Finite Group Databasebeta

Subgroups — #34011725

Order 18688.

234 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 120 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	#22625478	C₂	1	2	yes	maximal
2	#22624956	C₂	1	2	yes	maximal, complemented
3	#22624955	C₂	1	2	yes	maximal, complemented
4	C₂×#17988362	C₂	1	1	yes	maximal, characteristic
5	#17989603	—	2	2	no	—
6	#17989450	—	2	2	no	—
7	#17989448	C₂²	1	4	yes	—
8	C₂²×#11924003	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
10	#17988362	C₄	1	1	yes	characteristic
11	#17988362	C₄	1	1	yes	characteristic
12	#17988362	C₂²	1	1	yes	characteristic
13	#17988362	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
15	C₄⋊#10596833	—	2	4	no	—
16	C₂²⋊#10596833	—	4	2	no	—
17	C₄×#10596833	—	4	2	no	—
18	C₂²×#10596831	Q₈	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
20	C₂×#11924003	—	2	2	no	—
21	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
22	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₇₃	D₄	1	1	yes	characteristic
25	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₃	C₈	1	4	yes	complemented
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#10596833	—	4	4	no	—
32	C₂²×Dic₇₃	SD₁₆	1	2	yes	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
36	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄×Dic₇₃	SD₁₆	1	2	yes	—
38	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
39	#11924003	—	2	1	no	—
40	#11924003	—	2	1	no	—
41	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
47	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
48	C₂³×C₇₃	#106	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
50	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#108	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×Dic₇₃	—	4	2	no	—
54	C₂×Dic₇₃	#106	1	2	yes	—
55	C₂×Dic₇₃	C₂×SD₁₆	1	2	yes	—
56	#10596833	—	8	4	no	—
57	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
58	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	#10596831	—	2	1	no	—
61	#10596831	—	2	1	no	—
62	C₂²×C₇₃	—	2	2	no	—
63	C₂²×C₇₃	#334	1	2	yes	—
64	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
65	C₄×C₇₃	#334	1	2	yes	—
66	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
68	Dic₇₃	—	2	2	no	—
69	Dic₇₃	—	2	2	no	—
70	#11999	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	—	2	2	no	—
72	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
73	C₂×C₇₃	#2155	1	2	yes	—
74	C₂×#346	—	73	1	no	—
75	#1564	—	73	2	no	—
76	#1368	—	73	2	no	—
77	#1368	—	73	2	no	—
78	C₇₃	#11999	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₁₆	—	73	1	no	—
80	C₂²×C₁₆	#64408	1	1	yes	characteristic
81	#431	—	146	2	no	—
82	C₂×C₄×C₈	—	73	1	no	—
83	#346	—	73	1	no	—
84	#346	—	73	1	no	—
85	#346	—	73	1	no	—
86	#346	—	73	1	no	—
87	#335	—	146	2	no	—
88	#334	—	73	4	no	—
89	C₂²×C₈	—	73	1	no	—
90	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
91	C₂×C₄²	—	73	1	no	—
92	C₂×C₁₆	—	73	1	no	—
93	C₂×C₁₆	—	73	1	no	—
94	C₂×C₁₆	—	146	2	no	—
95	C₂×C₁₆	#10596833	1	4	yes	complemented
96	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
97	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
98	C₄⋊C₈	—	146	4	no	—
99	C₂²⋊C₈	—	292	2	no	—
100	C₄×C₈	—	292	2	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	—	73	1	no	—
105	C₂²×C₄	—	73	2	no	—
106	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
107	C₂×C₈	—	292	4	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	146	2	no	—
111	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
112	C₂×C₈	C₂×#10596833	1	2	yes	—
113	C₂×C₈	#11924038	1	2	yes	—
114	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
115	C₄²	—	73	2	no	—
116	C₄²	—	73	1	no	—
117	C₄²	—	73	1	no	—
118	C₁₆	—	146	1	no	—
119	C₁₆	—	146	1	no	—
120	C₁₆	C₂×#10596833	1	2	yes	—
121	C₁₆	C₂×#10596833	1	2	yes	—
122	C₂³	#14556659	1	1	yes	characteristic
123	C₂×C₄	—	292	2	no	—
124	C₂×C₄	—	73	2	no	—
125	C₂×C₄	—	73	2	no	—
126	C₂×C₄	C₄⋊#10596833	1	2	yes	—
127	C₂×C₄	#14556659	1	1	yes	characteristic
128	C₂×C₄	#14556661	1	2	yes	—
129	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
130	C₈	—	584	4	no	—
131	C₈	—	146	1	no	—
132	C₈	—	146	1	no	—
133	C₈	C₄⋊#10596833	1	2	yes	—
134	C₈	C₂²×#10596833	1	1	yes	characteristic
135	C₈	C₂×#11924038	1	1	yes	characteristic
136	C₂²	—	2	2	no	—
137	C₂²	#17989448	1	2	yes	—
138	C₂²	C₂×#14556659	1	1	yes	characteristic
139	C₄	—	146	2	no	—
140	C₄	—	146	2	no	—
141	C₄	#17989448	1	2	yes	—
142	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
143	C₄	C₂×#14556661	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₂×#17989448	1	1	yes	characteristic
146	C₂	#22626278	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556659
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	#106
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#11999
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22624956	C₂	semidirect
3. #22624955	C₂	semidirect
28. C₂×C₁₆×C₇₃	C₈	semidirect
78. C₇₃	#11999	semidirect
95. C₂×C₁₆	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625478	2
#22624956	2
#22624955	2
C₂×#17988362	2
#11999	73