Subgroups — C₂³×#14556059 #32861093

Order 18176.

3614 conjugacy classes of subgroups.

The subgroup lattice has 160 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	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14556059	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₇₁	—	2	1	no	—
6	C₂³×C₇₁⋊C₈	C₄	1	16	yes	complemented
7	C₂³×C₇₁⋊C₈	C₂²	1	14	yes	—
8	C₂²×C₄⋊Dic₇₁	—	2	7	no	—
9	C₂²×C₄⋊Dic₇₁	C₂²	1	14	yes	—
10	C₂×#14556059	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₄×C₇₁	—	2	1	no	—
12	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₇₁	—	2	7	no	—
14	C₂²×C₄⋊C₄×C₇₁	C₂²	1	14	yes	—
15	C₂³×Dic₇₁	—	2	14	no	—
16	C₂³×Dic₇₁	—	4	8	no	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	112	yes	complemented
18	C₂²×C₇₁⋊C₈	C₂³	1	28	yes	—
19	C₂×C₄⋊Dic₇₁	—	2	42	no	—
20	C₂×C₄⋊Dic₇₁	C₂³	1	28	yes	—
21	#14556059	C₂³	1	64	yes	direct product factor
22	C₂⁵×C₇₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₁	—	2	14	no	—
24	C₂³×C₄×C₇₁	—	4	8	no	—
25	C₂³×C₄×C₇₁	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₇₁	D₄	1	7	yes	—
27	C₂³×C₄×C₇₁	Q₈	1	8	yes	—
28	C₂³×C₄×C₇₁	C₂³	1	7	yes	—
29	C₂×C₄⋊C₄×C₇₁	—	2	42	no	—
30	C₂×C₄⋊C₄×C₇₁	C₂³	1	28	yes	—
31	C₂²×Dic₇₁	—	2	28	no	—
32	C₂²×Dic₇₁	—	4	56	no	—
33	C₂×C₇₁⋊C₈	C₂²×C₄	1	112	yes	complemented
34	C₂×C₇₁⋊C₈	C₂⁴	1	8	yes	—
35	C₄⋊Dic₇₁	—	2	28	no	—
36	C₄⋊Dic₇₁	C₂⁴	1	8	yes	—
37	C₂⁴×C₇₁	C₄⋊C₄	1	8	yes	—
38	C₂⁴×C₇₁	SD₁₆	1	8	yes	—
39	C₂⁴×C₇₁	SD₁₆	1	8	yes	—
40	C₂⁴×C₇₁	C₂×D₄	1	7	yes	—
41	C₂²×C₄×C₇₁	—	2	28	no	—
42	C₂²×C₄×C₇₁	—	4	56	no	—
43	C₂²×C₄×C₇₁	C₄⋊C₄	1	56	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	28	yes	—
45	C₂²×C₄×C₇₁	C₂×D₄	1	21	yes	—
46	C₂²×C₄×C₇₁	C₂×Q₈	1	28	yes	—
47	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
48	C₄⋊C₄×C₇₁	—	2	28	no	—
49	C₄⋊C₄×C₇₁	C₂⁴	1	8	yes	—
50	C₂³×C₇₁	#106	1	64	yes	complemented
51	C₂³×C₇₁	C₂×C₄⋊C₄	1	28	yes	—
52	C₂³×C₇₁	C₂×SD₁₆	1	28	yes	—
53	C₂³×C₇₁	C₂×SD₁₆	1	28	yes	—
54	C₂³×C₇₁	C₂²×D₄	1	7	yes	—
55	C₂×C₄×C₇₁	—	2	8	no	—
56	C₂×C₄×C₇₁	—	4	56	no	—
57	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	84	yes	—
58	C₂×C₄×C₇₁	C₂³×C₄	1	14	yes	—
59	C₂×C₄×C₇₁	C₂²×D₄	1	7	yes	—
60	C₂×C₄×C₇₁	C₂²×Q₈	1	14	yes	—
61	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
62	C₂×Dic₇₁	—	2	8	no	—
63	C₂×Dic₇₁	—	4	56	no	—
64	C₇₁⋊C₈	C₂³×C₄	1	16	yes	complemented
65	C₂²×C₇₁	C₂×#106	1	112	yes	complemented
66	C₂²×C₇₁	C₂²×C₄⋊C₄	1	14	yes	—
67	C₂²×C₇₁	C₂²×SD₁₆	1	14	yes	—
68	C₂²×C₇₁	C₂²×SD₁₆	1	14	yes	—
69	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
70	C₄×C₇₁	—	4	8	no	—
71	C₄×C₇₁	C₂²×C₄⋊C₄	1	14	yes	—
72	C₄×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
73	C₄×C₇₁	C₂³×Q₈	1	1	yes	characteristic
74	Dic₇₁	—	4	8	no	—
75	C₂³×#106	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	C₂²×#106	1	28	yes	complemented
77	C₂×C₇₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₇₁	C₂³×SD₁₆	1	1	yes	characteristic
79	C₂×C₇₁	C₂³×SD₁₆	1	1	yes	characteristic
80	C₂⁴×C₈	—	71	1	no	—
81	C₂³×C₄⋊C₄	—	71	1	no	—
82	C₂³×C₄⋊C₄	D₇₁	1	1	yes	characteristic
83	C₂²×#106	—	71	28	no	—
84	C₇₁	C₂³×#106	1	1	yes	characteristic, complemented, Sylow
85	C₂⁴×C₄	—	2	1	no	—
86	C₂⁴×C₄	—	142	1	no	—
87	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
88	C₂³×C₈	—	71	16	no	—
89	C₂³×C₈	—	71	14	no	—
90	C₂²×C₄⋊C₄	—	2	7	no	—
91	C₂²×C₄⋊C₄	—	71	14	no	—
92	C₂²×C₄⋊C₄	—	142	7	no	—
93	C₂²×C₄⋊C₄	C₂×D₇₁	1	14	yes	—
94	C₂×#106	—	71	112	no	—
95	C₂⁵	C₂²⋊D₇₁	1	1	yes	characteristic
96	C₂³×C₄	—	2	14	no	—
97	C₂³×C₄	—	4	8	no	—
98	C₂³×C₄	—	284	8	no	—
99	C₂³×C₄	—	142	14	no	—
100	C₂³×C₄	Dic₁₄₂	1	8	yes	—
101	C₂³×C₄	C₄×D₇₁	1	8	yes	—
102	C₂³×C₄	C₂²⋊D₇₁	1	7	yes	—
103	C₂³×C₄	C₂²×D₇₁	1	7	yes	—
104	C₂²×C₈	—	71	112	no	—
105	C₂²×C₈	—	71	28	no	—
106	C₂×C₄⋊C₄	—	2	42	no	—
107	C₂×C₄⋊C₄	—	71	28	no	—
108	C₂×C₄⋊C₄	—	142	42	no	—
109	C₂×C₄⋊C₄	C₂²×D₇₁	1	28	yes	—
110	#106	—	71	64	no	—
111	C₂⁴	Dic₇₁⋊C₄	1	8	yes	—
112	C₂⁴	#11765852	1	8	yes	—
113	C₂⁴	#11765853	1	8	yes	—
114	C₂⁴	C₂×C₂²⋊D₇₁	1	7	yes	—
115	C₂²×C₄	—	2	28	no	—
116	C₂²×C₄	—	4	56	no	—
117	C₂²×C₄	—	284	56	no	—
118	C₂²×C₄	—	142	28	no	—
119	C₂²×C₄	Dic₇₁⋊C₄	1	56	yes	—
120	C₂²×C₄	C₂×C₄×D₇₁	1	28	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₇₁	1	21	yes	—
122	C₂²×C₄	C₂×Dic₁₄₂	1	28	yes	—
123	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
124	C₂×C₈	—	71	112	no	—
125	C₂×C₈	—	71	8	no	—
126	C₄⋊C₄	—	2	28	no	—
127	C₄⋊C₄	—	71	8	no	—
128	C₄⋊C₄	—	142	28	no	—
129	C₄⋊C₄	C₂³×D₇₁	1	8	yes	—
130	C₂³	#14556059	1	64	yes	direct product factor
131	C₂³	C₂×Dic₇₁⋊C₄	1	28	yes	—
132	C₂³	C₂×#11765852	1	28	yes	—
133	C₂³	C₂×#11765853	1	28	yes	—
134	C₂³	C₂²×C₂²⋊D₇₁	1	7	yes	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	4	56	no	—
137	C₂×C₄	—	284	56	no	—
138	C₂×C₄	—	142	8	no	—
139	C₂×C₄	C₂×Dic₇₁⋊C₄	1	84	yes	—
140	C₂×C₄	C₂²×C₄×D₇₁	1	14	yes	—
141	C₂×C₄	C₂²×C₂²⋊D₇₁	1	7	yes	—
142	C₂×C₄	C₂²×Dic₁₄₂	1	14	yes	—
143	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
144	C₈	—	71	16	no	—
145	C₂²	C₂×#14556059	1	112	yes	direct product factor
146	C₂²	C₂²×Dic₇₁⋊C₄	1	14	yes	—
147	C₂²	C₂²×#11765852	1	14	yes	—
148	C₂²	C₂²×#11765853	1	14	yes	—
149	C₂²	C₂³×C₂²⋊D₇₁	1	1	yes	characteristic
150	C₄	—	4	8	no	—
151	C₄	—	284	8	no	—
152	C₄	C₂²×Dic₇₁⋊C₄	1	14	yes	—
153	C₄	C₂³×C₄×D₇₁	1	1	yes	characteristic
154	C₄	C₂³×Dic₁₄₂	1	1	yes	characteristic
155	C₂	C₂²×#14556059	1	28	yes	direct product factor
156	C₂	C₂³×Dic₇₁⋊C₄	1	1	yes	characteristic
157	C₂	C₂³×#11765852	1	1	yes	characteristic
158	C₂	C₂³×#11765853	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₁
Fitting	C₂³×C₄⋊C₄×C₇₁	C₂
socle, prime radical	C₂⁵×C₇₁	D₄
cosocle	C₂×C₄×C₇₁	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₇₁
hypocenter	C₇₁	C₂³×#106
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×#14556059	C₂	direct (N × H)
6. C₂³×C₇₁⋊C₈	C₄	semidirect
10. C₂×#14556059	C₂²	direct (N × H)
17. C₂²×C₇₁⋊C₈	C₂×C₄	semidirect
21. #14556059	C₂³	direct (N × H)
33. C₂×C₇₁⋊C₈	C₂²×C₄	semidirect
50. C₂³×C₇₁	#106	semidirect
64. C₇₁⋊C₈	C₂³×C₄	semidirect
65. C₂²×C₇₁	C₂×#106	semidirect
76. C₂×C₇₁	C₂²×#106	semidirect
84. C₇₁	C₂³×#106	semidirect
130. C₂³	#14556059	direct (N × H)
145. C₂²	C₂×#14556059	direct (N × H)
155. C₂	C₂²×#14556059	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
C₂³×C₄⋊Dic₇₁	2
C₂²×#14556059	2
C₂³×C₄⋊C₄×C₇₁	2
C₂³×#106	71

Subgroups — C23×#14556059 #32861093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556059 #32861093