Finite Group Databasebeta

Subgroups — #17989384

Order 4672.

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

normal subgroups only

This group has 262 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	#41×D₇₃	C₂	1	2	yes	maximal, complemented
2	#14556623	C₂	1	2	yes	maximal, complemented
3	#14556620	C₂	1	2	yes	maximal, complemented
4	#14556618	C₂	1	2	yes	maximal, complemented
5	#14556615	C₂	1	2	yes	maximal, complemented
6	#14556613	C₂	1	1	yes	maximal, characteristic, complemented
7	#14556611	C₂	1	2	yes	maximal, complemented
8	C₂×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
9	#135×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#11924029	C₂²	1	2	yes	—
11	#11924028	—	2	2	no	—
12	#11924028	C₂²	1	2	yes	complemented
13	#11924027	—	2	2	no	—
14	#11924027	C₂²	1	2	yes	—
15	Q₈×D₇₃	C₂²	1	2	yes	complemented
16	D₄×D₇₃	—	2	2	no	—
17	D₄×D₇₃	C₂²	1	2	yes	complemented
18	C₂×C₄×D₇₃	—	2	2	no	—
19	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
20	Dic₂₉₂	C₂²	1	1	yes	characteristic, complemented
21	#11924018	C₂²	1	2	yes	complemented
22	#11924017	C₂²	1	2	yes	complemented
23	#11924016	C₂²	1	2	yes	complemented
24	#11924015	C₂²	1	2	yes	complemented
25	D₅₈₄	C₂²	1	1	yes	characteristic, complemented
26	#11924013	C₂²	1	2	yes	complemented
27	#11924011	C₂²	1	2	yes	complemented
28	#11924011	C₂²	1	2	yes	complemented
29	C₂×#10596831	C₂²	1	1	yes	characteristic, complemented
30	#41×C₇₃	C₂²	1	2	yes	complemented
31	Q₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
32	SD₁₆×C₇₃	C₂²	1	2	yes	complemented
33	D₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
35	C₂²×D₇₃	—	2	2	no	—
36	C₂²×D₇₃	D₄	1	1	yes	characteristic
37	Q₈×C₇₃	C₂³	1	2	yes	—
38	D₄×C₇₃	—	2	2	no	—
39	D₄×C₇₃	C₂³	1	2	yes	—
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	C₂²⋊D₇₃	—	2	2	no	—
43	C₂²⋊D₇₃	—	2	2	no	—
44	C₂²⋊D₇₃	—	2	2	no	—
45	C₂×Dic₇₃	—	2	2	no	—
46	C₂×Dic₇₃	D₄	1	1	yes	characteristic
47	D₂₉₂	—	2	2	no	—
48	D₂₉₂	C₂³	1	2	yes	—
49	C₄×D₇₃	—	2	2	no	—
50	C₄×D₇₃	—	2	2	no	—
51	C₄×D₇₃	—	2	2	no	—
52	C₄×D₇₃	D₄	1	2	yes	—
53	C₄×D₇₃	C₂³	1	2	yes	—
54	Dic₁₄₆	—	2	2	no	—
55	Dic₁₄₆	C₂³	1	2	yes	—
56	C₈×C₇₃	C₂³	1	1	yes	characteristic
57	C₈×C₇₃	C₂³	1	1	yes	characteristic
58	#10596831	C₂³	1	2	yes	—
59	C₂²×C₇₃	—	2	2	no	—
60	C₂²×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×D₇₃	—	2	2	no	—
62	C₂×D₇₃	—	4	1	no	—
63	C₂×D₇₃	—	4	4	no	—
64	C₂×D₇₃	C₂×D₄	1	2	yes	—
65	C₄×C₇₃	—	2	2	no	—
66	C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄×C₇₃	C₂⁴	1	1	yes	characteristic
68	Dic₇₃	—	2	2	no	—
69	Dic₇₃	C₂×D₄	1	2	yes	—
70	C₂×C₇₃	—	2	1	no	—
71	C₂×C₇₃	—	4	2	no	—
72	C₂×C₇₃	C₂²×D₄	1	1	yes	characteristic
73	D₇₃	—	2	2	no	—
74	D₇₃	—	4	2	no	—
75	C₇₃	#575	1	1	yes	characteristic, complemented, Sylow
76	#575	—	73	1	no	maximal, Sylow
77	C₂×#41	—	73	2	no	—
78	#137	—	73	2	no	—
79	#137	—	73	2	no	—
80	AGL(1,ℤ/8)	—	73	2	no	—
81	AGL(1,ℤ/8)	—	73	2	no	—
82	#135	—	73	2	no	—
83	#135	—	73	1	no	—
84	#135	D₇₃	1	1	yes	characteristic, complemented
85	C₂×M₁₆	—	73	1	no	—
86	#41	—	73	2	no	—
87	#41	—	73	2	no	—
88	#41	—	73	2	no	—
89	#41	—	146	2	no	—
90	#41	—	146	2	no	—
91	#41	C₂×D₇₃	1	2	yes	complemented
92	C₂×Q₈	—	73	2	no	—
93	C₂×D₄	—	73	2	no	—
94	C₂×D₄	—	146	2	no	—
95	C₂²×C₄	—	73	1	no	—
96	C₂²×C₄	—	146	2	no	—
97	Q₁₆	—	73	2	no	—
98	Q₁₆	—	73	1	no	—
99	Q₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
100	SD₁₆	—	73	2	no	—
101	SD₁₆	—	73	2	no	—
102	SD₁₆	—	73	2	no	—
103	SD₁₆	C₂×D₇₃	1	2	yes	complemented
104	D₈	—	73	2	no	—
105	D₈	—	73	1	no	—
106	D₈	C₂×D₇₃	1	1	yes	characteristic, complemented
107	M₁₆	—	73	2	no	—
108	M₁₆	—	73	2	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	C₂×D₇₃	1	1	yes	characteristic, complemented
111	C₂³	—	73	1	no	—
112	C₂³	—	146	2	no	—
113	Q₈	—	73	2	no	—
114	Q₈	—	146	2	no	—
115	Q₈	C₂²×D₇₃	1	2	yes	—
116	D₄	—	2	2	no	—
117	D₄	—	73	2	no	—
118	D₄	—	146	2	no	—
119	D₄	—	146	2	no	—
120	D₄	—	146	2	no	—
121	D₄	—	146	2	no	—
122	D₄	C₂²×D₇₃	1	2	yes	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	2	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	146	2	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	—	146	2	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
132	C₈	—	73	2	no	—
133	C₈	C₂²×D₇₃	1	1	yes	characteristic
134	C₈	C₂²×D₇₃	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	292	1	no	—
137	C₂²	—	292	4	no	—
138	C₂²	—	73	2	no	—
139	C₂²	—	146	2	no	—
140	C₂²	D₄×D₇₃	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	73	2	no	—
143	C₄	—	146	2	no	—
144	C₄	D₄×D₇₃	1	1	yes	characteristic
145	C₄	C₂³×D₇₃	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	—	292	2	no	—
149	C₂	—	146	2	no	—
150	C₂	C₂×D₄×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₇₃
commutator, cosocle	C₄×C₇₃	C₂⁴
Frattini	C₄	C₂³×D₇₃
Fitting	#135×C₇₃	C₂
socle	C₂×C₇₃	C₂²×D₄
hypercenter	#135	D₇₃
hypocenter	C₇₃	#575
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×D₇₃	C₂	semidirect
2. #14556623	C₂	semidirect
3. #14556620	C₂	semidirect
4. #14556618	C₂	semidirect
5. #14556615	C₂	semidirect
6. #14556613	C₂	semidirect
7. #14556611	C₂	semidirect
8. C₂×#11924011	C₂	semidirect
9. #135×C₇₃	C₂	semidirect
12. #11924028	C₂²	semidirect
15. Q₈×D₇₃	C₂²	semidirect
17. D₄×D₇₃	C₂²	semidirect
20. Dic₂₉₂	C₂²	semidirect
21. #11924018	C₂²	semidirect
22. #11924017	C₂²	semidirect
23. #11924016	C₂²	semidirect
24. #11924015	C₂²	semidirect
25. D₅₈₄	C₂²	semidirect
26. #11924013	C₂²	semidirect
27. #11924011	C₂²	semidirect
28. #11924011	C₂²	semidirect
29. C₂×#10596831	C₂²	semidirect
30. #41×C₇₃	C₂²	semidirect
31. Q₁₆×C₇₃	C₂²	semidirect
32. SD₁₆×C₇₃	C₂²	semidirect
33. D₈×C₇₃	C₂²	semidirect
34. C₂×C₈×C₇₃	C₂²	semidirect
75. C₇₃	#575	semidirect
84. #135	D₇₃	semidirect
91. #41	C₂×D₇₃	semidirect
99. Q₁₆	C₂×D₇₃	semidirect
103. SD₁₆	C₂×D₇₃	semidirect
106. D₈	C₂×D₇₃	semidirect
110. C₂×C₈	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#41×D₇₃	2
#14556623	2
#14556620	2
#14556618	2
#14556615	2
#14556613	2
#14556611	2
C₂×#11924011	2
#135×C₇₃	2
#575	73