Finite Group Databasebeta

Subgroups — #33006879

Order 18688.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	#22612205	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#22608131	C₂	1	1	yes	maximal, characteristic
4	#22608130	C₂	1	1	yes	maximal, characteristic
5	#22608126	C₂	1	1	yes	maximal, characteristic
6	#22608112	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1416	C₂	1	1	yes	maximal, characteristic
8	#17988991	—	2	1	no	—
9	Q₁₆×Dic₇₃	—	2	1	no	—
10	C₂×#14556497	C₂²	1	1	yes	characteristic
11	#17988413	C₂²	1	1	yes	characteristic
12	#17988413	C₂²	1	1	yes	characteristic
13	#17988412	—	2	1	no	—
14	#17988391	—	2	1	no	—
15	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
18	#366×C₇₃	C₂²	1	1	yes	characteristic
19	#14556590	—	4	1	no	—
20	Q₈×Dic₇₃	—	4	1	no	—
21	C₂×#11924003	D₄	1	1	yes	characteristic
22	#14556525	—	2	1	no	—
23	#14556525	—	2	1	no	—
24	#14556524	—	4	1	no	—
25	#14556515	—	4	1	no	—
26	C₈×Dic₇₃	D₄	1	1	yes	characteristic
27	#14556497	—	2	1	no	—
28	C₂×Q₁₆×C₇₃	—	2	1	no	—
29	Q₃₂×C₇₃	—	2	1	no	—
30	Q₃₂×C₇₃	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
32	#107×C₇₃	—	2	1	no	—
33	C₃₂×C₇₃	—	2	1	no	—
34	C₂×Dic₁₄₆	—	8	1	no	—
35	C₂×#10596831	D₈	1	1	yes	characteristic
36	C₄⋊Dic₇₃	—	4	1	no	—
37	C₄⋊Dic₇₃	—	4	1	no	—
38	C₄⋊Dic₇₃	—	8	1	no	—
39	#11924007	—	8	1	no	—
40	C₄×Dic₇₃	—	8	1	no	—
41	C₄×Dic₇₃	D₈	1	1	yes	characteristic
42	#11924003	—	2	1	no	—
43	C₂×Q₈×C₇₃	—	4	1	no	—
44	Q₁₆×C₇₃	—	2	2	no	—
45	Q₁₆×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₃	—	4	1	no	—
48	C₁₆×C₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₇₃	#41	1	1	yes	characteristic
50	Q₈×C₇₃	—	4	2	no	—
51	Q₈×C₇₃	—	8	1	no	—
52	C₂×C₄×C₇₃	—	8	1	no	—
53	C₂×C₄×C₇₃	—	8	1	no	—
54	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₃	—	8	1	no	—
56	C₂×Dic₇₃	—	8	1	no	—
57	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
58	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
59	Dic₁₄₆	—	16	2	no	—
60	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₇₃	#137	1	1	yes	characteristic
62	#10596831	—	2	1	no	—
63	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₇₃	—	8	2	no	—
65	C₄×C₇₃	—	16	1	no	—
66	C₄×C₇₃	#474	1	1	yes	characteristic
67	C₄×C₇₃	#510	1	1	yes	characteristic
68	Dic₇₃	—	2	1	no	—
69	Dic₇₃	—	16	1	no	—
70	Dic₇₃	—	16	1	no	—
71	Dic₇₃	Q₆₄	1	2	yes	—
72	#13706	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#2224	1	1	yes	characteristic
74	C₂×C₇₃	C₂×Q₆₄	1	1	yes	characteristic
75	C₂×C₇₃	#2263	1	1	yes	characteristic
76	#2252	—	73	1	no	—
77	C₄×Q₃₂	—	73	1	no	—
78	#1423	—	73	1	no	—
79	#1423	—	73	1	no	—
80	C₄⋊C₃₂	—	73	1	no	—
81	#1416	—	73	1	no	—
82	#1416	D₇₃	1	1	yes	characteristic
83	C₇₃	#13706	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
85	#500	—	146	1	no	—
86	C₄×Q₁₆	—	146	1	no	—
87	C₂×C₃₂	—	73	1	no	—
88	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
89	#366	—	73	1	no	—
90	#366	—	73	1	no	—
91	#366	—	146	1	no	—
92	#366	C₂×D₇₃	1	1	yes	characteristic
93	#358	—	146	1	no	—
94	C₄×C₁₆	—	73	1	no	—
95	C₂×Q₁₆	—	2	1	no	—
96	C₄⋊Q₈	—	292	1	no	—
97	C₄×Q₈	—	292	1	no	—
98	Q₃₂	—	2	1	no	—
99	Q₃₂	Dic₁₄₆	1	2	yes	complemented
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	292	1	no	—
104	#107	—	146	1	no	—
105	#107	—	146	1	no	—
106	#103	—	292	1	no	—
107	C₄×C₈	—	73	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	146	1	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	584	1	no	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	292	1	no	—
118	C₄⋊C₄	—	292	1	no	—
119	C₄⋊C₄	—	584	1	no	—
120	C₄⋊C₄	—	584	1	no	—
121	C₄²	—	584	1	no	—
122	C₄²	—	73	1	no	—
123	C₁₆	—	146	1	no	—
124	C₁₆	C₂×Dic₁₄₆	1	1	yes	characteristic
125	C₁₆	#11924027	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	1168	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	584	1	no	—
132	C₂×C₄	—	584	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
136	C₈	—	146	1	no	—
137	C₈	#14556568	1	1	yes	characteristic
138	C₈	#14556620	1	1	yes	characteristic
139	C₂²	D₁₆×D₇₃	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	73	2	no	—
143	C₄	—	1168	1	no	—
144	C₄	—	1168	1	no	—
145	C₄	—	146	1	no	—
146	C₄	#17988865	1	1	yes	characteristic
147	C₄	#17989026	1	1	yes	characteristic
148	C₂	#22612066	1	1	yes	characteristic
149	C₂	Q₆₄×D₇₃	1	1	yes	characteristic
150	C₂	#22612245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₃
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1416	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
hypercenter	#1416	D₇₃
hypocenter	C₇₃	#13706
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
30. Q₃₂×C₇₃	Q₈	semidirect
83. C₇₃	#13706	semidirect
99. Q₃₂	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
#22612205	2
Q₃₂×Dic₇₃	2
#22608131	2
#22608130	2
#22608126	2
#22608112	2
C₇₃×#1416	2
#13706	73