Finite Group Databasebeta

Subgroups — #32997661

Order 18688.

200 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22608694	C₂	1	1	yes	maximal, characteristic
2	#22608679	C₂	1	1	yes	maximal, characteristic
3	#22607997	C₂	1	4	yes	maximal, complemented
4	C₇₃×#2163	C₂	1	1	yes	maximal, characteristic
5	#17988644	—	2	1	no	—
6	#17988640	—	2	1	no	—
7	#17988340	—	2	1	no	—
8	#17988340	—	2	1	no	—
9	#17988338	C₂²	1	2	yes	—
10	#17988338	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₃	C₄	1	2	yes	—
13	C₂²⋊C₁₆×C₇₃	C₄	1	2	yes	—
14	#346×C₇₃	C₂²	1	2	yes	—
15	C₄⋊#10596831	—	2	2	no	—
16	C₄⋊#10596831	—	2	2	no	—
17	C₂²⋊#10596831	—	4	1	no	—
18	C₂²⋊#10596831	—	4	1	no	—
19	#14556502	—	4	1	no	—
20	C₄×#10596831	—	4	1	no	—
21	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×#10596831	—	4	2	no	—
30	C₂×#10596831	—	4	2	no	—
31	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	—	4	1	no	—
36	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
42	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	4	no	—
45	C₂³×C₇₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	—	2	1	no	—
58	C₈×C₇₃	—	2	1	no	—
59	C₈×C₇₃	#131	1	2	yes	—
60	#10596831	—	8	2	no	—
61	#10596831	—	8	2	no	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	#424	1	1	yes	characteristic
72	C₄×C₇₃	#430	1	1	yes	characteristic
73	#12031	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	—	4	1	no	—
75	C₂×C₇₃	#1570	1	1	yes	characteristic
76	C₂×C₇₃	#2155	1	1	yes	characteristic
77	C₂×C₇₃	#2155	1	1	yes	characteristic
78	#2163	D₇₃	1	1	yes	characteristic
79	#1568	—	73	1	no	—
80	#1564	—	73	1	no	—
81	#1368	—	73	4	no	—
82	C₇₃	#12031	1	1	yes	characteristic, complemented, Sylow
83	#432	—	146	1	no	—
84	#431	—	146	1	no	—
85	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
86	C₄⋊C₁₆	Dic₇₃	1	2	yes	—
87	C₂²⋊C₁₆	Dic₇₃	1	2	yes	—
88	#346	C₂×D₇₃	1	2	yes	—
89	#335	—	146	1	no	—
90	#335	—	146	1	no	—
91	#334	—	73	2	no	—
92	#334	—	73	2	no	—
93	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
95	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
96	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
97	C₄⋊C₈	—	146	2	no	—
98	C₄⋊C₈	—	146	2	no	—
99	C₂²⋊C₈	—	292	1	no	—
100	C₂²⋊C₈	—	292	1	no	—
101	#97	—	292	1	no	—
102	C₄×C₈	—	292	1	no	—
103	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
104	C₄×C₈	D₂₉₂	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
107	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
108	C₂²×C₄	#11924017	1	1	yes	characteristic
109	C₂²×C₄	#11924017	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	292	2	no	—
113	C₂×C₈	—	292	2	no	—
114	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
116	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
117	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
118	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
119	C₄²	#11924017	1	1	yes	characteristic
120	C₄²	#11924017	1	1	yes	characteristic
121	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
122	C₁₆	—	2	4	no	—
123	C₂³	#14556523	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#14556523	1	1	yes	characteristic
129	C₂×C₄	#14556523	1	1	yes	characteristic
130	C₂×C₄	#14556523	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
132	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
133	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	584	2	no	—
138	C₈	—	584	2	no	—
139	C₈	#14556598	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#14556523	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#17988614	1	1	yes	characteristic
150	C₄	#17988636	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#22608702	1	1	yes	characteristic
153	C₂	#22611704	1	1	yes	characteristic
154	C₂	#22611704	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556523
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2163	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	#106
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#2163	D₇₃
hypocenter	C₇₃	#12031
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #22607997	C₂	semidirect
82. C₇₃	#12031	semidirect

Maximal subgroups

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

subgroup	index
#22608694	2
#22608679	2
#22607997	2
C₇₃×#2163	2
#12031	73