Finite Group Databasebeta

Subgroups — #25017221

Order 11008.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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	#19136910	C₂	1	1	yes	maximal, characteristic
2	#19136480	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136470	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132784	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132781	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132730	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1426	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583331	C₂²	1	1	yes	characteristic
9	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
10	#14583015	—	2	1	no	—
11	#14582981	—	2	1	no	—
12	#14582738	C₄	1	2	yes	complemented
13	D₁₃₇₆	C₄	1	2	yes	complemented
14	#14582733	C₂²	1	1	yes	characteristic
15	#14582705	—	2	1	no	—
16	#14582699	—	2	1	no	—
17	#14582674	C₂²	1	1	yes	characteristic
18	C₄₃×M₆₄	C₂²	1	1	yes	characteristic
19	C₄₃×#368	C₂²	1	1	yes	characteristic
20	C₄₃×#367	C₂²	1	1	yes	characteristic
21	#13063347	—	2	1	no	—
22	C₂×D₃₄₄	—	2	1	no	—
23	#13063333	—	4	1	no	—
24	#13063287	—	4	1	no	—
25	Dic₃₄₄	C₂×C₄	1	1	yes	characteristic, complemented
26	#13063270	—	2	1	no	—
27	D₆₈₈	—	2	1	no	—
28	D₆₈₈	C₂×C₄	1	2	yes	—
29	D₆₈₈	C₂×C₄	1	1	yes	characteristic, complemented
30	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
31	#13063251	—	4	1	no	—
32	#13063245	—	4	1	no	—
33	C₈×Dic₄₃	D₄	1	1	yes	characteristic
34	C₄₃⋊C₃₂	—	2	1	no	—
35	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
36	#108×C₄₃	—	2	1	no	—
37	#107×C₄₃	—	2	1	no	—
38	C₃₂×C₄₃	C₂×C₄	1	2	yes	complemented
39	#10621898	—	4	1	no	—
40	C₂×D₁₇₂	—	4	1	no	—
41	C₂×C₄×D₄₃	—	8	1	no	—
42	M₁₆×C₄₃	—	4	1	no	—
43	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₄₃	—	4	1	no	—
45	(C₂×D₄₃)⋊C₄	—	8	1	no	—
46	C₄×Dic₄₃	D₈	1	1	yes	characteristic
47	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
48	Dic₁₇₂	—	2	1	no	—
49	D₃₄₄	—	2	2	no	—
50	D₃₄₄	—	2	1	no	—
51	D₃₄₄	—	4	1	no	—
52	#10621873	—	4	1	no	—
53	#10621872	—	8	1	no	—
54	C₈×D₄₃	—	8	1	no	—
55	C₁₆×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₄₃	#41	1	1	yes	characteristic
57	C₄₃⋊C₁₆	—	2	1	no	—
58	C₂²×D₄₃	—	8	1	no	—
59	C₂×C₄×C₄₃	—	8	1	no	—
60	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
61	C₂²⋊D₄₃	—	8	1	no	—
62	C₂×Dic₄₃	—	2	1	no	—
63	D₁₇₂	—	4	1	no	—
64	D₁₇₂	—	4	2	no	—
65	D₁₇₂	—	8	1	no	—
66	C₄×D₄₃	—	8	1	no	—
67	C₄×D₄₃	—	16	2	no	—
68	Dic₈₆	—	4	1	no	—
69	C₈×C₄₃	—	8	1	no	—
70	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
71	C₈×C₄₃	#135	1	1	yes	characteristic
72	C₄₃⋊C₈	—	2	1	no	—
73	#13658	—	43	1	no	maximal, Sylow
74	C₂²×C₄₃	#509	1	1	yes	characteristic
75	C₂×D₄₃	—	8	2	no	—
76	C₂×D₄₃	—	8	1	no	—
77	C₂×D₄₃	—	16	1	no	—
78	C₂×D₄₃	—	16	1	no	—
79	C₄×C₄₃	—	16	1	no	—
80	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
81	C₄×C₄₃	#509	1	1	yes	characteristic
82	Dic₄₃	—	4	1	no	—
83	Dic₄₃	—	8	1	no	—
84	#2263	—	43	1	no	—
85	#2179	—	43	1	no	—
86	#2177	—	43	1	no	—
87	#1426	D₄₃	1	1	yes	characteristic, complemented
88	#1419	—	43	1	no	—
89	#1418	—	43	1	no	—
90	#1398	—	43	1	no	—
91	C₂×C₄₃	—	2	1	no	—
92	C₂×C₄₃	#2177	1	1	yes	characteristic
93	D₄₃	—	16	1	no	—
94	D₄₃	—	16	2	no	—
95	#508	—	43	1	no	—
96	C₂×D₁₆	—	43	1	no	—
97	#443	—	86	1	no	—
98	C₄×D₈	—	86	1	no	—
99	SD₆₄	—	43	2	no	—
100	D₃₂	—	43	2	no	—
101	M₆₄	—	43	1	no	—
102	M₆₄	C₂×D₄₃	1	1	yes	characteristic
103	#368	C₂×D₄₃	1	1	yes	characteristic
104	#367	C₂×D₄₃	1	1	yes	characteristic
105	#359	—	86	1	no	—
106	#357	—	86	1	no	—
107	#346	—	43	1	no	—
108	C₄₃	#13658	1	1	yes	characteristic, complemented, Sylow
109	#135	—	86	1	no	—
110	C₂×D₈	—	86	1	no	—
111	#131	—	172	1	no	—
112	C₄×D₄	—	172	1	no	—
113	Q₃₂	—	43	1	no	—
114	SD₃₂	—	86	1	no	—
115	D₁₆	—	43	2	no	—
116	D₁₆	—	43	1	no	—
117	D₁₆	—	86	1	no	—
118	C₂×C₁₆	—	43	1	no	—
119	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
120	#108	—	2	1	no	—
121	#107	—	2	1	no	—
122	#104	—	172	1	no	—
123	#102	—	172	1	no	—
124	C₄×C₈	—	43	1	no	—
125	C₃₂	—	86	1	no	—
126	C₃₂	C₄×D₄₃	1	2	yes	complemented
127	#41	—	172	1	no	—
128	C₂×D₄	—	172	1	no	—
129	C₂²×C₄	—	344	1	no	—
130	Q₁₆	—	86	1	no	—
131	SD₁₆	—	172	1	no	—
132	D₈	—	86	2	no	—
133	D₈	—	86	1	no	—
134	D₈	—	172	1	no	—
135	M₁₆	—	4	1	no	—
136	M₁₆	—	344	1	no	—
137	C₂×C₈	—	43	1	no	—
138	C₂×C₈	—	344	1	no	—
139	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
140	C₄⋊C₄	—	4	1	no	—
141	C₂²⋊C₄	—	344	1	no	—
142	C₄²	—	43	1	no	—
143	C₁₆	—	86	1	no	—
144	C₁₆	C₂×C₄×D₄₃	1	1	yes	characteristic
145	C₁₆	#10621902	1	1	yes	characteristic
146	C₂³	—	344	1	no	—
147	Q₈	—	172	1	no	—
148	D₄	—	344	1	no	—
149	D₄	—	344	1	no	—
150	D₄	—	172	2	no	—
151	D₄	—	172	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	86	1	no	—
154	C₂×C₄	—	344	1	no	—
155	C₂×C₄	—	688	2	no	—
156	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
157	C₈	—	8	1	no	—
158	C₈	—	86	1	no	—
159	C₈	#13063287	1	1	yes	characteristic
160	C₈	#13063348	1	1	yes	characteristic
161	C₂²	—	344	1	no	—
162	C₂²	—	344	2	no	—
163	C₂²	—	688	1	no	—
164	C₂²	—	688	1	no	—
165	C₂²	#14583336	1	1	yes	characteristic
166	C₄	—	16	1	no	—
167	C₄	—	344	1	no	—
168	C₄	—	172	1	no	—
169	C₄	#14582981	1	1	yes	characteristic
170	C₄	#14583336	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	688	2	no	—
173	C₂	—	688	1	no	—
174	C₂	#19136470	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19136470
commutator	C₁₆×C₄₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#1426	C₂
socle	C₂×C₄₃	#2177
prime radical	#19136910	C₂
cosocle	C₂×C₁₆×C₄₃	C₂³
hypercenter	#1426	D₄₃
hypocenter	C₄₃	#13658
LCS	G ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1426

Complements

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

N	H ≅ G/N	type
2. #19136480	C₂	semidirect
3. #19136470	C₂	semidirect
4. #19132784	C₂	semidirect
5. #19132781	C₂	semidirect
6. #19132730	C₂	semidirect
7. C₄₃×#1426	C₂	semidirect
12. #14582738	C₄	semidirect
13. D₁₃₇₆	C₄	semidirect
25. Dic₃₄₄	C₂×C₄	semidirect
29. D₆₈₈	C₂×C₄	semidirect
38. C₃₂×C₄₃	C₂×C₄	semidirect
87. #1426	D₄₃	semidirect
108. C₄₃	#13658	semidirect
126. C₃₂	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19136910	2
#19136480	2
#19136470	2
#19132784	2
#19132781	2
#19132730	2
C₄₃×#1426	2
#13658	43