C₄	C₄×D₇₁	1	14	yes	—
138	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
139	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
140	#131	—	71	512	no	—
141	C₂×M₁₆	—	71	224	no	—
142	C₂×M₁₆	C₂²×D₇₁	1	112	yes	complemented
143	C₂²×C₈	—	71	14	no	—
144	C₂²×C₈	—	71	14	no	—
145	C₂²×C₈	—	71	112	no	—
146	C₂²×C₈	—	71	224	no	—
147	C₂²×C₈	C₂²×D₇₁	1	28	yes	—
148	C₂⁴	—	2	8	no	—
149	C₂⁴	—	71	28	no	—
150	C₂⁴	—	142	16	no	—
151	C₂⁴	C₂×C₄×D₇₁	1	14	yes	—
152	C₂⁴	C₂×C₄×D₇₁	1	1	yes	characteristic
153	#41	—	71	448	no	—
154	#41	—	71	64	no	—
155	C₂×Q₈	—	71	112	no	—
156	C₂×D₄	—	71	224	no	—
157	C₂×D₄	—	71	112	no	—
158	C₂²×C₄	—	71	28	no	—
159	C₂²×C₄	—	71	168	no	—
160	C₂²×C₄	—	71	56	no	—
161	C₂²×C₄	C₂×C₄×D₇₁	1	7	yes	—
162	C₂²×C₄	C₂×C₄×D₇₁	1	14	yes	—
163	C₂²×C₄	C₂×C₄×D₇₁	1	84	yes	—
164	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
165	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
166	M₁₆	—	71	128	no	—
167	M₁₆	C₂³×D₇₁	1	64	yes	complemented
168	C₂×C₈	—	71	28	no	—
169	C₂×C₈	—	71	28	no	—
170	C₂×C₈	—	71	64	no	—
171	C₂×C₈	—	71	128	no	—
172	C₂×C₈	C₂³×D₇₁	1	56	yes	—
173	C₂³	—	2	56	no	—
174	C₂³	—	71	56	no	—
175	C₂³	—	142	112	no	—
176	C₂³	#14556136	1	8	yes	direct product factor
177	C₂³	C₂²×C₄×D₇₁	1	28	yes	—
178	C₂³	C₂²×C₄×D₇₁	1	7	yes	—
179	Q₈	—	71	64	no	—
180	D₄	—	71	128	no	—
181	D₄	—	71	64	no	—
182	C₂×C₄	—	71	56	no	—
183	C₂×C₄	—	71	112	no	—
184	C₂×C₄	—	71	16	no	—
185	C₂×C₄	C₂²×C₄×D₇₁	1	21	yes	—
186	C₂×C₄	C₂²×C₄×D₇₁	1	28	yes	—
187	C₂×C₄	C₂²×C₄×D₇₁	1	56	yes	—
188	C₂×C₄	C₂⁴×D₇₁	1	7	yes	—
189	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
190	C₈	—	71	8	no	—
191	C₈	—	71	8	no	—
192	C₈	C₂⁴×D₇₁	1	16	yes	—
193	C₂²	—	2	56	no	—
194	C₂²	—	71	16	no	—
195	C₂²	—	142	112	no	—
196	C₂²	C₂×#14556136	1	28	yes	direct product factor
197	C₂²	C₂³×C₄×D₇₁	1	8	yes	—
198	C₂²	C₂³×C₄×D₇₁	1	7	yes	—
199	C₄	—	71	16	no	—
200	C₄	C₂³×C₄×D₇₁	1	7	yes	—
201	C₄	C₂³×C₄×D₇₁	1	8	yes	—
202	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
203	C₂	—	2	8	no	—
204	C₂	—	142	16	no	—
205	C₂	C₂²×#14556136	1	14	yes	direct product factor
206	C₂	C₂⁴×C₄×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₁
commutator	C₂×C₇₁	C₂⁵×C₄
Frattini	C₄	C₂⁵×D₇₁
Fitting	C₂³×M₁₆×C₇₁	C₂
socle	C₂⁴×C₇₁	C₂²×C₄
prime radical	C₂³×#11765865	C₂
cosocle	C₄×C₇₁	C₂⁶
hypercenter	C₂³×M₁₆	D₇₁
hypocenter	C₇₁	C₂³×#131
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×#14556136	C₂	direct (N × H)
3. C₂³×#11765848	C₂	semidirect
4. C₂⁴×C₇₁⋊C₈	C₂	semidirect
5. C₂³×C₈×D₇₁	C₂	semidirect
6. C₂³×M₁₆×C₇₁	C₂	semidirect
13. C₂×#14556136	C₂²	direct (N × H)
14. C₂²×#11765848	C₂²	semidirect
17. C₂³×C₇₁⋊C₈	C₂²	semidirect
18. C₂²×C₈×D₇₁	C₂²	semidirect
20. C₂²×M₁₆×C₇₁	C₂²	semidirect
31. #14556136	C₂³	direct (N × H)
32. C₂×#11765848	C₂³	semidirect
35. C₂²×C₇₁⋊C₈	C₂³	semidirect
36. C₂×C₈×D₇₁	C₂³	semidirect
42. C₂×M₁₆×C₇₁	C₂³	semidirect
54. #11765848	C₂⁴	semidirect
57. C₂×C₇₁⋊C₈	C₂⁴	semidirect
58. C₈×D₇₁	C₂⁴	semidirect
67. M₁₆×C₇₁	C₂⁴	semidirect
70. C₂³×C₇₁	#131	semidirect
90. C₂²×C₇₁	C₂×#131	semidirect
101. C₂×C₇₁	C₂²×#131	semidirect
107. C₂³×M₁₆	D₇₁	semidirect
110. C₇₁	C₂³×#131	semidirect
120. C₂²×M₁₆	C₂×D₇₁	semidirect
142. C₂×M₁₆	C₂²×D₇₁	semidirect
167. M₁₆	C₂³×D₇₁	semidirect
176. C₂³	#14556136	direct (N × H)
196. C₂²	C₂×#14556136	direct (N × H)
205. C₂	C₂²×#14556136	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×#11765865	2
C₂²×#14556136	2
C₂³×#11765848	2
C₂⁴×C₇₁⋊C₈	2
C₂³×C₈×D₇₁	2
C₂³×M₁₆×C₇₁	2
C₂³×#131	71

Subgroups — C23×#14556136 #32911213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556136 #32911213