Finite Group Databasebeta

Subgroups — #33006903

Order 18688.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22612212	C₂	1	1	yes	maximal, characteristic, complemented
2	#22611812	C₂	1	1	yes	maximal, characteristic
3	#22608137	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608136	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608128	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608118	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1418	C₂	1	1	yes	maximal, characteristic
8	#17988991	—	2	1	no	—
9	#17988704	—	2	1	no	—
10	#17988421	C₂²	1	1	yes	characteristic
11	#17988417	C₂²	1	1	yes	characteristic
12	#17988415	C₂²	1	1	yes	characteristic
13	#17988414	—	2	1	no	—
14	#17988394	—	2	1	no	—
15	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
16	#508×C₇₃	C₂²	1	1	yes	characteristic
17	M₆₄×C₇₃	C₂²	1	1	yes	characteristic
18	#367×C₇₃	C₂²	1	1	yes	characteristic
19	#14556598	—	4	1	no	—
20	#14556590	—	4	1	no	—
21	C₂×#11924003	D₄	1	1	yes	characteristic
22	#14556527	—	2	1	no	—
23	#14556525	—	2	1	no	—
24	#14556524	—	4	1	no	—
25	#14556518	—	4	1	no	—
26	C₈×Dic₇₃	D₄	1	1	yes	characteristic
27	#14556497	—	2	1	no	—
28	#135×C₇₃	—	2	1	no	—
29	Q₃₂×C₇₃	Q₈	1	1	yes	characteristic, complemented
30	SD₃₂×C₇₃	—	2	1	no	—
31	D₁₆×C₇₃	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
33	#107×C₇₃	—	2	1	no	—
34	C₃₂×C₇₃	—	2	1	no	—
35	C₂×Dic₁₄₆	—	8	1	no	—
36	#11924012	—	4	1	no	—
37	#11924012	—	8	1	no	—
38	C₂×#10596831	—	8	1	no	—
39	C₂×#10596831	D₈	1	1	yes	characteristic
40	C₄⋊Dic₇₃	—	4	1	no	—
41	#11924007	—	8	1	no	—
42	C₄×Dic₇₃	D₈	1	1	yes	characteristic
43	#11924003	—	2	1	no	—
44	#41×C₇₃	—	4	1	no	—
45	Q₁₆×C₇₃	—	2	1	no	—
46	SD₁₆×C₇₃	—	4	1	no	—
47	D₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₇₃	—	4	1	no	—
50	C₁₆×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₁₆×C₇₃	#41	1	1	yes	characteristic
52	Q₈×C₇₃	—	4	1	no	—
53	D₄×C₇₃	—	4	1	no	—
54	D₄×C₇₃	—	8	1	no	—
55	C₂×C₄×C₇₃	—	8	1	no	—
56	C₂×C₄×C₇₃	—	8	1	no	—
57	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	8	1	no	—
60	Dic₁₄₆	—	16	2	no	—
61	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₇₃	#137	1	1	yes	characteristic
63	#10596831	—	8	1	no	—
64	#10596831	SD₃₂	1	1	yes	characteristic
65	#10596831	SD₃₂	1	1	yes	characteristic
66	C₂²×C₇₃	—	8	1	no	—
67	C₂²×C₇₃	#509	1	1	yes	characteristic
68	C₄×C₇₃	—	8	1	no	—
69	C₄×C₇₃	—	16	1	no	—
70	C₄×C₇₃	#474	1	1	yes	characteristic
71	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
72	Dic₇₃	—	2	2	no	—
73	Dic₇₃	—	16	1	no	—
74	#13710	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	—	16	1	no	—
77	C₂×C₇₃	#2226	1	1	yes	characteristic
78	#2254	—	73	1	no	—
79	#2178	—	73	1	no	—
80	#1426	—	73	1	no	—
81	#1426	—	73	1	no	—
82	#1422	—	73	1	no	—
83	#1418	—	73	1	no	—
84	#1418	D₇₃	1	1	yes	characteristic
85	C₇₃	#13710	1	1	yes	characteristic, complemented, Sylow
86	#508	C₂×D₇₃	1	1	yes	characteristic
87	#500	—	146	1	no	—
88	#443	—	146	1	no	—
89	M₆₄	—	73	1	no	—
90	M₆₄	C₂×D₇₃	1	1	yes	characteristic
91	#368	—	73	1	no	—
92	#367	—	73	1	no	—
93	#367	—	146	1	no	—
94	#367	C₂×D₇₃	1	1	yes	characteristic
95	#359	—	146	1	no	—
96	C₄×C₁₆	—	73	1	no	—
97	#135	—	2	1	no	—
98	#131	—	292	1	no	—
99	C₄⋊Q₈	—	292	1	no	—
100	Q₃₂	Dic₁₄₆	1	1	yes	characteristic, complemented
101	SD₃₂	—	2	1	no	—
102	D₁₆	Dic₁₄₆	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	73	1	no	—
104	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
105	#108	—	146	1	no	—
106	#107	—	2	1	no	—
107	#107	—	292	1	no	—
108	#107	—	146	1	no	—
109	#104	—	292	1	no	—
110	C₄×C₈	—	73	1	no	—
111	C₃₂	—	2	1	no	—
112	C₃₂	—	146	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	584	1	no	—
115	Q₁₆	—	2	1	no	—
116	SD₁₆	—	4	1	no	—
117	D₈	—	2	1	no	—
118	M₁₆	—	292	1	no	—
119	M₁₆	—	584	1	no	—
120	C₂×C₈	—	584	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	292	1	no	—
125	C₄⋊C₄	—	584	1	no	—
126	C₄²	—	73	1	no	—
127	C₁₆	—	146	1	no	—
128	C₁₆	C₂×Dic₁₄₆	1	1	yes	characteristic
129	C₁₆	#11924027	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	1168	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	584	1	no	—
137	C₂×C₄	—	146	1	no	—
138	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
139	C₈	—	584	1	no	—
140	C₈	—	73	1	no	—
141	C₈	—	73	1	no	—
142	C₈	#14556568	1	1	yes	characteristic
143	C₈	#14556620	1	1	yes	characteristic
144	C₂²	—	8	1	no	—
145	C₂²	#17989021	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	16	1	no	—
148	C₄	—	1168	1	no	—
149	C₄	—	146	2	no	—
150	C₄	#17988865	1	1	yes	characteristic
151	C₄	SD₃₂×D₇₃	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	16	1	no	—
154	C₂	#22612076	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22612076
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1418	C₂
socle	C₂×C₇₃	#2226
prime radical	#508×C₇₃	C₂²
hypercenter	#1418	D₇₃
hypocenter	C₇₃	#13710
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
1. #22612212	C₂	semidirect
3. #22608137	C₂	semidirect
4. #22608136	C₂	semidirect
5. #22608128	C₂	semidirect
29. Q₃₂×C₇₃	Q₈	semidirect
31. D₁₆×C₇₃	Q₈	semidirect
85. C₇₃	#13710	semidirect
100. Q₃₂	Dic₁₄₆	semidirect
102. D₁₆	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
#22612212	2
#22611812	2
#22608137	2
#22608136	2
#22608128	2
#22608118	2
C₇₃×#1418	2
#13710	73