Finite Group Databasebeta

Subgroups — #21426076

Order 7552.

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

normal subgroups only

This group has 206 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	#16838219	C₂	1	1	yes	maximal, characteristic
2	#16837845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837677	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837219	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837210	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273774	—	2	1	no	—
9	#14273760	—	2	1	no	—
10	#14273760	—	2	2	no	—
11	#14273757	—	2	1	no	—
12	#14273749	—	2	1	no	—
13	#14273749	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₃₆	—	2	1	no	—
15	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
17	Dic₄₇₂	—	2	2	no	—
18	Dic₄₇₂	—	2	1	no	—
19	#14273672	—	2	1	no	—
20	#14273663	C₂²	1	1	yes	characteristic
21	#14273645	—	2	1	no	—
22	#131×C₅₉	C₂²	1	1	yes	characteristic
23	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	#41×C₅₉	D₄	1	1	yes	characteristic
26	Q₈×D₅₉	—	4	1	no	—
27	#11745687	—	4	2	no	—
28	#11745687	—	4	1	no	—
29	#11745685	—	2	2	no	—
30	#11745685	—	2	1	no	—
31	C₂×Dic₁₁₈	—	2	2	no	—
32	C₂×Dic₁₁₈	—	2	1	no	—
33	C₂×Dic₁₁₈	—	2	1	no	—
34	M₁₆×C₅₉	—	2	1	no	—
35	M₁₆×C₅₉	D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
38	#11745672	—	4	1	no	—
39	#11745670	—	4	1	no	—
40	#11745664	—	2	1	no	—
41	Dic₂₃₆	—	2	2	no	—
42	Dic₂₃₆	—	2	1	no	—
43	Dic₂₃₆	—	2	1	no	—
44	Dic₂₃₆	—	2	1	no	—
45	Dic₂₃₆	D₄	1	1	yes	characteristic
46	Dic₂₃₆	D₄	1	2	yes	—
47	D₄₇₂	D₄	1	1	yes	characteristic
48	#11745660	—	2	2	no	—
49	#11745660	—	2	1	no	—
50	C₁₆×C₅₉	—	2	1	no	—
51	C₁₆×C₅₉	—	2	1	no	—
52	Q₈×C₅₉	D₈	1	1	yes	characteristic
53	D₄×C₅₉	—	2	1	no	—
54	D₄×C₅₉	D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	1	no	—
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₅₉	—	4	2	no	—
58	C₂²⋊D₅₉	—	4	1	no	—
59	C₂×Dic₅₉	—	4	2	no	—
60	C₂×Dic₅₉	—	4	1	no	—
61	C₂×Dic₅₉	—	4	1	no	—
62	D₂₃₆	—	2	1	no	—
63	C₄×D₅₉	—	4	2	no	—
64	C₄×D₅₉	—	4	1	no	—
65	Dic₁₁₈	—	2	2	no	—
66	Dic₁₁₈	—	2	1	no	—
67	Dic₁₁₈	—	4	2	no	—
68	Dic₁₁₈	—	4	1	no	—
69	Dic₁₁₈	—	4	1	no	—
70	Dic₁₁₈	—	4	1	no	—
71	C₈×C₅₉	—	2	1	no	—
72	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₅₉⋊C₈	—	4	1	no	—
75	C₂²×C₅₉	—	2	1	no	—
76	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₅₉	—	4	1	no	—
78	C₄×C₅₉	—	2	1	no	—
79	C₄×C₅₉	#120	1	1	yes	characteristic
80	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
81	Dic₅₉	—	4	1	no	—
82	Dic₅₉	—	4	2	no	—
83	Dic₅₉	—	8	1	no	—
84	#2195	—	59	1	no	maximal, Sylow
85	C₂×C₅₉	—	2	1	no	—
86	C₂×C₅₉	—	4	1	no	—
87	C₂×C₅₉	#447	1	1	yes	characteristic
88	D₅₉	—	8	1	no	—
89	#578	—	59	1	no	—
90	#510	—	59	1	no	—
91	C₂×Q₃₂	—	59	1	no	—
92	#473	—	59	1	no	—
93	#362	—	59	1	no	—
94	#359	—	59	1	no	—
95	#350	D₅₉	1	1	yes	characteristic, complemented
96	C₅₉	#2195	1	1	yes	characteristic, complemented, Sylow
97	2₋¹⁺⁴	—	118	1	no	—
98	#137	—	118	2	no	—
99	#137	—	118	1	no	—
100	#137	—	118	1	no	—
101	#135	—	59	1	no	—
102	#135	—	118	1	no	—
103	C₂×Q₁₆	—	59	1	no	—
104	C₂×Q₁₆	—	59	1	no	—
105	C₂×Q₁₆	—	118	1	no	—
106	#131	C₂×D₅₉	1	1	yes	characteristic
107	Q₃₂	—	118	2	no	—
108	Q₃₂	—	118	1	no	—
109	SD₃₂	—	118	1	no	—
110	M₃₂	C₂×D₅₉	1	1	yes	characteristic
111	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
112	#108	—	59	1	no	—
113	#101	—	118	1	no	—
114	#41	—	118	2	no	—
115	#41	—	118	1	no	—
116	#41	—	236	2	no	—
117	#41	—	236	1	no	—
118	#41	D₂₃₆	1	1	yes	characteristic
119	C₂×Q₈	—	118	2	no	—
120	C₂×Q₈	—	118	1	no	—
121	C₂×Q₈	—	118	1	no	—
122	C₂×Q₈	—	236	1	no	—
123	Q₁₆	—	59	2	no	—
124	Q₁₆	—	59	1	no	—
125	Q₁₆	—	118	2	no	—
126	Q₁₆	—	118	1	no	—
127	Q₁₆	—	118	1	no	—
128	Q₁₆	—	118	1	no	—
129	Q₁₆	—	236	1	no	—
130	SD₁₆	—	118	2	no	—
131	SD₁₆	—	118	1	no	—
132	SD₁₆	—	236	1	no	—
133	D₈	—	59	1	no	—
134	M₁₆	—	2	1	no	—
135	M₁₆	—	118	1	no	—
136	M₁₆	D₂₃₆	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	1	no	—
141	Q₈	—	118	2	no	—
142	Q₈	—	118	1	no	—
143	Q₈	—	236	2	no	—
144	Q₈	—	236	1	no	—
145	Q₈	—	236	1	no	—
146	Q₈	—	236	1	no	—
147	Q₈	D₄₇₂	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	118	1	no	—
150	D₄	—	236	2	no	—
151	D₄	—	236	1	no	—
152	D₄	D₄₇₂	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	236	2	no	—
155	C₂×C₄	—	236	1	no	—
156	C₂×C₄	—	236	2	no	—
157	C₂×C₄	—	236	1	no	—
158	C₂×C₄	—	236	1	no	—
159	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	236	1	no	—
162	C₈	D₄×D₅₉	1	1	yes	characteristic
163	C₈	D₄×D₅₉	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	236	1	no	—
166	C₂²	#14273755	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	472	1	no	—
169	C₄	—	236	2	no	—
170	C₄	—	236	1	no	—
171	C₄	#14273697	1	1	yes	characteristic
172	C₄	C₂×D₄₇₂	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	472	1	no	—
176	C₂	#16837538	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16837538
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#350	C₂
socle	C₂×C₅₉	#447
hypercenter	#350	D₅₉
hypocenter	C₅₉	#2195
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
2. #16837845	C₂	semidirect
3. C₂×Dic₄₇₂	C₂	semidirect
4. #16837677	C₂	semidirect
5. #16837219	C₂	semidirect
6. #16837210	C₂	semidirect
7. C₅₉×#350	C₂	semidirect
95. #350	D₅₉	semidirect
96. C₅₉	#2195	semidirect