Finite Group Databasebeta

Subgroups — #21426061

Order 7552.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16838216	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14273672	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837673	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837217	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837185	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273774	—	2	1	no	—
9	#14273757	—	2	1	no	—
10	#14273756	—	2	2	no	—
11	#14273755	—	2	1	no	—
12	#14273746	—	2	1	no	—
13	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
14	Q₁₆×D₅₉	—	2	1	no	—
15	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
16	#14273733	C₂²	1	1	yes	characteristic
17	#14273672	—	2	2	no	—
18	#14273670	—	2	1	no	—
19	#14273668	—	2	1	no	—
20	#14273667	C₂²	1	1	yes	characteristic
21	#14273643	—	2	1	no	—
22	#135×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	#108×C₅₉	C₂²	1	1	yes	characteristic
25	#41×C₅₉	—	2	1	no	—
26	Q₈×D₅₉	—	2	2	no	—
27	#11745687	—	2	2	no	—
28	#11745687	—	4	2	no	—
29	#11745685	—	4	1	no	—
30	#11745685	D₄	1	1	yes	characteristic
31	C₂×D₂₃₆	—	2	1	no	—
32	C₂×Dic₁₁₈	—	2	1	no	—
33	C₂×Dic₁₁₈	—	4	1	no	—
34	Q₁₆×C₅₉	D₄	1	1	yes	characteristic
35	SD₁₆×C₅₉	—	2	1	no	—
36	D₈×C₅₉	D₄	1	1	yes	characteristic
37	M₁₆×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	#11745672	—	2	2	no	—
40	#11745670	—	2	2	no	—
41	#11745664	D₄	1	1	yes	characteristic
42	Dic₂₃₆	—	2	1	no	—
43	Dic₂₃₆	D₄	1	2	yes	—
44	D₄₇₂	—	2	1	no	—
45	D₄₇₂	—	2	1	no	—
46	D₄₇₂	—	4	1	no	—
47	#11745660	—	4	1	no	—
48	#11745659	—	2	1	no	—
49	C₈×D₅₉	—	2	1	no	—
50	C₁₆×C₅₉	—	2	1	no	—
51	C₅₉⋊C₁₆	—	2	1	no	—
52	C₂²×D₅₉	—	4	1	no	—
53	Q₈×C₅₉	—	2	1	no	—
54	D₄×C₅₉	—	2	1	no	—
55	D₄×C₅₉	—	4	1	no	—
56	C₂×C₄×C₅₉	—	4	1	no	—
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₅₉	—	2	1	no	—
59	C₂²⋊D₅₉	—	4	2	no	—
60	C₂×Dic₅₉	—	4	2	no	—
61	C₂×Dic₅₉	—	4	1	no	—
62	D₂₃₆	—	4	1	no	—
63	D₂₃₆	—	4	1	no	—
64	D₂₃₆	D₈	1	1	yes	characteristic
65	C₄×D₅₉	—	2	1	no	—
66	C₄×D₅₉	—	4	2	no	—
67	Dic₁₁₈	—	2	2	no	—
68	Dic₁₁₈	—	4	2	no	—
69	Dic₁₁₈	—	4	1	no	—
70	Dic₁₁₈	D₈	1	1	yes	characteristic
71	C₈×C₅₉	—	4	1	no	—
72	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₅₉⋊C₈	—	2	1	no	—
75	C₂²×C₅₉	—	4	1	no	—
76	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₅₉	—	2	1	no	—
78	C₂×D₅₉	—	8	1	no	—
79	C₂×D₅₉	—	16	1	no	—
80	C₄×C₅₉	—	4	1	no	—
81	C₄×C₅₉	#120	1	1	yes	characteristic
82	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
83	Dic₅₉	—	2	1	no	—
84	Dic₅₉	—	4	2	no	—
85	#2193	—	59	1	no	maximal, Sylow
86	C₂×C₅₉	—	2	1	no	—
87	C₂×C₅₉	—	8	1	no	—
88	C₂×C₅₉	#447	1	1	yes	characteristic
89	D₅₉	—	4	1	no	—
90	D₅₉	—	16	1	no	—
91	#578	—	59	1	no	—
92	#509	—	59	1	no	—
93	C₂×SD₃₂	—	59	1	no	—
94	#472	—	59	1	no	—
95	#362	—	59	1	no	—
96	#359	D₅₉	1	1	yes	characteristic, complemented
97	#350	—	59	1	no	—
98	C₅₉	#2193	1	1	yes	characteristic, complemented, Sylow
99	2₋¹⁺⁴	—	118	1	no	—
100	#137	—	118	1	no	—
101	#137	—	118	2	no	—
102	AGL(1,ℤ/8)	—	118	1	no	—
103	#135	—	118	1	no	—
104	#135	C₂×D₅₉	1	1	yes	characteristic
105	C₂×Q₁₆	—	59	1	no	—
106	C₂×Q₁₆	—	118	1	no	—
107	C₂×D₈	—	59	1	no	—
108	#131	—	59	1	no	—
109	SD₃₂	—	118	2	no	—
110	SD₃₂	—	118	1	no	—
111	D₁₆	—	118	1	no	—
112	M₃₂	—	59	1	no	—
113	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
114	#108	C₂×D₅₉	1	1	yes	characteristic
115	#100	—	118	1	no	—
116	#41	—	2	1	no	—
117	#41	—	59	1	no	—
118	#41	—	118	2	no	—
119	#41	—	236	1	no	—
120	#41	—	236	2	no	—
121	C₂×Q₈	—	118	2	no	—
122	C₂×Q₈	—	118	1	no	—
123	C₂×Q₈	—	236	1	no	—
124	C₂×D₄	—	118	1	no	—
125	Q₁₆	—	59	2	no	—
126	Q₁₆	—	118	2	no	—
127	Q₁₆	—	118	1	no	—
128	Q₁₆	D₂₃₆	1	1	yes	characteristic
129	SD₁₆	—	2	1	no	—
130	SD₁₆	—	118	2	no	—
131	SD₁₆	—	236	1	no	—
132	D₈	—	118	1	no	—
133	D₈	—	118	1	no	—
134	D₈	—	236	1	no	—
135	D₈	D₂₃₆	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	59	1	no	—
138	M₁₆	—	118	1	no	—
139	C₂×C₈	—	118	1	no	—
140	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	118	1	no	—
143	C₂³	—	236	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	59	1	no	—
146	Q₈	—	118	2	no	—
147	Q₈	—	236	2	no	—
148	Q₈	—	236	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	59	1	no	—
152	D₄	—	118	1	no	—
153	D₄	—	236	2	no	—
154	D₄	—	236	1	no	—
155	D₄	—	236	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	118	1	no	—
158	C₂×C₄	—	236	2	no	—
159	C₂×C₄	—	236	2	no	—
160	C₂×C₄	—	236	1	no	—
161	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
162	C₈	—	4	1	no	—
163	C₈	—	118	1	no	—
164	C₈	C₂×D₂₃₆	1	1	yes	characteristic
165	C₈	D₄×D₅₉	1	1	yes	characteristic
166	C₂²	—	4	1	no	—
167	C₂²	—	118	1	no	—
168	C₂²	—	944	1	no	—
169	C₂²	—	472	1	no	—
170	C₂²	#14273754	1	1	yes	characteristic
171	C₄	—	4	1	no	—
172	C₄	—	118	1	no	—
173	C₄	—	236	2	no	—
174	C₄	#14273697	1	1	yes	characteristic
175	C₄	D₈×D₅₉	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	—	944	1	no	—
179	C₂	—	236	1	no	—
180	C₂	#16837537	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16837537
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#359	C₂
socle	C₂×C₅₉	#447
hypercenter	#359	D₅₉
hypocenter	C₅₉	#2193
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
1. #16838216	C₂	semidirect
2. #16837837	C₂	semidirect
3. C₂×#14273672	C₂	semidirect
4. #16837673	C₂	semidirect
5. #16837217	C₂	semidirect
6. #16837185	C₂	semidirect
7. C₅₉×#359	C₂	semidirect
96. #359	D₅₉	semidirect
98. C₅₉	#2193	semidirect