Subgroups — C₂³×#14556119 #32904119

Order 18176.

8414 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556119	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₇₁	C₂	1	2	yes	maximal
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	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₇₁	C₂²	1	2	yes	—
10	C₂×#14556119	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊Dic₇₁	C₂²	1	56	yes	—
12	C₂²×C₄⋊Dic₇₁	C₂²	1	28	yes	—
13	C₂²×Dic₇₁⋊C₄	C₂²	1	56	yes	—
14	C₂²×C₄×Dic₇₁	C₂²	1	28	yes	—
15	C₂⁴×C₄×C₇₁	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₇₁	C₂²	1	28	yes	—
18	C₂³×Dic₇₁	—	2	8	no	—
19	C₂³×Dic₇₁	—	2	8	no	—
20	C₂³×Dic₇₁	—	2	16	no	—
21	C₂³×Dic₇₁	C₂³	1	14	yes	—
22	C₂³×Dic₇₁	C₂³	1	14	yes	—
23	C₂³×Dic₇₁	C₂³	1	28	yes	—
24	#14556119	C₂³	1	512	yes	direct product factor
25	C₂×C₄⋊Dic₇₁	C₂³	1	224	yes	—
26	C₂×C₄⋊Dic₇₁	C₂³	1	112	yes	—
27	C₂×Dic₇₁⋊C₄	C₂³	1	224	yes	—
28	C₂×C₄×Dic₇₁	C₂³	1	112	yes	—
29	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₇₁	—	2	16	no	—
31	C₂³×C₄×C₇₁	Q₈	1	16	yes	—
32	C₂³×C₄×C₇₁	C₂³	1	28	yes	—
33	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
34	C₂×C₄⋊C₄×C₇₁	C₂³	1	112	yes	—
35	C₂²×Dic₇₁	—	2	56	no	—
36	C₂²×Dic₇₁	—	2	56	no	—
37	C₂²×Dic₇₁	—	2	112	no	—
38	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
39	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
40	C₂²×Dic₇₁	C₂⁴	1	56	yes	—
41	C₄⋊Dic₇₁	C₂⁴	1	128	yes	—
42	C₄⋊Dic₇₁	C₂⁴	1	64	yes	—
43	Dic₇₁⋊C₄	C₂⁴	1	128	yes	—
44	C₄×Dic₇₁	C₂⁴	1	64	yes	—
45	C₂⁴×C₇₁	C₂×Q₈	1	8	yes	—
46	C₂⁴×C₇₁	#41	1	8	yes	—
47	C₂⁴×C₇₁	#41	1	8	yes	—
48	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
49	C₂²×C₄×C₇₁	—	2	112	no	—
50	C₂²×C₄×C₇₁	C₂×Q₈	1	112	yes	—
51	C₂²×C₄×C₇₁	C₂⁴	1	56	yes	—
52	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
53	C₄⋊C₄×C₇₁	C₂⁴	1	64	yes	—
54	C₂³×C₇₁	#125	1	64	yes	complemented
55	C₂³×C₇₁	C₂²×Q₈	1	28	yes	—
56	C₂³×C₇₁	C₂×#41	1	28	yes	—
57	C₂³×C₇₁	C₂×#41	1	28	yes	—
58	C₂³×C₇₁	C₂⁵	1	7	yes	—
59	C₂×C₄×C₇₁	—	2	112	no	—
60	C₂×C₄×C₇₁	C₂²×Q₈	1	112	yes	—
61	C₂×C₄×C₇₁	C₂⁵	1	16	yes	—
62	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
63	C₂×Dic₇₁	—	2	56	no	—
64	C₂×Dic₇₁	—	2	56	no	—
65	C₂×Dic₇₁	—	2	112	no	—
66	C₂×Dic₇₁	C₂⁵	1	8	yes	—
67	C₂×Dic₇₁	C₂⁵	1	8	yes	—
68	C₂×Dic₇₁	C₂⁵	1	16	yes	—
69	C₂²×C₇₁	C₂×#125	1	112	yes	complemented
70	C₂²×C₇₁	C₂³×Q₈	1	14	yes	—
71	C₂²×C₇₁	C₂²×#41	1	14	yes	—
72	C₂²×C₇₁	C₂²×#41	1	14	yes	—
73	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
74	C₄×C₇₁	—	2	16	no	—
75	C₄×C₇₁	C₂³×Q₈	1	16	yes	—
76	Dic₇₁	—	2	8	no	—
77	Dic₇₁	—	2	8	no	—
78	Dic₇₁	—	2	16	no	—
79	C₂³×#125	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	C₂²×#125	1	28	yes	complemented
81	C₂×C₇₁	C₂⁴×Q₈	1	1	yes	characteristic
82	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
83	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
84	C₂²×#125	—	71	56	no	—
85	C₂³×C₄⋊C₄	—	71	2	no	—
86	C₂³×C₄⋊C₄	—	71	2	no	—
87	C₂³×C₄⋊C₄	—	71	1	no	—
88	C₂³×C₄⋊C₄	D₇₁	1	1	yes	characteristic
89	C₂³×C₄²	—	71	1	no	—
90	C₇₁	C₂³×#125	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴×C₄	—	71	1	no	—
92	C₂⁴×C₄	—	71	1	no	—
93	C₂⁴×C₄	—	71	2	no	—
94	C₂⁴×C₄	C₂×D₇₁	1	2	yes	—
95	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
96	C₂×#125	—	71	448	no	—
97	C₂²×C₄⋊C₄	—	71	56	no	—
98	C₂²×C₄⋊C₄	—	71	28	no	—
99	C₂²×C₄⋊C₄	—	71	56	no	—
100	C₂²×C₄⋊C₄	C₂×D₇₁	1	28	yes	—
101	C₂²×C₄²	—	71	28	no	—
102	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
103	C₂³×C₄	—	2	16	no	—
104	C₂³×C₄	—	71	28	no	—
105	C₂³×C₄	—	71	14	no	—
106	C₂³×C₄	—	71	14	no	—
107	C₂³×C₄	—	142	8	no	—
108	C₂³×C₄	—	142	8	no	—
109	C₂³×C₄	—	142	16	no	—
110	C₂³×C₄	Dic₁₄₂	1	16	yes	—
111	C₂³×C₄	C₂²×D₇₁	1	28	yes	—
112	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
113	#125	—	71	512	no	—
114	C₂×C₄⋊C₄	—	71	224	no	—
115	C₂×C₄⋊C₄	—	71	224	no	—
116	C₂×C₄⋊C₄	—	71	112	no	—
117	C₂×C₄⋊C₄	C₂²×D₇₁	1	112	yes	—
118	C₂×C₄²	—	71	112	no	—
119	C₂⁴	C₂×Dic₁₄₂	1	8	yes	—
120	C₂⁴	#11765864	1	8	yes	—
121	C₂⁴	#11765866	1	8	yes	—
122	C₂⁴	C₂³×D₇₁	1	7	yes	—
123	C₂²×C₄	—	2	112	no	—
124	C₂²×C₄	—	71	28	no	—
125	C₂²×C₄	—	71	28	no	—
126	C₂²×C₄	—	71	56	no	—
127	C₂²×C₄	—	142	56	no	—
128	C₂²×C₄	—	142	112	no	—
129	C₂²×C₄	—	142	56	no	—
130	C₂²×C₄	C₂×Dic₁₄₂	1	112	yes	—
131	C₂²×C₄	C₂³×D₇₁	1	56	yes	—
132	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
133	C₄⋊C₄	—	71	128	no	—
134	C₄⋊C₄	—	71	128	no	—
135	C₄⋊C₄	—	71	64	no	—
136	C₄⋊C₄	C₂³×D₇₁	1	64	yes	—
137	C₄²	—	71	64	no	—
138	C₂³	#14556119	1	64	yes	direct product factor
139	C₂³	C₂²×Dic₁₄₂	1	28	yes	—
140	C₂³	C₂×#11765864	1	28	yes	—
141	C₂³	C₂×#11765866	1	28	yes	—
142	C₂³	C₂⁴×D₇₁	1	7	yes	—
143	C₂×C₄	—	2	112	no	—
144	C₂×C₄	—	71	8	no	—
145	C₂×C₄	—	71	8	no	—
146	C₂×C₄	—	71	16	no	—
147	C₂×C₄	—	142	56	no	—
148	C₂×C₄	—	142	56	no	—
149	C₂×C₄	—	142	112	no	—
150	C₂×C₄	C₂²×Dic₁₄₂	1	112	yes	—
151	C₂×C₄	C₂⁴×D₇₁	1	16	yes	—
152	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
153	C₂²	C₂×#14556119	1	112	yes	direct product factor
154	C₂²	C₂³×Dic₁₄₂	1	14	yes	—
155	C₂²	C₂²×#11765864	1	14	yes	—
156	C₂²	C₂²×#11765866	1	14	yes	—
157	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
158	C₄	—	2	16	no	—
159	C₄	—	142	8	no	—
160	C₄	—	142	8	no	—
161	C₄	—	142	16	no	—
162	C₄	C₂³×Dic₁₄₂	1	16	yes	—
163	C₂	C₂²×#14556119	1	28	yes	direct product factor
164	C₂	C₂⁴×Dic₁₄₂	1	1	yes	characteristic
165	C₂	C₂³×#11765864	1	1	yes	characteristic
166	C₂	C₂³×#11765866	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×#125
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
1. C₂²×#14556119	C₂	direct (N × H)
10. C₂×#14556119	C₂²	direct (N × H)
24. #14556119	C₂³	direct (N × H)
54. C₂³×C₇₁	#125	semidirect
69. C₂²×C₇₁	C₂×#125	semidirect
80. C₂×C₇₁	C₂²×#125	semidirect
90. C₇₁	C₂³×#125	semidirect
138. C₂³	#14556119	direct (N × H)
153. C₂²	C₂×#14556119	direct (N × H)
163. C₂	C₂²×#14556119	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×#14556119	2
C₂³×C₄⋊Dic₇₁	2
C₂³×Dic₇₁⋊C₄	2
C₂³×C₄×Dic₇₁	2
C₂³×C₄⋊C₄×C₇₁	2
C₂³×#125	71

Subgroups — C23×#14556119 #32904119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556119 #32904119