Finite Group Databasebeta

Subgroups — #23773931

Order 10112.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19099191	C₂	1	2	yes	maximal
2	#19098972	C₂	1	4	yes	maximal, complemented
3	#19098965	C₂	1	2	yes	maximal, complemented
4	#19098961	C₂	1	2	yes	maximal, complemented
5	#19098605	C₂	1	2	yes	maximal
6	#19098601	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579093	C₂	1	1	yes	maximal, characteristic
8	#405×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#14579161	C₂²	1	4	yes	—
10	#14579160	—	2	2	no	—
11	#14579153	C₂²	1	4	yes	—
12	#14579152	—	2	2	no	—
13	#14579138	—	2	4	no	—
14	#14579120	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
16	#14579095	C₂²	1	4	yes	—
17	#14579095	C₂²	1	2	yes	—
18	#14579093	C₂²	1	1	yes	characteristic
19	#14579093	C₂²	1	2	yes	—
20	#14579093	C₂²	1	4	yes	—
21	#14579093	C₂²	1	1	yes	characteristic
22	#14579093	C₂²	1	2	yes	—
23	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	#117×C₇₉	C₂²	1	1	yes	characteristic
26	#97×C₇₉	C₂²	1	2	yes	—
27	C₄×C₈×C₇₉	C₂²	1	2	yes	—
28	C₂×Dic₁₅₈	—	2	4	no	—
29	C₂²×Dic₇₉	—	2	2	no	—
30	C₄⋊Dic₇₉	—	2	2	no	—
31	C₄⋊Dic₇₉	—	2	4	no	—
32	C₄⋊Dic₇₉	C₂³	1	4	yes	—
33	C₄⋊Dic₇₉	C₂³	1	2	yes	—
34	C₄⋊Dic₇₉	C₂³	1	2	yes	—
35	Dic₇₉⋊C₄	—	2	4	no	—
36	Dic₇₉⋊C₄	—	2	4	no	—
37	C₂²⋊Dic₇₉	—	2	2	no	—
38	C₄×Dic₇₉	—	2	2	no	—
39	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₉	Q₈	1	4	yes	complemented
41	C₂×C₈×C₇₉	Q₈	1	4	yes	complemented
42	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂³	1	2	yes	—
44	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
46	C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
47	C₄²×C₇₉	C₂³	1	2	yes	—
48	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	—	2	2	no	—
56	C₂×Dic₇₉	—	2	4	no	—
57	C₂×Dic₇₉	—	8	2	no	—
58	Dic₁₅₈	—	8	4	no	—
59	C₈×C₇₉	C₂×Q₈	1	2	yes	—
60	C₈×C₇₉	C₂×Q₈	1	2	yes	—
61	C₈×C₇₉	C₂×Q₈	1	4	yes	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₇₉	—	2	4	no	—
66	C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
67	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
69	Dic₇₉	—	4	2	no	—
70	Dic₇₉	—	4	4	no	—
71	Dic₇₉	—	4	2	no	—
72	C₂×C₇₉	—	2	2	no	—
73	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₇₉	#577	1	2	yes	—
75	#3163	—	79	1	no	maximal, Sylow
76	C₇₉	#3163	1	1	yes	characteristic, complemented, Sylow
77	#544	—	79	2	no	—
78	#501	—	79	4	no	—
79	#499	—	79	2	no	—
80	#498	—	79	2	no	—
81	#428	—	79	2	no	—
82	#427	—	79	1	no	—
83	C₂×#106	—	79	1	no	—
84	#405	D₇₉	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
87	C₄⋊Q₈	—	79	4	no	—
88	C₄⋊Q₈	—	158	2	no	—
89	#125	—	79	4	no	—
90	#125	—	158	2	no	—
91	C₂²⋊Q₈	—	158	4	no	—
92	#117	—	79	2	no	—
93	#117	C₂×D₇₉	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	79	2	no	—
95	#107	—	79	2	no	—
96	#107	—	79	4	no	—
97	#106	—	79	2	no	—
98	#106	—	79	2	no	—
99	#106	—	79	1	no	—
100	#106	—	79	1	no	—
101	#106	—	79	4	no	—
102	#97	C₂×D₇₉	1	2	yes	—
103	C₄×C₈	C₂×D₇₉	1	2	yes	—
104	C₂×Q₈	—	158	4	no	—
105	C₂²×C₄	—	158	2	no	—
106	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
107	M₁₆	Dic₁₅₈	1	4	yes	complemented
108	C₂×C₈	Dic₁₅₈	1	4	yes	complemented
109	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₇₉	1	2	yes	—
111	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
112	C₄⋊C₄	—	79	4	no	—
113	C₄⋊C₄	—	79	2	no	—
114	C₄⋊C₄	—	79	2	no	—
115	C₄⋊C₄	—	158	4	no	—
116	C₄⋊C₄	—	158	4	no	—
117	C₄⋊C₄	—	158	2	no	—
118	C₄⋊C₄	—	158	4	no	—
119	C₄⋊C₄	D₃₁₆	1	2	yes	—
120	C₂²⋊C₄	—	158	2	no	—
121	C₂²⋊C₄	D₃₁₆	1	2	yes	—
122	C₄²	—	158	2	no	—
123	C₄²	C₂²×D₇₉	1	2	yes	—
124	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
125	Q₈	—	632	4	no	—
126	C₂×C₄	—	632	2	no	—
127	C₂×C₄	—	158	2	no	—
128	C₂×C₄	—	158	2	no	—
129	C₂×C₄	—	158	4	no	—
130	C₂×C₄	C₂×D₃₁₆	1	4	yes	—
131	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
132	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
133	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
134	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
135	C₈	C₂×Dic₁₅₈	1	2	yes	—
136	C₈	C₂×Dic₁₅₈	1	2	yes	—
137	C₈	C₂×Dic₁₅₈	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#14579161	1	2	yes	—
140	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	316	4	no	—
143	C₄	—	316	2	no	—
144	C₄	—	316	2	no	—
145	C₄	#14579161	1	2	yes	—
146	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
147	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#14579161	1	1	yes	characteristic
150	C₂	#19099375	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#405×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#405	D₇₉
hypocenter	C₇₉	#3163
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #19098972	C₂	semidirect
3. #19098965	C₂	semidirect
4. #19098961	C₂	semidirect
40. M₁₆×C₇₉	Q₈	semidirect
41. C₂×C₈×C₇₉	Q₈	semidirect
76. C₇₉	#3163	semidirect
107. M₁₆	Dic₁₅₈	semidirect
108. C₂×C₈	Dic₁₅₈	semidirect