Finite Group Databasebeta

Subgroups — #34001086

Order 18688.

178 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22626352	C₂	1	1	yes	maximal, characteristic
2	#22626282	C₂	1	1	yes	maximal, characteristic
3	#367×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989597	—	2	1	no	—
5	#17989520	C₄	1	2	yes	complemented
6	#17989519	—	2	1	no	—
7	#17989518	—	2	1	no	—
8	#17989514	—	2	1	no	—
9	#17989513	C₄	1	2	yes	complemented
10	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
11	#107×D₇₃	—	2	1	no	—
12	#17988415	C₄	1	1	yes	characteristic, complemented
13	#17988414	—	2	1	no	—
14	#367×C₇₃	C₄	1	1	yes	characteristic, complemented
15	#14556670	—	4	1	no	—
16	#14556660	—	2	2	no	—
17	#14556659	—	4	1	no	—
18	#14556641	—	2	2	no	—
19	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
20	C₄⋊C₄×D₇₃	—	4	1	no	—
21	C₂×#11924003	D₄	1	1	yes	characteristic
22	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₁₆×D₇₃	D₄	1	1	yes	characteristic
25	C₁₆×D₇₃	Q₈	1	1	yes	characteristic
26	#14556525	—	2	1	no	—
27	#14556524	—	4	1	no	—
28	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#107×C₇₃	—	2	1	no	—
30	C₄⋊#64408	—	4	2	no	—
31	C₂²⋊#64408	—	8	1	no	—
32	C₄×#64408	—	8	1	no	—
33	C₂×C₄×D₇₃	—	8	1	no	—
34	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
35	C₂×#10596831	SD₁₆	1	1	yes	characteristic
36	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
39	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
40	C₄⋊Dic₇₃	—	4	1	no	—
41	#11924007	—	8	1	no	—
42	#11924003	C₂²⋊C₄	1	1	yes	characteristic
43	#11924003	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₇₃	—	4	1	no	—
46	C₁₆×C₇₃	C₄²	1	1	yes	characteristic
47	C₁₆×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×D₇₃	D₁₆	1	1	yes	characteristic
49	C₂×#64408	—	8	2	no	—
50	C₂×C₄×C₇₃	—	8	1	no	—
51	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
52	C₂×Dic₇₃	—	8	1	no	—
53	C₂×Dic₇₃	SD₃₂	1	1	yes	characteristic
54	C₄×D₇₃	—	16	2	no	—
55	C₄×D₇₃	#102	1	1	yes	characteristic
56	C₄×D₇₃	#107	1	1	yes	characteristic
57	C₄×D₇₃	SD₃₂	1	1	yes	characteristic
58	C₄×D₇₃	Q₃₂	1	1	yes	characteristic
59	C₈×C₇₃	#95	1	1	yes	characteristic
60	C₈×C₇₃	#103	1	1	yes	characteristic
61	#10596831	#103	1	1	yes	characteristic
62	#10596831	#106	1	1	yes	characteristic
63	C₂²×C₇₃	#357	1	1	yes	characteristic
64	C₂×D₇₃	—	2	2	no	—
65	C₂×D₇₃	#357	1	1	yes	characteristic
66	C₂×D₇₃	#366	1	1	yes	characteristic
67	#64408	—	16	2	no	—
68	C₄×C₇₃	—	16	1	no	—
69	C₄×C₇₃	#340	1	1	yes	characteristic
70	C₄×C₇₃	#358	1	1	yes	characteristic
71	Dic₇₃	—	16	1	no	—
72	Dic₇₃	#358	1	1	yes	characteristic
73	Dic₇₃	#367	1	1	yes	characteristic
74	#7477	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	#1380	1	1	yes	characteristic
77	D₇₃	—	2	1	no	—
78	D₇₃	#1418	1	2	yes	—
79	C₂×M₆₄	—	73	1	no	—
80	#2158	—	73	1	no	—
81	C₂×#367	—	73	1	no	—
82	C₇₃	#7477	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	—	73	1	no	—
84	#427	—	146	1	no	—
85	C₂×#107	—	146	1	no	—
86	M₆₄	—	73	2	no	—
87	M₆₄	—	146	1	no	—
88	C₂×C₃₂	—	146	1	no	—
89	#367	—	73	1	no	—
90	#367	—	146	1	no	—
91	#367	—	146	1	no	—
92	#367	#64408	1	1	yes	characteristic, complemented
93	#366	—	73	2	no	—
94	C₂²×C₈	—	73	1	no	—
95	#117	—	292	1	no	—
96	C₂×C₄⋊C₄	—	292	1	no	—
97	C₂×C₁₆	—	73	1	no	—
98	C₂×C₁₆	—	73	1	no	—
99	C₂×C₁₆	—	73	1	no	—
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	—	73	1	no	—
102	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
103	#107	—	2	1	no	—
104	#107	—	292	1	no	—
105	#107	—	146	2	no	—
106	#107	—	146	1	no	—
107	#106	—	292	1	no	—
108	C₃₂	—	146	2	no	—
109	C₂²×C₄	—	584	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	292	2	no	—
119	C₄⋊C₄	—	292	1	no	—
120	C₄⋊C₄	—	584	1	no	—
121	C₂²⋊C₄	—	584	1	no	—
122	C₄²	—	584	1	no	—
123	C₁₆	—	73	1	no	—
124	C₁₆	—	73	1	no	—
125	C₁₆	C₄×#64408	1	1	yes	characteristic
126	C₁₆	C₄⋊#64408	1	1	yes	characteristic
127	C₂³	—	73	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	584	2	no	—
130	C₂×C₄	—	584	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	73	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	1168	2	no	—
137	C₂×C₄	#14556653	1	1	yes	characteristic
138	C₈	—	73	1	no	—
139	C₈	—	73	1	no	—
140	C₈	(C₂×#64408)⋊C₄	1	1	yes	characteristic
141	C₈	#14556654	1	1	yes	characteristic
142	C₂²	—	73	1	no	—
143	C₂²	—	73	1	no	—
144	C₂²	—	146	2	no	—
145	C₂²	#17989499	1	1	yes	characteristic
146	C₄	—	16	1	no	—
147	C₄	—	73	1	no	—
148	C₄	—	73	1	no	—
149	C₄	—	1168	1	no	—
150	C₄	—	1168	2	no	—
151	C₄	#17989460	1	1	yes	characteristic
152	C₄	#17989500	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	73	2	no	—
155	C₂	—	146	1	no	—
156	C₂	#22624988	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22624988
commutator	C₁₆×C₇₃	C₄²
Frattini	C₂×C₁₆	C₂×#64408
Fitting	#367×C₇₃	C₄
socle	C₂×C₇₃	#1380
prime radical	C₂²×D₇₃	D₁₆
cosocle	C₂×C₁₆×D₇₃	C₂²
hypercenter	#367	#64408
hypocenter	C₇₃	#7477
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

Complements

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

N	H ≅ G/N	type
5. #17989520	C₄	semidirect
9. #17989513	C₄	semidirect
12. #17988415	C₄	semidirect
14. #367×C₇₃	C₄	semidirect
82. C₇₃	#7477	semidirect
92. #367	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626352	2
#22626282	2
#367×D₇₃	2
#7477	73