Subgroups — C₂³×#14556150 #32911239

Order 18176.

11422 conjugacy classes of subgroups.

The subgroup lattice has 206 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	2	yes	maximal, complemented
2	C₂²×#14556150	C₂	1	56	yes	maximal, direct product factor
3	C₂³×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11765854	C₂	1	2	yes	maximal, complemented
5	C₂³×D₅₆₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11765865	—	2	28	no	—
8	C₂²×#11765865	C₂²	1	56	yes	complemented
9	C₂³×Dic₁₄₂	C₂²	1	2	yes	—
10	C₂³×D₂₈₄	C₂²	1	2	yes	—
11	C₂³×C₂²⋊D₇₁	—	2	2	no	—
12	C₂³×C₄×D₇₁	—	2	2	no	—
13	C₂×#14556150	C₂²	1	448	yes	direct product factor
14	C₂²×Dic₂₈₄	C₂²	1	28	yes	complemented
15	C₂²×#11765854	C₂²	1	56	yes	complemented
16	C₂²×D₅₆₈	C₂²	1	28	yes	complemented
17	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₇₁	C₂²	1	28	yes	complemented
21	C₂⁴×D₇₁	—	2	2	no	—
22	C₂×#11765865	—	2	336	no	—
23	C₂×#11765865	C₂³	1	224	yes	complemented
24	C₂²×Dic₁₄₂	—	2	14	no	—
25	C₂²×Dic₁₄₂	C₂³	1	28	yes	—
26	C₂²×D₂₈₄	—	2	14	no	—
27	C₂²×D₂₈₄	C₂³	1	28	yes	—
28	C₂²×C₂²⋊D₇₁	—	2	56	no	—
29	C₂³×Dic₇₁	—	2	2	no	—
30	C₂²×C₄×D₇₁	—	2	56	no	—
31	#14556150	C₂³	1	512	yes	direct product factor
32	C₂×Dic₂₈₄	C₂³	1	112	yes	complemented
33	C₂×#11765854	C₂³	1	224	yes	complemented
34	C₂×D₅₆₈	C₂³	1	112	yes	complemented
35	C₂⁵×C₇₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₇₁	D₄	1	14	yes	—
37	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
40	C₂²×C₈×C₇₁	C₂³	1	14	yes	—
41	C₂²×C₈×C₇₁	C₂³	1	14	yes	—
42	C₂²×C₈×C₇₁	C₂³	1	112	yes	complemented
43	C₂³×D₇₁	—	2	28	no	—
44	C₂³×D₇₁	—	4	16	no	—
45	#11765865	—	2	448	no	—
46	#11765865	C₂⁴	1	128	yes	complemented
47	C₂×Dic₁₄₂	—	2	84	no	—
48	C₂×Dic₁₄₂	C₂⁴	1	56	yes	—
49	C₂×D₂₈₄	—	2	84	no	—
50	C₂×D₂₈₄	C₂⁴	1	56	yes	—
51	C₂×C₂²⋊D₇₁	—	2	224	no	—
52	C₂²×Dic₇₁	—	2	28	no	—
53	C₂×C₄×D₇₁	—	2	224	no	—
54	Dic₂₈₄	C₂⁴	1	64	yes	complemented
55	#11765854	C₂⁴	1	128	yes	complemented
56	D₅₆₈	C₂⁴	1	64	yes	complemented
57	C₂⁴×C₇₁	—	2	8	no	—
58	C₂⁴×C₇₁	C₂×D₄	1	14	yes	—
59	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×D₄	1	14	yes	—
61	C₂²×C₄×C₇₁	C₂×D₄	1	7	yes	—
62	C₂²×C₄×C₇₁	C₂×D₄	1	84	yes	—
63	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
64	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
65	C₂×C₈×C₇₁	C₂⁴	1	28	yes	—
66	C₂×C₈×C₇₁	C₂⁴	1	28	yes	—
67	C₂×C₈×C₇₁	C₂⁴	1	64	yes	complemented
68	C₂³×C₇₁	—	2	56	no	—
69	C₂³×C₇₁	#135	1	8	yes	complemented
70	C₂³×C₇₁	C₂²×D₄	1	28	yes	—
71	C₂³×C₇₁	C₂²×D₄	1	7	yes	—
72	C₂²×D₇₁	—	2	56	no	—
73	C₂²×D₇₁	—	4	112	no	—
74	C₂×C₄×C₇₁	C₂²×D₄	1	28	yes	—
75	C₂×C₄×C₇₁	C₂²×D₄	1	21	yes	—
76	C₂×C₄×C₇₁	C₂²×D₄	1	56	yes	—
77	C₂×C₄×C₇₁	C₂⁵	1	7	yes	—
78	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
79	C₂²⋊D₇₁	—	2	128	no	—
80	C₂×Dic₇₁	—	2	56	no	—
81	D₂₈₄	—	2	56	no	—
82	D₂₈₄	C₂⁵	1	16	yes	—
83	C₄×D₇₁	—	2	128	no	—
84	Dic₁₄₂	—	2	56	no	—
85	Dic₁₄₂	C₂⁵	1	16	yes	—
86	C₈×C₇₁	C₂⁵	1	8	yes	—
87	C₈×C₇₁	C₂⁵	1	8	yes	—
88	C₂²×C₇₁	—	2	56	no	—
89	C₂²×C₇₁	C₂×#135	1	28	yes	complemented
90	C₂²×C₇₁	C₂³×D₄	1	8	yes	—
91	C₂²×C₇₁	C₂³×D₄	1	7	yes	—
92	C₂×D₇₁	—	2	16	no	—
93	C₂×D₇₁	—	4	112	no	—
94	C₄×C₇₁	C₂³×D₄	1	7	yes	—
95	C₄×C₇₁	C₂³×D₄	1	8	yes	—
96	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
97	Dic₇₁	—	2	16	no	—
98	C₂³×#135	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	—	2	8	no	—
100	C₂×C₇₁	C₂²×#135	1	14	yes	complemented
101	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
102	D₇₁	—	4	16	no	—
103	C₂³×#41	—	71	2	no	—
104	C₂²×#135	—	71	56	no	—
105	C₂³×Q₁₆	—	71	1	no	—
106	C₂³×SD₁₆	—	71	2	no	—
107	C₂³×D₈	—	71	1	no	—
108	C₂⁴×C₈	D₇₁	1	1	yes	characteristic, complemented
109	C₇₁	C₂³×#135	1	1	yes	characteristic, complemented, Sylow
110	C₂²×#41	—	71	56	no	—
111	C₂²×#41	—	142	28	no	—
112	C₂³×Q₈	—	71	2	no	—
113	C₂³×D₄	—	71	2	no	—
114	C₂³×D₄	—	142	2	no	—
115	C₂⁴×C₄	—	142	2	no	—
116	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
117	C₂×#135	—	71	448	no	—
118	C₂²×Q₁₆	—	71	28	no	—
119	C₂²×SD₁₆	—	71	56	no	—
120	C₂²×D₈	—	71	28	no	—
121	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
123	C₂³×C₈	C₂×D₇₁	1	28	yes	complemented
124	C₂⁵	—	142	2	no	—
125	C₂⁵	D₂₈₄	1	1	yes	characteristic
126	C₂×#41	—	71	224	no	—
127	C₂×#41	—	142	336	no	—
128	C₂²×Q₈	—	71	28	no	—
129	C₂²×Q₈	—	142	14	no	—
130	C₂²×D₄	—	71	28	no	—
131	C₂²×D₄	—	142	56	no	—
132	C₂²×D₄	—	142	14	no	—
133	C₂³×C₄	—	142	2	no	—
134	C₂³×C₄	—	142	56	no	—
135	C₂³×C₄	D₂₈₄	1	14	yes	—
136	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
137	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
138	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
139	#135	—	71	512	no	—
140	C₂×Q₁₆	—	71	112	no	—
141	C₂×SD₁₆	—	71	224	no	—
142	C₂×D₈	—	71	112	no	—
143	C₂²×C₈	C₂²×D₇₁	1	14	yes	—
144	C₂²×C₈	C₂²×D₇₁	1	14	yes	—
145	C₂²×C₈	C₂²×D₇₁	1	112	yes	complemented
146	C₂⁴	—	2	8	no	—
147	C₂⁴	—	284	16	no	—
148	C₂⁴	—	142	28	no	—
149	C₂⁴	C₂×D₂₈₄	1	14	yes	—
150	C₂⁴	C₂×D₂₈₄	1	1	yes	characteristic
151	#41	—	71	128	no	—
152	#41	—	142	448	no	—
153	C₂×Q₈	—	71	56	no	—
154	C₂×Q₈	—	142	84	no	—
155	C₂×D₄	—	71	56	no	—
156	C₂×D₄	—	142	224	no	—
157	C₂×D₄	—	142	84	no	—
158	C₂²×C₄	—	142	28	no	—
159	C₂²×C₄	—	142	224	no	—
160	C₂²×C₄	C₂×D₂₈₄	1	14	yes	—
161	C₂²×C₄	C₂×D₂₈₄	1	7	yes	—
162	C₂²×C₄	C₂×D₂₈₄	1	84	yes	—
163	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
164	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
165	Q₁₆	—	71	64	no	—
166	SD₁₆	—	71	128	no	—
167	D₈	—	71	64	no	—
168	C₂×C₈	C₂³×D₇₁	1	28	yes	—
169	C₂×C₈	C₂³×D₇₁	1	28	yes	—
170	C₂×C₈	C₂³×D₇₁	1	64	yes	complemented
171	C₂³	—	2	56	no	—
172	C₂³	—	284	112	no	—
173	C₂³	—	142	56	no	—
174	C₂³	#14556150	1	8	yes	direct product factor
175	C₂³	C₂²×D₂₈₄	1	28	yes	—
176	C₂³	C₂²×D₂₈₄	1	7	yes	—
177	Q₈	—	71	16	no	—
178	Q₈	—	142	56	no	—
179	D₄	—	71	16	no	—
180	D₄	—	142	128	no	—
181	D₄	—	142	56	no	—
182	C₂×C₄	—	142	56	no	—
183	C₂×C₄	—	142	128	no	—
184	C₂×C₄	C₂²×D₂₈₄	1	28	yes	—
185	C₂×C₄	C₂²×D₂₈₄	1	21	yes	—
186	C₂×C₄	C₂²×D₂₈₄	1	56	yes	—
187	C₂×C₄	C₂⁴×D₇₁	1	7	yes	—
188	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
189	C₈	C₂⁴×D₇₁	1	8	yes	—
190	C₈	C₂⁴×D₇₁	1	8	yes	—
191	C₂²	—	2	56	no	—
192	C₂²	—	284	112	no	—
193	C₂²	—	142	16	no	—
194	C₂²	C₂×#14556150	1	28	yes	direct product factor
195	C₂²	C₂³×D₂₈₄	1	8	yes	—
196	C₂²	C₂³×D₂₈₄	1	7	yes	—
197	C₄	—	142	16	no	—
198	C₄	C₂³×D₂₈₄	1	8	yes	—
199	C₄	C₂³×D₂₈₄	1	7	yes	—
200	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
201	C₂	—	2	8	no	—
202	C₂	—	284	16	no	—
203	C₂	C₂²×#14556150	1	14	yes	direct product factor
204	C₂	C₂⁴×D₂₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₈₄
commutator, cosocle	C₄×C₇₁	C₂⁶
Frattini	C₄	C₂⁵×D₇₁
Fitting	C₂⁴×C₈×C₇₁	C₂
socle	C₂⁴×C₇₁	C₂×D₄
hypercenter	C₂⁴×C₈	D₇₁
hypocenter	C₇₁	C₂³×#135
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11765865	C₂	semidirect
2. C₂²×#14556150	C₂	direct (N × H)
3. C₂³×Dic₂₈₄	C₂	semidirect
4. C₂³×#11765854	C₂	semidirect
5. C₂³×D₅₆₈	C₂	semidirect
6. C₂⁴×C₈×C₇₁	C₂	semidirect
8. C₂²×#11765865	C₂²	semidirect
13. C₂×#14556150	C₂²	direct (N × H)
14. C₂²×Dic₂₈₄	C₂²	semidirect
15. C₂²×#11765854	C₂²	semidirect
16. C₂²×D₅₆₈	C₂²	semidirect
20. C₂³×C₈×C₇₁	C₂²	semidirect
23. C₂×#11765865	C₂³	semidirect
31. #14556150	C₂³	direct (N × H)
32. C₂×Dic₂₈₄	C₂³	semidirect
33. C₂×#11765854	C₂³	semidirect
34. C₂×D₅₆₈	C₂³	semidirect
42. C₂²×C₈×C₇₁	C₂³	semidirect
46. #11765865	C₂⁴	semidirect
54. Dic₂₈₄	C₂⁴	semidirect
55. #11765854	C₂⁴	semidirect
56. D₅₆₈	C₂⁴	semidirect
67. C₂×C₈×C₇₁	C₂⁴	semidirect
69. C₂³×C₇₁	#135	semidirect
89. C₂²×C₇₁	C₂×#135	semidirect
100. C₂×C₇₁	C₂²×#135	semidirect
108. C₂⁴×C₈	D₇₁	semidirect
109. C₇₁	C₂³×#135	semidirect
123. C₂³×C₈	C₂×D₇₁	semidirect
145. C₂²×C₈	C₂²×D₇₁	semidirect
170. C₂×C₈	C₂³×D₇₁	semidirect
174. C₂³	#14556150	direct (N × H)
194. C₂²	C₂×#14556150	direct (N × H)
203. C₂	C₂²×#14556150	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×#14556150	2
C₂³×Dic₂₈₄	2
C₂³×#11765854	2
C₂³×D₅₆₈	2
C₂⁴×C₈×C₇₁	2
C₂³×#135	71

Subgroups — C23×#14556150 #32911239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556150 #32911239