Subgroups — #34182010

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763552	C₂	1	2	yes	maximal
2	C₂×#19098316	C₂	1	4	yes	maximal
3	C₄²×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
5	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
6	#19098316	—	2	8	no	—
7	C₂×C₄×C₈×C₇₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
11	C₄³×C₇₉	C₂²	1	1	yes	characteristic
12	C₂²×C₇₉⋊C₈	—	2	4	no	—
13	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
14	C₄⋊(C₇₉⋊C₈)	—	4	8	no	—
15	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇₉	D₄	1	2	yes	—
20	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₇₉	—	2	4	no	—
23	C₄×C₈×C₇₉	—	2	2	no	—
24	C₄×C₈×C₇₉	C₈	1	4	yes	complemented
25	C₄×C₈×C₇₉	D₄	1	4	yes	—
26	C₄×C₈×C₇₉	Q₈	1	4	yes	—
27	C₂×C₇₉⋊C₈	—	4	4	no	—
28	C₂×C₇₉⋊C₈	—	4	8	no	—
29	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
30	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₇₉	—	2	2	no	—
35	C₂×C₈×C₇₉	—	2	2	no	—
36	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
37	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
40	C₄²×C₇₉	—	2	4	no	—
41	C₄²×C₇₉	—	2	2	no	—
42	C₄²×C₇₉	C₂×C₈	1	2	yes	—
43	C₄²×C₇₉	M₁₆	1	2	yes	—
44	C₄²×C₇₉	D₈	1	4	yes	—
45	C₄²×C₇₉	Q₁₆	1	4	yes	—
46	C₄²×C₇₉	C₂×D₄	1	2	yes	—
47	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
48	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	C₄⋊C₈	1	4	yes	—
53	C₂×C₄×C₇₉	#107	1	4	yes	—
54	C₂×C₄×C₇₉	#107	1	4	yes	—
55	C₂×C₄×C₇₉	#108	1	4	yes	—
56	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
65	C₈×C₇₉	—	2	4	no	—
66	C₈×C₇₉	C₄⋊C₈	1	8	yes	complemented
67	C₇₉⋊C₈	—	8	4	no	—
68	C₂²×C₇₉	#384	1	1	yes	characteristic
69	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₇₉	#423	1	1	yes	characteristic
71	C₂²×C₇₉	C₂×#107	1	2	yes	—
72	C₂²×C₇₉	C₂×#108	1	2	yes	—
73	C₄×C₇₉	—	2	4	no	—
74	C₄×C₇₉	—	2	2	no	—
75	C₄×C₇₉	#335	1	8	yes	complemented
76	C₄×C₇₉	C₂×C₄⋊C₈	1	2	yes	—
77	C₄×C₇₉	#423	1	2	yes	—
78	C₄×C₇₉	#493	1	1	yes	characteristic
79	C₄×C₇₉	#494	1	1	yes	characteristic
80	C₄×C₇₉	#500	1	2	yes	—
81	#8806	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	C₂×#335	1	2	yes	—
83	C₂×C₇₉	#1567	1	2	yes	—
84	C₂×C₇₉	#1842	1	1	yes	characteristic
85	C₂×C₇₉	#1845	1	1	yes	characteristic
86	C₂×C₇₉	#1848	1	1	yes	characteristic
87	#1842	—	79	2	no	—
88	C₄²×C₈	D₇₉	1	1	yes	characteristic
89	C₂×#335	—	79	4	no	—
90	C₇₉	#8806	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄⋊C₈	—	79	4	no	—
92	C₂×C₄⋊C₈	—	158	4	no	—
93	C₂×C₄×C₈	Dic₇₉	1	2	yes	—
94	C₂×C₄×C₈	Dic₇₉	1	1	yes	characteristic
95	C₂×C₄×C₈	Dic₇₉	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
97	C₄³	C₂×D₇₉	1	1	yes	characteristic
98	#335	—	158	8	no	—
99	C₂²×C₈	—	158	4	no	—
100	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
101	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
102	C₂×C₄²	Dic₁₅₈	1	2	yes	—
103	C₂×C₄²	D₃₁₆	1	2	yes	—
104	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
106	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
107	C₄⋊C₈	—	316	8	no	—
108	C₄⋊C₈	—	158	8	no	—
109	C₄×C₈	—	2	4	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₇₉⋊C₈	1	4	yes	complemented
112	C₄×C₈	Dic₁₅₈	1	4	yes	—
113	C₄×C₈	D₃₁₆	1	4	yes	—
114	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
115	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
116	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
118	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	316	4	no	—
122	C₂×C₈	—	316	8	no	—
123	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
124	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
125	C₂×C₈	C₂×C₇₉⋊C₈	1	4	yes	—
126	C₂×C₈	#11929608	1	4	yes	—
127	C₄²	—	2	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
130	C₄²	#11929608	1	2	yes	—
131	C₄²	D₆₃₂	1	4	yes	—
132	C₄²	Dic₃₁₆	1	4	yes	—
133	C₄²	C₂×D₃₁₆	1	2	yes	—
134	C₄²	C₂×Dic₁₅₈	1	2	yes	—
135	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	4	yes	—
140	C₂×C₄	#14579095	1	4	yes	—
141	C₂×C₄	#14579095	1	4	yes	—
142	C₂×C₄	#14579097	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
144	C₂×C₄	#14579158	1	1	yes	characteristic
145	C₂×C₄	#14579161	1	2	yes	—
146	C₂×C₄	#14579161	1	1	yes	characteristic
147	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
148	C₂×C₄	C₂×#11929608	1	2	yes	—
149	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
150	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
151	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	—	632	4	no	—
154	C₈	C₄⋊(C₇₉⋊C₈)	1	8	yes	complemented
155	C₂²	#19098433	1	1	yes	characteristic
156	C₂²	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
157	C₂²	#19098587	1	1	yes	characteristic
158	C₂²	C₂×#14579095	1	2	yes	—
159	C₂²	C₂×#14579097	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	4	no	—
162	C₄	#19098316	1	8	yes	complemented
163	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	2	yes	—
164	C₄	#19098587	1	2	yes	—
165	C₄	#19098940	1	1	yes	characteristic
166	C₄	#19098943	1	1	yes	characteristic
167	C₄	#19098968	1	2	yes	—
168	C₂	C₂×#19098316	1	2	yes	—
169	C₂	#23762159	1	2	yes	—
170	C₂	#23763552	1	1	yes	characteristic
171	C₂	#23763560	1	1	yes	characteristic
172	C₂	#23763572	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄²×C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄²×C₈	D₇₉
hypocenter	C₇₉	#8806
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₇₉	C₈	semidirect
66. C₈×C₇₉	C₄⋊C₈	semidirect
75. C₄×C₇₉	#335	semidirect
90. C₇₉	#8806	semidirect
111. C₄×C₈	C₇₉⋊C₈	semidirect
154. C₈	C₄⋊(C₇₉⋊C₈)	semidirect
162. C₄	#19098316	semidirect

Maximal subgroups

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

subgroup	index
#23763552	2
C₂×#19098316	2
C₄²×C₈×C₇₉	2
#8806	79