Finite Group Databasebeta

Subgroups — #17989122

Order 4672.

234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14556586	C₂	1	2	yes	maximal, complemented
2	#14556579	C₂	1	1	yes	maximal, characteristic, complemented
3	#14556576	C₂	1	2	yes	maximal, complemented
4	#14556571	C₂	1	1	yes	maximal, characteristic, complemented
5	#14556566	C₂	1	1	yes	maximal, characteristic
6	#14556556	C₂	1	1	yes	maximal, characteristic, complemented
7	#14556553	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556553	C₂	1	2	yes	maximal, complemented
9	#14556549	C₂	1	2	yes	maximal, complemented
10	C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#125×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₁₄₆	C₂²	1	1	yes	characteristic
13	C₂×D₂₉₂	C₂²	1	2	yes	—
14	C₂×D₂₉₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₇₃	C₂²	1	2	yes	—
19	C₄⋊Dic₇₃	C₂²	1	2	yes	—
20	#11924007	C₂²	1	2	yes	—
21	#11924005	C₂²	1	2	yes	—
22	#11924005	C₂²	1	2	yes	—
23	#11924005	C₂²	1	2	yes	—
24	#11924005	C₂²	1	2	yes	—
25	#11924005	C₂²	1	2	yes	—
26	C₄×Dic₇₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₇₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₇₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₇₃	C₂²	1	2	yes	—
30	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
31	C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
33	C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
34	C₄²×C₇₃	C₂²	1	1	yes	characteristic
35	C₂²×D₇₃	C₂³	1	2	yes	—
36	C₂²×D₇₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂³	1	2	yes	—
40	C₂×C₄×C₇₃	C₂³	1	2	yes	—
41	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₇₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₇₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂³	1	2	yes	—
45	C₂×Dic₇₃	C₂³	1	1	yes	characteristic
46	D₂₉₂	—	2	4	no	—
47	D₂₉₂	—	2	2	no	—
48	C₄×D₇₃	—	2	2	no	—
49	C₄×D₇₃	—	2	2	no	—
50	C₄×D₇₃	—	2	2	no	—
51	C₄×D₇₃	—	2	4	no	—
52	Dic₁₄₆	—	2	2	no	—
53	C₂²×C₇₃	C₂⁴	1	1	yes	characteristic
54	C₂×D₇₃	—	2	1	no	—
55	C₂×D₇₃	—	2	1	no	—
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	—	2	2	no	—
58	C₂×D₇₃	—	2	2	no	—
59	C₂×D₇₃	#41	1	2	yes	—
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	—	2	2	no	—
63	C₄×C₇₃	—	2	2	no	—
64	C₄×C₇₃	#41	1	2	yes	—
65	Dic₇₃	—	2	2	no	—
66	Dic₇₃	—	2	1	no	—
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	#41	1	2	yes	—
69	C₂×C₇₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₇₃	C₂×#41	1	1	yes	characteristic
71	C₂×C₇₃	C₂×#41	1	1	yes	characteristic
72	D₇₃	—	2	2	no	—
73	D₇₃	—	4	2	no	—
74	C₇₃	#529	1	1	yes	characteristic, complemented, Sylow
75	#529	—	73	1	no	maximal, Sylow
76	#126	—	73	2	no	—
77	#125	D₇₃	1	1	yes	characteristic, complemented
78	#124	—	73	1	no	—
79	#123	—	73	2	no	—
80	C₂²⋊Q₈	—	73	1	no	—
81	#121	—	73	1	no	—
82	C₄×Q₈	—	73	1	no	—
83	C₄×D₄	—	73	1	no	—
84	C₄×D₄	—	73	2	no	—
85	#117	—	73	2	no	—
86	C₂×C₄²	—	73	1	no	—
87	C₂×Q₈	—	73	1	no	—
88	C₂×D₄	—	73	2	no	—
89	C₂×D₄	—	73	1	no	—
90	C₂²×C₄	—	73	1	no	—
91	C₂²×C₄	—	73	1	no	—
92	C₂²×C₄	—	73	1	no	—
93	C₂²×C₄	—	73	2	no	—
94	C₄⋊C₄	—	73	2	no	—
95	C₄⋊C₄	—	73	2	no	—
96	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
97	C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
99	C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
100	C₂²⋊C₄	—	73	2	no	—
101	C₂²⋊C₄	—	73	2	no	—
102	C₂²⋊C₄	—	73	2	no	—
103	C₂²⋊C₄	—	73	2	no	—
104	C₂²⋊C₄	—	73	2	no	—
105	C₄²	—	73	1	no	—
106	C₄²	—	73	1	no	—
107	C₄²	—	73	1	no	—
108	C₄²	—	73	2	no	—
109	C₄²	C₂×D₇₃	1	1	yes	characteristic
110	C₂³	—	73	1	no	—
111	C₂³	—	73	2	no	—
112	Q₈	—	146	2	no	—
113	D₄	—	146	4	no	—
114	D₄	—	146	2	no	—
115	C₂×C₄	—	73	1	no	—
116	C₂×C₄	—	73	1	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	73	2	no	—
119	C₂×C₄	—	146	4	no	—
120	C₂×C₄	—	146	2	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	—	146	2	no	—
123	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₇₃	1	2	yes	—
126	C₂×C₄	C₂²×D₇₃	1	2	yes	—
127	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
128	C₂²	—	73	2	no	—
129	C₂²	—	146	2	no	—
130	C₂²	—	146	2	no	—
131	C₂²	—	146	2	no	—
132	C₂²	—	146	1	no	—
133	C₂²	—	146	1	no	—
134	C₂²	C₂³×D₇₃	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	2	no	—
139	C₄	—	73	2	no	—
140	C₄	—	146	1	no	—
141	C₄	—	146	1	no	—
142	C₄	—	146	2	no	—
143	C₄	#11924029	1	2	yes	—
144	C₂	—	292	2	no	—
145	C₂	—	146	2	no	—
146	C₂	#41×D₇₃	1	1	yes	characteristic
147	C₂	#41×D₇₃	1	1	yes	characteristic
148	C₂	C₂×#11924029	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₃
commutator, socle, cosocle	C₂²×C₇₃	C₂⁴
Fitting	#125×C₇₃	C₂
prime radical	#14556566	C₂
hypercenter	#125	D₇₃
hypocenter	C₇₃	#529
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #14556586	C₂	semidirect
2. #14556579	C₂	semidirect
3. #14556576	C₂	semidirect
4. #14556571	C₂	semidirect
6. #14556556	C₂	semidirect
7. #14556553	C₂	semidirect
8. #14556553	C₂	semidirect
9. #14556549	C₂	semidirect
10. C₄²×D₇₃	C₂	semidirect
11. #125×C₇₃	C₂	semidirect
74. C₇₃	#529	semidirect
77. #125	D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#14556586	2
#14556579	2
#14556576	2
#14556571	2
#14556566	2
#14556556	2
#14556553	2
#14556549	2
C₄²×D₇₃	2
#125×C₇₃	2
#529	73