Finite Group Databasebeta

Subgroups — #25017488

Order 11008.

176 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19136869	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136441	C₂	1	1	yes	maximal, characteristic
3	#19132796	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₄₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132774	C₂	1	1	yes	maximal, characteristic
6	#19132772	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
9	#14583303	—	2	1	no	—
10	#14582981	—	2	1	no	—
11	C₂×C₄₃⋊C₃₂	C₂²	1	1	yes	characteristic
12	#14582725	C₂²	1	1	yes	characteristic
13	#14582724	—	2	1	no	—
14	#14582699	—	2	1	no	—
15	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₃₂×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#366	C₂²	1	1	yes	characteristic
18	C₄₃×#366	C₂²	1	1	yes	characteristic
19	C₂×D₃₄₄	—	2	1	no	—
20	#13063324	—	4	1	no	—
21	#13063287	—	4	1	no	—
22	D₆₈₈	—	2	1	no	—
23	D₆₈₈	Q₈	1	2	yes	complemented
24	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
25	#13063259	—	2	1	no	—
26	#13063258	—	4	1	no	—
27	#13063245	—	4	1	no	—
28	C₈×Dic₄₃	D₄	1	1	yes	characteristic
29	C₄₃⋊C₃₂	—	2	1	no	—
30	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
31	#107×C₄₃	—	2	1	no	—
32	#107×C₄₃	—	2	1	no	—
33	C₃₂×C₄₃	—	2	1	no	—
34	C₂×D₁₇₂	—	4	1	no	—
35	C₂×C₄×D₄₃	—	8	1	no	—
36	C₂×Dic₈₆	—	8	1	no	—
37	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₄₃	—	4	1	no	—
39	C₄⋊C₄×C₄₃	—	4	1	no	—
40	(C₂×D₄₃)⋊C₄	—	8	1	no	—
41	C₄⋊Dic₄₃	—	4	1	no	—
42	Dic₄₃⋊C₄	—	8	1	no	—
43	C₄×Dic₄₃	D₈	1	1	yes	characteristic
44	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
45	D₃₄₄	—	2	2	no	—
46	D₃₄₄	—	4	1	no	—
47	C₁₆×C₄₃	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₄₃	#41	1	1	yes	characteristic
49	C₄₃⋊C₁₆	—	2	1	no	—
50	C₂²×D₄₃	—	8	1	no	—
51	C₂×C₄×C₄₃	—	8	1	no	—
52	C₂×C₄×C₄₃	—	8	1	no	—
53	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₄₃	—	8	1	no	—
55	C₂×Dic₄₃	D₁₆	1	1	yes	characteristic
56	C₂×Dic₄₃	D₁₆	1	1	yes	characteristic
57	D₁₇₂	—	4	2	no	—
58	D₁₇₂	—	8	1	no	—
59	C₄×D₄₃	—	16	2	no	—
60	Dic₈₆	—	16	2	no	—
61	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₄₃	#137	1	1	yes	characteristic
63	C₄₃⋊C₈	—	2	1	no	—
64	#13705	—	43	1	no	maximal, Sylow
65	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
66	C₂×D₄₃	—	8	2	no	—
67	C₂×D₄₃	—	16	1	no	—
68	C₂×D₄₃	—	16	1	no	—
69	C₄×C₄₃	—	16	1	no	—
70	C₄×C₄₃	—	16	1	no	—
71	C₄×C₄₃	#474	1	1	yes	characteristic
72	C₄×C₄₃	#510	1	1	yes	characteristic
73	Dic₄₃	—	2	1	no	—
74	Dic₄₃	—	16	1	no	—
75	Dic₄₃	SD₆₄	1	2	yes	—
76	#2252	—	43	1	no	—
77	C₄×D₁₆	—	43	1	no	—
78	#1424	—	43	1	no	—
79	#1424	D₄₃	1	1	yes	characteristic, complemented
80	C₄⋊C₃₂	—	43	1	no	—
81	#1415	—	43	1	no	—
82	#1415	—	43	1	no	—
83	C₂×C₄₃	#2224	1	1	yes	characteristic
84	C₂×C₄₃	C₂×SD₆₄	1	1	yes	characteristic
85	C₂×C₄₃	#2263	1	1	yes	characteristic
86	D₄₃	—	16	2	no	—
87	C₂×D₁₆	—	43	1	no	—
88	#500	—	86	1	no	—
89	C₄×D₈	—	86	1	no	—
90	C₂×C₃₂	—	43	1	no	—
91	C₂×C₃₂	C₂×D₄₃	1	1	yes	characteristic
92	#366	—	43	1	no	—
93	#366	—	86	1	no	—
94	#366	C₂×D₄₃	1	1	yes	characteristic
95	#366	C₂×D₄₃	1	1	yes	characteristic
96	#357	—	86	1	no	—
97	C₄×C₁₆	—	43	1	no	—
98	C₄₃	#13705	1	1	yes	characteristic, complemented, Sylow
99	C₂×D₈	—	86	1	no	—
100	C₄⋊Q₈	—	172	1	no	—
101	C₄×D₄	—	172	1	no	—
102	D₁₆	—	43	2	no	—
103	D₁₆	—	86	1	no	—
104	C₂×C₁₆	—	43	1	no	—
105	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
106	#107	—	2	1	no	—
107	#107	—	2	1	no	—
108	#107	—	86	1	no	—
109	#107	—	172	1	no	—
110	#102	—	172	1	no	—
111	C₄×C₈	—	43	1	no	—
112	C₃₂	—	2	1	no	—
113	C₃₂	—	86	1	no	—
114	C₂×Q₈	—	344	1	no	—
115	C₂×D₄	—	172	1	no	—
116	C₂²×C₄	—	344	1	no	—
117	D₈	—	86	2	no	—
118	D₈	—	172	1	no	—
119	C₂×C₈	—	43	1	no	—
120	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	344	1	no	—
124	C₄⋊C₄	—	172	1	no	—
125	C₂²⋊C₄	—	344	1	no	—
126	C₄²	—	43	1	no	—
127	C₁₆	—	86	1	no	—
128	C₁₆	#10621898	1	1	yes	characteristic
129	C₁₆	Q₈×D₄₃	1	1	yes	characteristic
130	C₂³	—	344	1	no	—
131	Q₈	—	688	2	no	—
132	D₄	—	344	1	no	—
133	D₄	—	172	2	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	43	1	no	—
137	C₂×C₄	—	43	1	no	—
138	C₂×C₄	—	344	1	no	—
139	C₂×C₄	—	688	2	no	—
140	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
141	C₈	—	86	1	no	—
142	C₈	#13063303	1	1	yes	characteristic
143	C₈	#13063357	1	1	yes	characteristic
144	C₂²	—	344	2	no	—
145	C₂²	—	688	1	no	—
146	C₂²	—	688	1	no	—
147	C₂²	D₁₆×D₄₃	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	43	2	no	—
151	C₄	—	86	1	no	—
152	C₄	—	688	1	no	—
153	C₄	#14583178	1	1	yes	characteristic
154	C₄	#14583341	1	1	yes	characteristic
155	C₂	—	688	2	no	—
156	C₂	#19136731	1	1	yes	characteristic
157	C₂	SD₆₄×D₄₃	1	1	yes	characteristic
158	C₂	#19136909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₄₃
commutator, cosocle	C₂×C₁₆×C₄₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#1424	C₂
socle	C₂²×C₄₃	C₂×D₁₆
prime radical	C₂×D₆₈₈	C₂²
hypercenter	#1424	D₄₃
hypocenter	C₄₃	#13705
LCS	G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
1. #19136869	C₂	semidirect
3. #19132796	C₂	semidirect
4. Dic₄₃⋊C₃₂	C₂	semidirect
7. C₄₃×#1424	C₂	semidirect
23. D₆₈₈	Q₈	semidirect
79. #1424	D₄₃	semidirect
98. C₄₃	#13705	semidirect

Maximal subgroups

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

subgroup	index
#19136869	2
#19136441	2
#19132796	2
Dic₄₃⋊C₃₂	2
#19132774	2
#19132772	2
C₄₃×#1424	2
#13705	43