Subgroups — C₂³×#14556127 #32904142

Order 18176.

12190 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₄₂	C₂	1	2	yes	maximal
2	C₂²×#14556127	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₇₁⋊C₄	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₄₂	—	2	32	no	—
8	C₂³×Dic₁₄₂	C₂²	1	56	yes	—
9	C₂⁴×Dic₇₁	C₂²	1	2	yes	—
10	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₂×#14556127	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊Dic₇₁	C₂²	1	28	yes	—
14	C₂²×Dic₇₁⋊C₄	C₂²	1	56	yes	—
15	C₂²×C₄×Dic₇₁	C₂²	1	28	yes	—
16	C₂⁴×C₄×C₇₁	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₇₁	C₂²	1	28	yes	—
19	C₂²×Dic₁₄₂	—	2	448	no	—
20	C₂²×Dic₁₄₂	C₂³	1	224	yes	—
21	C₂³×Dic₇₁	—	2	16	no	—
22	C₂³×Dic₇₁	D₄	1	16	yes	—
23	C₂³×Dic₇₁	Q₈	1	16	yes	complemented
24	C₂³×Dic₇₁	C₂³	1	28	yes	—
25	C₂³×Dic₇₁	C₂³	1	14	yes	—
26	C₂³×Dic₇₁	C₂³	1	14	yes	—
27	#14556127	C₂³	1	512	yes	direct product factor
28	C₂×C₄⋊Dic₇₁	C₂³	1	112	yes	—
29	C₂×Dic₇₁⋊C₄	C₂³	1	224	yes	—
30	C₂×C₄×Dic₇₁	C₂³	1	112	yes	—
31	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₇₁	—	2	16	no	—
33	C₂³×C₄×C₇₁	Q₈	1	16	yes	complemented
34	C₂³×C₄×C₇₁	C₂³	1	28	yes	—
35	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
36	C₂×C₄⋊C₄×C₇₁	C₂³	1	112	yes	—
37	C₂×Dic₁₄₂	—	2	896	no	—
38	C₂×Dic₁₄₂	C₂⁴	1	128	yes	—
39	C₂²×Dic₇₁	—	2	112	no	—
40	C₂²×Dic₇₁	C₂×D₄	1	112	yes	—
41	C₂²×Dic₇₁	C₂×Q₈	1	112	yes	complemented
42	C₂²×Dic₇₁	C₂⁴	1	56	yes	—
43	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
44	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
45	C₄⋊Dic₇₁	C₂⁴	1	64	yes	—
46	Dic₇₁⋊C₄	C₂⁴	1	128	yes	—
47	C₄×Dic₇₁	C₂⁴	1	64	yes	—
48	C₂⁴×C₇₁	C₂×D₄	1	8	yes	—
49	C₂⁴×C₇₁	C₂×Q₈	1	8	yes	—
50	C₂⁴×C₇₁	C₂×Q₈	1	8	yes	—
51	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
52	C₂²×C₄×C₇₁	—	2	112	no	—
53	C₂²×C₄×C₇₁	C₂×Q₈	1	112	yes	complemented
54	C₂²×C₄×C₇₁	C₂⁴	1	56	yes	—
55	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
56	C₄⋊C₄×C₇₁	C₂⁴	1	64	yes	—
57	C₂³×C₇₁	C₄⋊Q₈	1	64	yes	complemented
58	C₂³×C₇₁	C₂²×D₄	1	28	yes	—
59	C₂³×C₇₁	C₂²×Q₈	1	28	yes	—
60	C₂³×C₇₁	C₂²×Q₈	1	28	yes	—
61	C₂³×C₇₁	C₂⁵	1	7	yes	—
62	C₂×C₄×C₇₁	—	2	112	no	—
63	C₂×C₄×C₇₁	C₂²×Q₈	1	112	yes	complemented
64	C₂×C₄×C₇₁	C₂⁵	1	16	yes	—
65	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
66	C₂×Dic₇₁	—	2	112	no	—
67	C₂×Dic₇₁	C₂²×D₄	1	112	yes	—
68	C₂×Dic₇₁	C₂²×Q₈	1	112	yes	complemented
69	C₂×Dic₇₁	C₂⁵	1	16	yes	—
70	C₂×Dic₇₁	C₂⁵	1	8	yes	—
71	C₂×Dic₇₁	C₂⁵	1	8	yes	—
72	Dic₁₄₂	—	2	256	no	—
73	C₂²×C₇₁	C₂×C₄⋊Q₈	1	112	yes	complemented
74	C₂²×C₇₁	C₂³×D₄	1	14	yes	—
75	C₂²×C₇₁	C₂³×Q₈	1	14	yes	—
76	C₂²×C₇₁	C₂³×Q₈	1	14	yes	—
77	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
78	C₄×C₇₁	—	2	16	no	—
79	C₄×C₇₁	C₂³×Q₈	1	16	yes	complemented
80	Dic₇₁	—	2	16	no	—
81	Dic₇₁	C₂³×D₄	1	16	yes	—
82	Dic₇₁	C₂³×Q₈	1	16	yes	complemented
83	C₂³×C₄⋊Q₈	—	71	1	no	maximal, Sylow
84	C₂×C₇₁	C₂²×C₄⋊Q₈	1	28	yes	complemented
85	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
86	C₂×C₇₁	C₂⁴×Q₈	1	1	yes	characteristic
87	C₂×C₇₁	C₂⁴×Q₈	1	1	yes	characteristic
88	C₂⁴×Q₈	—	71	2	no	—
89	C₂²×C₄⋊Q₈	—	71	56	no	—
90	C₂³×C₄⋊C₄	—	71	2	no	—
91	C₂³×C₄⋊C₄	—	71	1	no	—
92	C₂³×C₄⋊C₄	D₇₁	1	1	yes	characteristic
93	C₂³×C₄²	—	71	1	no	—
94	C₇₁	C₂³×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
95	C₂³×Q₈	—	71	56	no	—
96	C₂³×Q₈	—	142	32	no	—
97	C₂⁴×C₄	—	71	1	no	—
98	C₂⁴×C₄	—	71	1	no	—
99	C₂⁴×C₄	—	71	2	no	—
100	C₂⁴×C₄	C₂×D₇₁	1	2	yes	—
101	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	—	71	448	no	—
103	C₂²×C₄⋊C₄	—	71	56	no	—
104	C₂²×C₄⋊C₄	—	71	28	no	—
105	C₂²×C₄⋊C₄	C₂×D₇₁	1	28	yes	—
106	C₂²×C₄²	—	71	28	no	—
107	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
108	C₂²×Q₈	—	71	224	no	—
109	C₂²×Q₈	—	142	448	no	—
110	C₂³×C₄	—	2	16	no	—
111	C₂³×C₄	—	71	16	no	—
112	C₂³×C₄	—	71	16	no	—
113	C₂³×C₄	—	71	14	no	—
114	C₂³×C₄	—	71	14	no	—
115	C₂³×C₄	—	71	28	no	—
116	C₂³×C₄	—	142	16	no	—
117	C₂³×C₄	Dic₁₄₂	1	16	yes	complemented
118	C₂³×C₄	C₂²×D₇₁	1	28	yes	—
119	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
120	C₄⋊Q₈	—	71	512	no	—
121	C₂×C₄⋊C₄	—	71	224	no	—
122	C₂×C₄⋊C₄	—	71	112	no	—
123	C₂×C₄⋊C₄	C₂²×D₇₁	1	112	yes	—
124	C₂×C₄²	—	71	112	no	—
125	C₂⁴	D₄×D₇₁	1	8	yes	—
126	C₂⁴	Q₈×D₇₁	1	8	yes	—
127	C₂⁴	C₂×Dic₁₄₂	1	8	yes	—
128	C₂⁴	C₂³×D₇₁	1	7	yes	—
129	C₂×Q₈	—	71	128	no	—
130	C₂×Q₈	—	142	896	no	—
131	C₂²×C₄	—	2	112	no	—
132	C₂²×C₄	—	71	112	no	—
133	C₂²×C₄	—	71	112	no	—
134	C₂²×C₄	—	71	28	no	—
135	C₂²×C₄	—	71	28	no	—
136	C₂²×C₄	—	71	56	no	—
137	C₂²×C₄	—	142	112	no	—
138	C₂²×C₄	C₂×Dic₁₄₂	1	112	yes	complemented
139	C₂²×C₄	C₂³×D₇₁	1	56	yes	—
140	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
141	C₄⋊C₄	—	71	128	no	—
142	C₄⋊C₄	—	71	64	no	—
143	C₄⋊C₄	C₂³×D₇₁	1	64	yes	—
144	C₄²	—	71	64	no	—
145	C₂³	#14556127	1	64	yes	direct product factor
146	C₂³	C₂×D₄×D₇₁	1	28	yes	—
147	C₂³	C₂×Q₈×D₇₁	1	28	yes	—
148	C₂³	C₂²×Dic₁₄₂	1	28	yes	—
149	C₂³	C₂⁴×D₇₁	1	7	yes	—
150	Q₈	—	142	256	no	—
151	C₂×C₄	—	2	112	no	—
152	C₂×C₄	—	71	112	no	—
153	C₂×C₄	—	71	112	no	—
154	C₂×C₄	—	71	8	no	—
155	C₂×C₄	—	71	8	no	—
156	C₂×C₄	—	71	16	no	—
157	C₂×C₄	—	142	112	no	—
158	C₂×C₄	C₂²×Dic₁₄₂	1	112	yes	complemented
159	C₂×C₄	C₂⁴×D₇₁	1	16	yes	—
160	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
161	C₂²	C₂×#14556127	1	112	yes	direct product factor
162	C₂²	C₂²×D₄×D₇₁	1	14	yes	—
163	C₂²	C₂²×Q₈×D₇₁	1	14	yes	—
164	C₂²	C₂³×Dic₁₄₂	1	14	yes	—
165	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
166	C₄	—	2	16	no	—
167	C₄	—	71	16	no	—
168	C₄	—	71	16	no	—
169	C₄	—	142	16	no	—
170	C₄	C₂³×Dic₁₄₂	1	16	yes	complemented
171	C₂	C₂²×#14556127	1	28	yes	direct product factor
172	C₂	C₂³×D₄×D₇₁	1	1	yes	characteristic
173	C₂	C₂³×Q₈×D₇₁	1	1	yes	characteristic
174	C₂	C₂⁴×Dic₁₄₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂³×C₄⋊C₄×C₇₁	C₂
socle, prime radical	C₂⁵×C₇₁	C₂³
hypercenter	C₂³×C₄⋊C₄	D₇₁
hypocenter	C₇₁	C₂³×C₄⋊Q₈
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#14556127	C₂	direct (N × H)
12. C₂×#14556127	C₂²	direct (N × H)
23. C₂³×Dic₇₁	Q₈	semidirect
27. #14556127	C₂³	direct (N × H)
33. C₂³×C₄×C₇₁	Q₈	semidirect
41. C₂²×Dic₇₁	C₂×Q₈	semidirect
53. C₂²×C₄×C₇₁	C₂×Q₈	semidirect
57. C₂³×C₇₁	C₄⋊Q₈	semidirect
63. C₂×C₄×C₇₁	C₂²×Q₈	semidirect
68. C₂×Dic₇₁	C₂²×Q₈	semidirect
73. C₂²×C₇₁	C₂×C₄⋊Q₈	semidirect
79. C₄×C₇₁	C₂³×Q₈	semidirect
82. Dic₇₁	C₂³×Q₈	semidirect
84. C₂×C₇₁	C₂²×C₄⋊Q₈	semidirect
94. C₇₁	C₂³×C₄⋊Q₈	semidirect
117. C₂³×C₄	Dic₁₄₂	semidirect
138. C₂²×C₄	C₂×Dic₁₄₂	semidirect
145. C₂³	#14556127	direct (N × H)
158. C₂×C₄	C₂²×Dic₁₄₂	semidirect
161. C₂²	C₂×#14556127	direct (N × H)
170. C₄	C₂³×Dic₁₄₂	semidirect
171. C₂	C₂²×#14556127	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×Dic₁₄₂	2
C₂²×#14556127	2
C₂³×C₄⋊Dic₇₁	2
C₂³×Dic₇₁⋊C₄	2
C₂³×C₄×Dic₇₁	2
C₂³×C₄⋊C₄×C₇₁	2
C₂³×C₄⋊Q₈	71

Subgroups — C23×#14556127 #32904142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556127 #32904142