Finite Group Databasebeta

Subgroups — #13102976

Order 1600.

496 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717250	C₂	1	1	yes	maximal, characteristic
3	#11717244	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87800	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716688	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716684	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊(C₅⋊D₅)	—	2	1	no	—
10	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
11	C₂×#6045	C₂²	1	1	yes	characteristic
12	C₂×C₅²×C₈	—	2	2	no	—
13	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
15	C₂²⋊#1039	—	2	1	no	—
16	#87808	—	2	1	no	—
17	C₄⋊#1039	C₂²	1	1	yes	characteristic
18	C₄⋊#1039	C₂²	1	1	yes	characteristic
19	#87806	—	2	1	no	—
20	#87800	—	2	2	no	—
21	#65403	—	5	6	no	maximal
22	C₂³×C₅²	D₄	1	1	yes	characteristic
23	C₂²×C₅⋊D₅	—	2	1	no	—
24	C₂×C₄×C₅²	—	2	1	no	—
25	C₂×C₄×C₅²	—	2	1	no	—
26	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂²⋊(C₅⋊D₅)	—	4	2	no	—
29	C₂×#1039	—	2	1	no	—
30	C₂×#1039	—	2	1	no	—
31	C₂×#1039	—	2	1	no	—
32	C₄⋊(C₅⋊D₅)	—	2	1	no	—
33	C₄⋊(C₅⋊D₅)	—	2	1	no	—
34	#6045	—	2	1	no	—
35	#6045	—	2	1	no	—
36	C₅²×C₈	—	2	1	no	—
37	C₅²×C₈	D₄	1	2	yes	complemented
38	C₂²×C₅×C₈	D₅	1	6	yes	complemented
39	#4076	—	5	6	no	—
40	#4070	—	5	6	no	—
41	C₂×#677	—	5	6	no	—
42	#3953	—	5	6	no	—
43	#3949	—	5	6	no	—
44	#3948	—	5	6	no	—
45	C₂²×C₅²	—	2	1	no	—
46	C₂²×C₅²	—	2	1	no	—
47	C₂²×C₅²	SD₁₆	1	2	yes	complemented
48	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅⋊D₅	—	4	1	no	—
50	C₂×C₅⋊D₅	—	4	1	no	—
51	C₂×C₅⋊D₅	—	4	1	no	—
52	C₄×C₅²	—	2	1	no	—
53	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
54	C₄×C₅²	#41	1	1	yes	characteristic
55	#1039	—	4	1	no	—
56	#1039	—	4	1	no	—
57	#1039	—	4	1	no	—
58	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
59	C₂×C₂²⋊D₅	—	10	6	no	—
60	C₂×D₂₀	—	5	6	no	—
61	C₂×Dic₁₀	—	5	6	no	—
62	C₂×C₅×C₈	—	2	12	no	—
63	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
64	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
65	C₂²⋊Dic₅	—	10	6	no	—
66	#685	—	10	6	no	—
67	C₄⋊Dic₅	—	5	6	no	—
68	C₄⋊Dic₅	—	5	6	no	—
69	#683	—	10	6	no	—
70	#677	—	10	12	no	—
71	#465	—	25	1	no	Sylow
72	C₂×C₅²	—	2	2	no	—
73	C₂×C₅²	#121	1	1	yes	characteristic
74	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₅²	#135	1	1	yes	characteristic
76	C₅⋊D₅	—	8	1	no	—
77	C₂³×C₅	D₂₀	1	6	yes	—
78	C₂²×D₅	—	10	6	no	—
79	C₂×C₄×C₅	—	2	6	no	—
80	C₂×C₄×C₅	—	2	6	no	—
81	C₂×C₄×C₅	D₂₀	1	6	yes	—
82	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
83	C₂²⋊D₅	—	20	12	no	—
84	C₂×Dic₅	—	10	6	no	—
85	C₂×Dic₅	—	10	6	no	—
86	C₂×Dic₅	—	10	6	no	—
87	D₂₀	—	10	6	no	—
88	D₂₀	—	10	6	no	—
89	Dic₁₀	—	10	6	no	—
90	Dic₁₀	—	10	6	no	—
91	C₅×C₈	—	2	6	no	—
92	C₅×C₈	C₂²⋊D₅	1	12	yes	complemented
93	C₂×SD₁₆	—	25	1	no	—
94	C₂²×C₈	C₅⋊D₅	1	1	yes	characteristic, complemented
95	C₂²⋊Q₈	—	25	1	no	—
96	#121	—	25	1	no	—
97	#106	—	25	1	no	—
98	#103	—	25	1	no	—
99	#102	—	25	1	no	—
100	C₅²	#465	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₅	—	2	6	no	—
102	C₂²×C₅	—	2	6	no	—
103	C₂²×C₅	#677	1	12	yes	complemented
104	C₂²×C₅	C₂×D₂₀	1	6	yes	—
105	C₂×D₅	—	20	6	no	—
106	C₂×D₅	—	20	6	no	—
107	C₂×D₅	—	20	6	no	—
108	C₄×C₅	—	2	6	no	—
109	C₄×C₅	#709	1	6	yes	—
110	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
111	Dic₅	—	20	6	no	—
112	Dic₅	—	20	6	no	—
113	Dic₅	—	20	6	no	—
114	C₂×Q₈	—	25	1	no	—
115	C₂×D₄	—	25	1	no	—
116	C₂×D₄	—	50	1	no	—
117	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
118	SD₁₆	—	50	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
121	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
122	C₄⋊C₄	—	25	1	no	—
123	C₄⋊C₄	—	25	1	no	—
124	C₄⋊C₄	—	50	1	no	—
125	C₂²⋊C₄	—	50	1	no	—
126	C₂²⋊C₄	—	50	1	no	—
127	C₂×C₅	—	2	12	no	—
128	C₂×C₅	C₂×#677	1	6	yes	—
129	C₂×C₅	#4050	1	6	yes	—
130	C₂×C₅	#4076	1	6	yes	—
131	D₅	—	40	6	no	—
132	C₂³	—	50	1	no	—
133	C₂³	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
134	Q₈	—	50	1	no	—
135	Q₈	—	50	1	no	—
136	D₄	—	50	1	no	—
137	D₄	—	50	1	no	—
138	D₄	—	100	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	50	1	no	—
142	C₂×C₄	—	50	1	no	—
143	C₂×C₄	—	50	1	no	—
144	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	C₂²⋊(C₅⋊D₅)	1	2	yes	complemented
148	C₅	#65403	1	6	yes	complemented
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	100	1	no	—
152	C₂²	—	100	1	no	—
153	C₂²	—	100	1	no	—
154	C₂²	#87800	1	2	yes	complemented
155	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	100	1	no	—
158	C₄	—	100	1	no	—
159	C₄	—	100	1	no	—
160	C₄	#87900	1	1	yes	characteristic
161	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
162	C₂	—	2	2	no	—
163	C₂	—	200	1	no	—
164	C₂	C₂×#87800	1	1	yes	characteristic
165	C₂	#11717224	1	1	yes	characteristic
166	C₂	#11717250	1	1	yes	characteristic

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×C₈	C₂	semidirect
3. #11717244	C₂	semidirect
4. C₂×#87800	C₂	semidirect
5. #11716688	C₂	semidirect
6. #11716684	C₂	semidirect
7. #11716683	C₂	semidirect
37. C₅²×C₈	D₄	semidirect
38. C₂²×C₅×C₈	D₅	semidirect
47. C₂²×C₅²	SD₁₆	semidirect
92. C₅×C₈	C₂²⋊D₅	semidirect
94. C₂²×C₈	C₅⋊D₅	semidirect
100. C₅²	#465	semidirect
103. C₂²×C₅	#677	semidirect
147. C₈	C₂²⋊(C₅⋊D₅)	semidirect
148. C₅	#65403	semidirect
154. C₂²	#87800	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₈	2
#11717250	2
#11717244	2
C₂×#87800	2
#11716688	2
#11716684	2
#11716683	2
#65403	5