Finite Group Databasebeta

Subgroups — #21426044

Order 7552.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16838202	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837840	C₂	1	2	yes	maximal, complemented
3	#16837655	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837213	C₂	1	2	yes	maximal, complemented
5	C₅₉×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	—	2	1	no	—
7	#14273760	—	2	2	no	—
8	#14273755	—	2	2	no	—
9	#14273749	—	2	1	no	—
10	#14273749	C₂²	1	2	yes	—
11	C₂×D₄₇₂	C₂²	1	1	yes	characteristic, complemented
12	#14273701	—	2	1	no	—
13	#14273672	—	2	2	no	—
14	D₉₄₄	—	2	2	no	—
15	#14273659	C₂²	1	1	yes	characteristic, complemented
16	#14273649	—	2	1	no	—
17	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	M₃₂×C₅₉	C₂²	1	2	yes	complemented
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₉	—	4	1	no	—
21	C₂×C₂²⋊D₅₉	—	4	1	no	—
22	#11745685	—	2	2	no	—
23	#11745685	—	2	2	no	—
24	#11745685	—	4	1	no	—
25	#11745685	—	4	1	no	—
26	C₂×D₂₃₆	—	2	1	no	—
27	C₂×D₂₃₆	—	2	1	no	—
28	C₂×C₄×D₅₉	—	4	1	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	M₁₆×C₅₉	—	2	2	no	—
31	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	(C₂×D₅₉)⋊C₄	—	4	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	Dic₂₃₆	—	2	1	no	—
36	Dic₂₃₆	D₄	1	2	yes	complemented
37	D₄₇₂	—	2	1	no	—
38	D₄₇₂	—	2	2	no	—
39	D₄₇₂	D₄	1	2	yes	complemented
40	#11745660	—	2	2	no	—
41	#11745660	—	2	2	no	—
42	C₁₆×C₅₉	—	2	2	no	—
43	C₂³×C₅₉	D₈	1	1	yes	characteristic
44	C₂²×D₅₉	—	4	1	no	—
45	C₂²×D₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₅₉	—	4	2	no	—
51	C₂²⋊D₅₉	—	4	2	no	—
52	C₂²⋊D₅₉	—	8	2	no	—
53	C₂×Dic₅₉	—	4	1	no	—
54	C₂×Dic₅₉	—	4	1	no	—
55	D₂₃₆	—	2	2	no	—
56	D₂₃₆	—	4	1	no	—
57	D₂₃₆	—	4	1	no	—
58	D₂₃₆	—	4	1	no	—
59	C₄×D₅₉	—	4	2	no	—
60	C₄×D₅₉	—	4	2	no	—
61	Dic₁₁₈	—	2	2	no	—
62	Dic₁₁₈	—	4	1	no	—
63	C₈×C₅₉	—	2	1	no	—
64	C₈×C₅₉	C₂×D₄	1	2	yes	—
65	C₂²×C₅₉	—	2	1	no	—
66	C₂²×C₅₉	—	4	1	no	—
67	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
68	C₂×D₅₉	—	4	2	no	—
69	C₂×D₅₉	—	8	2	no	—
70	C₂×D₅₉	—	8	1	no	—
71	C₂×D₅₉	—	8	1	no	—
72	C₂×D₅₉	—	8	1	no	—
73	C₄×C₅₉	—	2	1	no	—
74	C₄×C₅₉	#120	1	1	yes	characteristic
75	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₅₉	—	4	2	no	—
77	Dic₅₉	—	8	1	no	—
78	#2190	—	59	1	no	maximal, Sylow
79	C₂×C₅₉	—	2	1	no	—
80	C₂×C₅₉	—	4	1	no	—
81	C₂×C₅₉	#447	1	1	yes	characteristic
82	D₅₉	—	8	2	no	—
83	D₅₉	—	16	1	no	—
84	#575	—	59	1	no	—
85	#509	—	59	2	no	—
86	#469	—	59	1	no	—
87	#360	—	59	2	no	—
88	#349	D₅₉	1	1	yes	characteristic, complemented
89	C₅₉	#2190	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	118	1	no	—
91	#137	—	118	2	no	—
92	AGL(1,ℤ/8)	—	118	2	no	—
93	#135	—	59	2	no	—
94	#135	—	118	1	no	—
95	C₂×D₈	—	59	1	no	—
96	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
97	#121	—	118	1	no	—
98	SD₃₂	—	118	2	no	—
99	D₁₆	—	118	2	no	—
100	M₃₂	C₂×D₅₉	1	2	yes	complemented
101	#106	—	59	1	no	—
102	#102	—	118	1	no	—
103	#41	—	118	2	no	—
104	#41	—	118	2	no	—
105	#41	—	236	1	no	—
106	#41	—	236	1	no	—
107	C₂×Q₈	—	118	1	no	—
108	C₂×D₄	—	118	1	no	—
109	C₂×D₄	—	118	1	no	—
110	C₂×D₄	—	236	1	no	—
111	C₂×D₄	—	236	1	no	—
112	C₂²×C₄	—	236	1	no	—
113	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
114	Q₁₆	—	59	2	no	—
115	Q₁₆	—	118	1	no	—
116	SD₁₆	—	118	2	no	—
117	SD₁₆	—	118	2	no	—
118	D₈	—	59	2	no	—
119	D₈	—	118	1	no	—
120	D₈	—	118	2	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	D₂₃₆	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊C₄	—	118	1	no	—
125	C₂²⋊C₄	—	236	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂³	—	236	1	no	—
128	C₂³	—	236	1	no	—
129	C₂³	D₄₇₂	1	1	yes	characteristic
130	Q₈	—	118	2	no	—
131	Q₈	—	236	1	no	—
132	D₄	—	118	2	no	—
133	D₄	—	472	2	no	—
134	D₄	—	236	1	no	—
135	D₄	—	236	2	no	—
136	D₄	—	236	2	no	—
137	D₄	—	236	1	no	—
138	D₄	—	236	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	236	2	no	—
142	C₂×C₄	—	236	1	no	—
143	C₂×C₄	—	236	1	no	—
144	C₂×C₄	—	236	2	no	—
145	C₂×C₄	D₄₇₂	1	1	yes	characteristic
146	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	D₄×D₅₉	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	472	1	no	—
152	C₂²	—	472	1	no	—
153	C₂²	—	472	2	no	—
154	C₂²	—	472	1	no	—
155	C₂²	—	236	2	no	—
156	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	472	1	no	—
159	C₄	—	236	2	no	—
160	C₄	#14273697	1	1	yes	characteristic
161	C₄	#14273755	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	944	1	no	—
165	C₂	—	472	2	no	—
166	C₂	#16837538	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16837538
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#349	C₂
socle	C₂×C₅₉	#447
hypercenter	#349	D₅₉
hypocenter	C₅₉	#2190
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
1. #16838202	C₂	semidirect
2. #16837840	C₂	semidirect
3. #16837655	C₂	semidirect
4. #16837213	C₂	semidirect
5. C₅₉×#349	C₂	semidirect
11. C₂×D₄₇₂	C₂²	semidirect
15. #14273659	C₂²	semidirect
18. M₃₂×C₅₉	C₂²	semidirect
36. Dic₂₃₆	D₄	semidirect
39. D₄₇₂	D₄	semidirect
88. #349	D₅₉	semidirect
89. C₅₉	#2190	semidirect
100. M₃₂	C₂×D₅₉	semidirect