Finite Group Databasebeta

Subgroups — #34033885

Order 18688.

270 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 105 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	#22627120	C₂	1	2	yes	maximal
2	#22626990	C₂	1	4	yes	maximal, complemented
3	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556688	—	2	2	no	—
5	C₄⋊#11924040	—	2	2	no	—
6	C₄⋊#11924040	C₂²	1	4	yes	—
7	C₄×#11924036	C₄	1	1	yes	characteristic
8	C₄×#11924036	C₂²	1	1	yes	characteristic
9	C₈×#10596833	C₄	1	2	yes	—
10	C₈×#10596833	C₂²	1	2	yes	—
11	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
12	#14556688	—	4	4	no	—
13	C₂×#11924040	—	2	4	no	—
14	C₂×#11924036	—	2	1	no	—
15	C₂×#11924036	D₄	1	1	yes	characteristic
16	C₂×#11924036	Q₈	1	1	yes	characteristic
17	C₄×#10596833	—	2	1	no	—
18	C₄×#10596833	C₂×C₄	1	2	yes	—
19	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
20	C₄×#10596833	D₄	1	1	yes	characteristic
21	C₄×#10596833	Q₈	1	1	yes	characteristic
22	C₄×#10596833	C₂³	1	1	yes	characteristic
23	C₂×C₈×D₇₃	C₈	1	2	yes	—
24	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₈	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
28	#11924040	—	4	4	no	—
29	#11924036	—	2	4	no	—
30	#11924036	—	2	2	no	—
31	#11924036	SD₁₆	1	4	yes	—
32	C₂×#10596833	—	2	2	no	—
33	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
34	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
36	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
40	C₂×#10596831	C₂×C₈	1	2	yes	—
41	C₈×D₇₃	—	2	4	no	—
42	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
44	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
46	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
54	C₄×D₇₃	—	2	2	no	—
55	C₄×D₇₃	—	2	2	no	—
56	C₄×D₇₃	#106	1	2	yes	—
57	C₄×D₇₃	#108	1	2	yes	—
58	#10596833	—	2	4	no	—
59	#10596833	—	2	1	no	—
60	#10596833	—	2	1	no	—
61	#10596833	#106	1	2	yes	—
62	#10596833	C₂×SD₁₆	1	2	yes	—
63	C₈×C₇₃	—	2	2	no	—
64	#10596831	M₃₂	1	4	yes	complemented
65	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
66	C₂×D₇₃	—	2	1	no	—
67	C₂×D₇₃	—	2	1	no	—
68	C₂×D₇₃	#334	1	2	yes	—
69	C₄×C₇₃	#334	1	2	yes	—
70	C₄×C₇₃	#364	1	2	yes	—
71	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
72	Dic₇₃	—	2	1	no	—
73	Dic₇₃	—	2	1	no	—
74	Dic₇₃	C₂×#106	1	1	yes	characteristic
75	Dic₇₃	C₂×#108	1	1	yes	characteristic
76	#11996	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
78	C₂×C₇₃	#2150	1	1	yes	characteristic
79	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
80	D₇₃	—	2	2	no	—
81	#2150	—	73	2	no	—
82	C₂×C₈²	—	73	1	no	—
83	#1367	—	73	4	no	—
84	C₇₃	#11996	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	146	2	no	—
86	C₂×C₄×C₈	—	73	1	no	—
87	C₂×C₄×C₈	—	73	1	no	—
88	C₂×C₄×C₈	—	73	1	no	—
89	C₄⋊C₁₆	—	73	4	no	—
90	C₄⋊C₁₆	—	146	2	no	—
91	C₈²	—	73	2	no	—
92	C₈²	—	73	2	no	—
93	C₂²×C₈	—	73	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂²×C₈	—	73	2	no	—
96	C₂²×C₈	—	146	1	no	—
97	C₂×C₄²	—	73	1	no	—
98	M₃₂	—	292	4	no	—
99	C₂×C₁₆	—	146	4	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	—	73	2	no	—
105	C₄×C₈	—	73	2	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	146	1	no	—
108	C₄×C₈	#10596842	1	1	yes	characteristic
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	4	no	—
115	C₂×C₈	—	73	2	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	146	4	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	—	146	2	no	—
121	C₂×C₈	—	146	4	no	—
122	C₂×C₈	C₂×#10596842	1	2	yes	—
123	C₄²	—	73	1	no	—
124	C₄²	—	73	1	no	—
125	C₄²	—	73	1	no	—
126	C₄²	C₂×#10596842	1	1	yes	characteristic
127	C₁₆	—	292	4	no	—
128	C₂³	—	73	1	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	2	no	—
131	C₂×C₄	—	73	2	no	—
132	C₂×C₄	—	73	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	146	2	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
137	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
138	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	73	2	no	—
141	C₈	—	73	2	no	—
142	C₈	—	73	4	no	—
143	C₈	—	146	1	no	—
144	C₈	—	146	1	no	—
145	C₈	—	146	4	no	—
146	C₂²	—	73	2	no	—
147	C₂²	—	146	1	no	—
148	C₂²	—	146	1	no	—
149	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
150	C₄	—	73	1	no	—
151	C₄	—	73	1	no	—
152	C₄	—	146	1	no	—
153	C₄	—	146	1	no	—
154	C₄	#17989676	1	2	yes	—
155	C₄	#17989692	1	2	yes	—
156	C₄	#17989706	1	2	yes	—
157	C₂	—	146	2	no	—
158	C₂	C₂×#17989676	1	1	yes	characteristic
159	C₂	#22627121	1	1	yes	characteristic
160	C₂	C₂×#17989692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10596842
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×#10596833	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#11996
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22626990	C₂	semidirect
64. #10596831	M₃₂	semidirect
84. C₇₃	#11996	semidirect

Maximal subgroups

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

subgroup	index
#22627120	2
#22626990	2
C₈×#11924036	2
#11996	73