Subgroups — #32904071

Order 18176.

11674 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×Dic₇₁	C₂	1	32	yes	maximal, complemented
2	#22601543	C₂	1	12	yes	maximal, complemented
3	C₂×#17986474	C₂	1	18	yes	maximal, complemented
4	C₂²×H(5,𝔽₂)×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986482	C₂²	1	96	yes	complemented
6	#17986477	C₂²	1	8	yes	complemented
7	#17986474	C₂²	1	72	yes	complemented
8	#17986474	C₂²	1	144	yes	complemented
9	#41×Dic₇₁	C₂²	1	96	yes	complemented
10	C₂×D₄×Dic₇₁	C₂²	1	144	yes	complemented
11	C₂×#14556087	C₂²	1	36	yes	complemented
12	C₂²×C₂²⋊Dic₇₁	C₂²	1	24	yes	complemented
13	C₂×H(5,𝔽₂)×C₇₁	C₄	1	32	yes	complemented
14	C₂×H(5,𝔽₂)×C₇₁	C₂²	1	16	yes	—
15	C₂²×#41×C₇₁	C₂²	1	6	yes	—
16	C₂³×D₄×C₇₁	C₂²	1	9	yes	—
17	C₂³×Dic₇₁	C₂³	1	48	yes	complemented
18	Q₈×Dic₇₁	C₂³	1	64	yes	complemented
19	D₄×Dic₇₁	C₂³	1	576	yes	complemented
20	#14556087	C₂³	1	48	yes	—
21	#14556087	C₂³	1	144	yes	complemented
22	C₂×C₄⋊Dic₇₁	C₂³	1	72	yes	complemented
23	C₂×C₂²⋊Dic₇₁	C₂³	1	72	yes	—
24	C₂×C₂²⋊Dic₇₁	C₂³	1	144	yes	complemented
25	C₂×C₄×Dic₇₁	C₂³	1	72	yes	complemented
26	C₂⁵×C₇₁	C₂³	1	6	yes	—
27	H(5,𝔽₂)×C₇₁	C₂×C₄	1	256	yes	complemented
28	C₂×#41×C₇₁	C₂×C₄	1	96	yes	complemented
29	C₂×#41×C₇₁	C₂³	1	48	yes	—
30	C₂²×Q₈×C₇₁	C₂³	1	2	yes	—
31	C₂²×D₄×C₇₁	C₂×C₄	1	144	yes	complemented
32	C₂²×D₄×C₇₁	C₂³	1	72	yes	—
33	C₂²×D₄×C₇₁	C₂³	1	18	yes	—
34	C₂³×C₄×C₇₁	C₂³	1	9	yes	—
35	C₂²×Dic₇₁	—	8	48	no	—
36	C₂²×Dic₇₁	C₂⁴	1	144	yes	—
37	C₄⋊Dic₇₁	C₂⁴	1	96	yes	—
38	C₂²⋊Dic₇₁	C₂⁴	1	16	yes	—
39	C₂²⋊Dic₇₁	C₂⁴	1	144	yes	—
40	C₄×Dic₇₁	C₂⁴	1	96	yes	—
41	C₂⁴×C₇₁	—	8	12	no	—
42	C₂⁴×C₇₁	C₂²×C₄	1	48	yes	complemented
43	C₂⁴×C₇₁	C₂⁴	1	24	yes	—
44	C₂⁴×C₇₁	C₂⁴	1	9	yes	—
45	#41×C₇₁	C₂²×C₄	1	384	yes	complemented
46	C₂×Q₈×C₇₁	C₂²×C₄	1	16	yes	complemented
47	C₂×Q₈×C₇₁	C₂⁴	1	8	yes	—
48	C₂×D₄×C₇₁	C₂²×C₄	1	144	yes	complemented
49	C₂×D₄×C₇₁	C₂²×C₄	1	576	yes	complemented
50	C₂×D₄×C₇₁	C₂⁴	1	72	yes	—
51	C₂²×C₄×C₇₁	C₂²×C₄	1	72	yes	complemented
52	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
53	C₂²×C₄×C₇₁	C₂⁴	1	36	yes	—
54	C₂³×C₇₁	—	4	18	no	—
55	C₂³×C₇₁	—	4	48	no	—
56	C₂³×C₇₁	—	8	48	no	—
57	C₂³×C₇₁	C₂³×C₄	1	36	yes	—
58	C₂³×C₇₁	C₂³×C₄	1	96	yes	—
59	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
60	C₂³×C₇₁	C₂⁵	1	18	yes	—
61	Q₈×C₇₁	C₂³×C₄	1	32	yes	—
62	D₄×C₇₁	C₂³×C₄	1	288	yes	—
63	C₂×C₄×C₇₁	C₂³×C₄	1	24	yes	—
64	C₂×C₄×C₇₁	C₂³×C₄	1	144	yes	—
65	C₂×C₄×C₇₁	C₂⁵	1	12	yes	—
66	C₂×Dic₇₁	—	4	144	no	—
67	C₂×Dic₇₁	C₂⁵	1	32	yes	—
68	C₂²×C₇₁	—	2	36	no	—
69	C₂²×C₇₁	—	2	2	no	—
70	C₂²×C₇₁	—	4	36	no	—
71	C₂²×C₇₁	—	4	96	no	—
72	C₂²×C₇₁	C₂⁴×C₄	1	2	yes	—
73	C₂²×C₇₁	C₂⁴×C₄	1	36	yes	—
74	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
75	C₄×C₇₁	C₂⁴×C₄	1	24	yes	—
76	Dic₇₁	—	2	32	no	—
77	#62652	—	71	1	no	maximal, Sylow
78	C₂×C₇₁	—	2	2	no	—
79	C₂×C₇₁	—	2	36	no	—
80	C₂×C₇₁	C₂⁵×C₄	1	1	yes	characteristic
81	C₂×C₇₁	H(7,𝔽₂)	1	2	yes	—
82	C₂²×H(5,𝔽₂)	D₇₁	1	1	yes	characteristic
83	C₄×H(5,𝔽₂)	—	71	32	no	—
84	#3428	—	71	12	no	—
85	C₂×#518	—	71	18	no	—
86	C₇₁	#62652	1	1	yes	characteristic, complemented, Sylow
87	C₂×H(5,𝔽₂)	Dic₇₁	1	32	yes	complemented
88	C₂×H(5,𝔽₂)	C₂×D₇₁	1	16	yes	—
89	C₂²×#41	C₂×D₇₁	1	6	yes	—
90	C₂³×D₄	C₂×D₇₁	1	9	yes	—
91	#520	—	71	96	no	—
92	#519	—	71	8	no	—
93	#518	—	71	72	no	—
94	#518	—	71	144	no	—
95	C₄×#41	—	71	96	no	—
96	C₂×C₄×D₄	—	71	144	no	—
97	C₂×#117	—	71	36	no	—
98	C₂²×C₂²⋊C₄	—	71	24	no	—
99	C₂⁵	C₂²×D₇₁	1	6	yes	—
100	H(5,𝔽₂)	C₂×Dic₇₁	1	256	yes	complemented
101	C₂×#41	C₂×Dic₇₁	1	96	yes	complemented
102	C₂×#41	C₂²×D₇₁	1	48	yes	—
103	C₂²×Q₈	C₂²×D₇₁	1	2	yes	—
104	C₂²×D₄	C₂×Dic₇₁	1	144	yes	complemented
105	C₂²×D₄	C₂²×D₇₁	1	72	yes	—
106	C₂²×D₄	C₂²×D₇₁	1	18	yes	—
107	C₂³×C₄	—	71	48	no	—
108	C₂³×C₄	C₂²×D₇₁	1	9	yes	—
109	C₄×Q₈	—	71	64	no	—
110	C₄×D₄	—	71	576	no	—
111	#117	—	71	48	no	—
112	#117	—	71	144	no	—
113	C₂×C₄⋊C₄	—	71	72	no	—
114	C₂×C₂²⋊C₄	—	71	72	no	—
115	C₂×C₂²⋊C₄	—	71	144	no	—
116	C₂×C₄²	—	71	72	no	—
117	C₂⁴	—	8	12	no	—
118	C₂⁴	C₂²×Dic₇₁	1	48	yes	complemented
119	C₂⁴	C₂³×D₇₁	1	9	yes	—
120	C₂⁴	C₂³×D₇₁	1	24	yes	—
121	#41	C₂²×Dic₇₁	1	384	yes	complemented
122	C₂×Q₈	C₂²×Dic₇₁	1	16	yes	complemented
123	C₂×Q₈	C₂³×D₇₁	1	8	yes	—
124	C₂×D₄	C₂²×Dic₇₁	1	144	yes	complemented
125	C₂×D₄	C₂²×Dic₇₁	1	576	yes	complemented
126	C₂×D₄	C₂³×D₇₁	1	72	yes	—
127	C₂²×C₄	—	568	48	no	—
128	C₂²×C₄	—	71	144	no	—
129	C₂²×C₄	C₂²×Dic₇₁	1	72	yes	complemented
130	C₂²×C₄	C₂³×D₇₁	1	36	yes	—
131	C₂²×C₄	C₂³×D₇₁	1	6	yes	—
132	C₄⋊C₄	—	71	96	no	—
133	C₂²⋊C₄	—	71	16	no	—
134	C₂²⋊C₄	—	71	144	no	—
135	C₄²	—	71	96	no	—
136	C₂³	—	4	48	no	—
137	C₂³	—	4	18	no	—
138	C₂³	—	8	48	no	—
139	C₂³	C₂³×Dic₇₁	1	36	yes	—
140	C₂³	C₂³×Dic₇₁	1	96	yes	—
141	C₂³	C₂⁴×D₇₁	1	1	yes	characteristic
142	C₂³	C₂⁴×D₇₁	1	18	yes	—
143	Q₈	C₂³×Dic₇₁	1	32	yes	—
144	D₄	C₂³×Dic₇₁	1	288	yes	—
145	C₂×C₄	—	284	144	no	—
146	C₂×C₄	—	71	32	no	—
147	C₂×C₄	C₂³×Dic₇₁	1	24	yes	—
148	C₂×C₄	C₂³×Dic₇₁	1	144	yes	—
149	C₂×C₄	C₂⁴×D₇₁	1	12	yes	—
150	C₂²	—	2	36	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	36	no	—
153	C₂²	—	4	96	no	—
154	C₂²	C₂⁴×Dic₇₁	1	2	yes	—
155	C₂²	C₂⁴×Dic₇₁	1	36	yes	—
156	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
157	C₄	—	142	32	no	—
158	C₄	C₂⁴×Dic₇₁	1	24	yes	—
159	C₂	—	2	2	no	—
160	C₂	—	2	36	no	—
161	C₂	C₂⁵×Dic₇₁	1	1	yes	characteristic
162	C₂	#22603150	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₇₁
commutator	C₂×C₇₁	C₂⁵×C₄
Fitting, prime radical	C₂²×H(5,𝔽₂)×C₇₁	C₂
socle, cosocle	C₂²×C₇₁	C₂⁶
hypercenter	C₂²×H(5,𝔽₂)	D₇₁
hypocenter	C₇₁	#62652
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×Dic₇₁	C₂	semidirect
2. #22601543	C₂	semidirect
3. C₂×#17986474	C₂	semidirect
5. #17986482	C₂²	semidirect
6. #17986477	C₂²	semidirect
7. #17986474	C₂²	semidirect
8. #17986474	C₂²	semidirect
9. #41×Dic₇₁	C₂²	semidirect
10. C₂×D₄×Dic₇₁	C₂²	semidirect
11. C₂×#14556087	C₂²	semidirect
12. C₂²×C₂²⋊Dic₇₁	C₂²	semidirect
13. C₂×H(5,𝔽₂)×C₇₁	C₄	semidirect
17. C₂³×Dic₇₁	C₂³	semidirect
18. Q₈×Dic₇₁	C₂³	semidirect
19. D₄×Dic₇₁	C₂³	semidirect
21. #14556087	C₂³	semidirect
22. C₂×C₄⋊Dic₇₁	C₂³	semidirect
24. C₂×C₂²⋊Dic₇₁	C₂³	semidirect
25. C₂×C₄×Dic₇₁	C₂³	semidirect
27. H(5,𝔽₂)×C₇₁	C₂×C₄	semidirect
28. C₂×#41×C₇₁	C₂×C₄	semidirect
31. C₂²×D₄×C₇₁	C₂×C₄	semidirect
42. C₂⁴×C₇₁	C₂²×C₄	semidirect
45. #41×C₇₁	C₂²×C₄	semidirect
46. C₂×Q₈×C₇₁	C₂²×C₄	semidirect
48. C₂×D₄×C₇₁	C₂²×C₄	semidirect
49. C₂×D₄×C₇₁	C₂²×C₄	semidirect
51. C₂²×C₄×C₇₁	C₂²×C₄	semidirect
86. C₇₁	#62652	semidirect
87. C₂×H(5,𝔽₂)	Dic₇₁	semidirect
100. H(5,𝔽₂)	C₂×Dic₇₁	semidirect
101. C₂×#41	C₂×Dic₇₁	semidirect
104. C₂²×D₄	C₂×Dic₇₁	semidirect
118. C₂⁴	C₂²×Dic₇₁	semidirect
121. #41	C₂²×Dic₇₁	semidirect
122. C₂×Q₈	C₂²×Dic₇₁	semidirect
124. C₂×D₄	C₂²×Dic₇₁	semidirect
125. C₂×D₄	C₂²×Dic₇₁	semidirect
129. C₂²×C₄	C₂²×Dic₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
H(5,𝔽₂)×Dic₇₁	2
#22601543	2
C₂×#17986474	2
C₂²×H(5,𝔽₂)×C₇₁	2
#62652	71