Finite Group Databasebeta

Subgroups — #23765401

Order 10112.

270 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19098749	C₂	1	2	yes	maximal, complemented
2	#19098740	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098358	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098350	C₂	1	2	yes	maximal, complemented
5	H(3,𝔽₄)×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579206	C₂²	1	1	yes	characteristic
7	#14579192	—	2	1	no	—
8	#14579130	—	2	2	no	—
9	#14579089	—	2	1	no	—
10	#14579078	C₂²	1	1	yes	characteristic
11	#14579076	C₂²	1	2	yes	—
12	#14579075	—	2	2	no	—
13	#124×C₇₉	—	2	1	no	—
14	#124×C₇₉	—	2	2	no	—
15	#124×C₇₉	—	2	1	no	—
16	#124×C₇₉	C₂²	1	1	yes	characteristic
17	#120×C₇₉	—	2	2	no	—
18	#120×C₇₉	C₂²	1	2	yes	—
19	#11929624	—	2	1	no	—
20	#11929623	—	4	1	no	—
21	C₂×C₂²⋊D₇₉	—	2	2	no	—
22	C₂×C₂²⋊D₇₉	—	4	2	no	—
23	D₄×D₇₉	—	4	1	no	—
24	#11929614	—	4	1	no	—
25	#11929611	—	4	1	no	—
26	#11929608	—	2	1	no	—
27	C₂²⋊Dic₇₉	—	2	2	no	—
28	C₂²⋊Dic₇₉	—	4	2	no	—
29	C₂⁴×C₇₉	D₄	1	2	yes	complemented
30	C₂×Q₈×C₇₉	—	2	1	no	—
31	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic, complemented
32	C₂×D₄×C₇₉	—	2	1	no	—
33	C₂×D₄×C₇₉	—	2	1	no	—
34	C₂×D₄×C₇₉	—	2	2	no	—
35	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂²⋊C₄×C₇₉	—	2	2	no	—
37	C₂²⋊C₄×C₇₉	—	2	2	no	—
38	C₂²⋊C₄×C₇₉	—	2	2	no	—
39	C₂²⋊C₄×C₇₉	—	2	2	no	—
40	C₂²⋊C₄×C₇₉	D₄	1	2	yes	complemented
41	C₄²×C₇₉	—	2	1	no	—
42	C₄²×C₇₉	D₄	1	1	yes	characteristic, complemented
43	C₂³×C₇₉	—	2	2	no	—
44	C₂³×C₇₉	—	4	2	no	—
45	C₂³×C₇₉	—	8	2	no	—
46	C₂³×C₇₉	C₂×D₄	1	2	yes	—
47	C₂²×D₇₉	—	2	2	no	—
48	Q₈×C₇₉	—	4	1	no	—
49	Q₈×C₇₉	—	8	1	no	—
50	D₄×C₇₉	—	4	1	no	—
51	D₄×C₇₉	—	8	2	no	—
52	D₄×C₇₉	—	8	1	no	—
53	D₄×C₇₉	—	8	1	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	2	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₇₉	—	4	2	no	—
59	C₂²⋊D₇₉	—	4	1	no	—
60	C₂²⋊D₇₉	—	8	4	no	—
61	C₂×Dic₇₉	—	2	2	no	—
62	C₂×Dic₇₉	—	4	2	no	—
63	D₃₁₆	—	2	1	no	—
64	C₄×D₇₉	—	4	1	no	—
65	Dic₁₅₈	—	2	1	no	—
66	C₇₉⋊C₈	—	4	1	no	—
67	C₂²×C₇₉	—	2	2	no	—
68	C₂²×C₇₉	—	4	2	no	—
69	C₂²×C₇₉	—	4	2	no	—
70	C₂²×C₇₉	—	4	2	no	—
71	C₂²×C₇₉	—	4	4	no	—
72	C₂²×C₇₉	—	4	2	no	—
73	C₂²×C₇₉	—	8	2	no	—
74	C₂²×C₇₉	#120	1	1	yes	characteristic
75	C₂×D₇₉	—	4	1	no	—
76	C₂×D₇₉	—	8	2	no	—
77	C₄×C₇₉	—	2	1	no	—
78	C₄×C₇₉	—	4	2	no	—
79	C₄×C₇₉	—	4	1	no	—
80	C₄×C₇₉	—	4	1	no	—
81	Dic₇₉	—	4	1	no	—
82	Dic₇₉	—	8	2	no	—
83	C₂×C₇₉	—	2	1	no	—
84	C₂×C₇₉	—	4	2	no	—
85	C₂×C₇₉	—	8	2	no	—
86	C₂×C₇₉	B(4,𝔽₂)	1	1	yes	characteristic
87	D₇₉	—	8	1	no	—
88	#2199	—	79	1	no	maximal, Sylow
89	C₇₉	#2199	1	1	yes	characteristic, complemented, Sylow
90	H(3,𝔽₄)	D₇₉	1	1	yes	characteristic, complemented
91	B(4,𝔽₂)	—	79	2	no	—
92	#455	—	79	1	no	—
93	#354	—	79	1	no	—
94	AΣL(1,𝔽₁₆)	—	79	2	no	—
95	H(5,𝔽₂)	—	79	1	no	—
96	#137	—	158	1	no	—
97	#124	—	2	1	no	—
98	#124	—	2	2	no	—
99	#124	—	2	1	no	—
100	#124	C₂×D₇₉	1	1	yes	characteristic
101	#120	—	2	2	no	—
102	#120	—	158	2	no	—
103	#120	C₂×D₇₉	1	2	yes	—
104	#104	—	158	1	no	—
105	#100	—	79	1	no	—
106	#99	—	79	2	no	—
107	#99	—	158	2	no	—
108	C₂⁴	C₂²⋊D₇₉	1	2	yes	complemented
109	#41	—	316	1	no	—
110	#41	—	158	1	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic, complemented
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	1	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	316	1	no	—
117	C₂×D₄	—	316	2	no	—
118	C₂×D₄	—	158	2	no	—
119	C₂×D₄	C₂²×D₇₉	1	1	yes	characteristic
120	Q₁₆	—	316	1	no	—
121	SD₁₆	—	316	1	no	—
122	M₁₆	—	158	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	316	2	no	—
128	C₂²⋊C₄	—	158	2	no	—
129	C₂²⋊C₄	C₂²⋊D₇₉	1	2	yes	complemented
130	C₄²	—	2	1	no	—
131	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic, complemented
132	C₂³	—	2	2	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	8	2	no	—
135	C₂³	—	158	2	no	—
136	C₂³	C₂×C₂²⋊D₇₉	1	2	yes	—
137	Q₈	—	4	1	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	158	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	8	2	no	—
142	D₄	—	8	1	no	—
143	D₄	—	8	1	no	—
144	D₄	—	316	2	no	—
145	D₄	—	316	1	no	—
146	D₄	—	632	4	no	—
147	D₄	—	158	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	316	1	no	—
152	C₂×C₄	—	316	2	no	—
153	C₂×C₄	—	158	2	no	—
154	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
155	C₈	—	316	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	4	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	8	2	no	—
163	C₂²	—	316	1	no	—
164	C₂²	—	632	2	no	—
165	C₂²	#14579130	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	316	1	no	—
171	C₄	—	632	2	no	—
172	C₂	—	2	1	no	—
173	C₂	—	4	2	no	—
174	C₂	—	8	2	no	—
175	C₂	—	632	1	no	—
176	C₂	#19098750	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19098750
commutator, cosocle	C₂×D₄×C₇₉	C₂³
Frattini	C₂×D₄	C₂²×D₇₉
Fitting	H(3,𝔽₄)×C₇₉	C₂
socle	C₂×C₇₉	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₇₉
hypocenter	C₇₉	#2199
LCS	G ▷ C₂×D₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #19098749	C₂	semidirect
2. #19098740	C₂	semidirect
3. #19098358	C₂	semidirect
4. #19098350	C₂	semidirect
5. H(3,𝔽₄)×C₇₉	C₂	semidirect
29. C₂⁴×C₇₉	D₄	semidirect
31. C₂×Q₈×C₇₉	D₄	semidirect
40. C₂²⋊C₄×C₇₉	D₄	semidirect
42. C₄²×C₇₉	D₄	semidirect
89. C₇₉	#2199	semidirect
90. H(3,𝔽₄)	D₇₉	semidirect
108. C₂⁴	C₂²⋊D₇₉	semidirect
112. C₂×Q₈	C₂²⋊D₇₉	semidirect
129. C₂²⋊C₄	C₂²⋊D₇₉	semidirect
131. C₄²	C₂²⋊D₇₉	semidirect