Finite Group Databasebeta

Subgroups — #33006874

Order 18688.

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

normal subgroups only

This group has 176 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, complemented
2	D₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#22608133	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608132	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608126	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608109	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1415	C₂	1	1	yes	maximal, characteristic
8	#17988991	—	2	1	no	—
9	D₈×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	#17988388	—	2	1	no	—
15	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×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	D₄×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	#14556512	—	4	1	no	—
26	C₈×Dic₇₃	D₄	1	1	yes	characteristic
27	#14556497	—	2	1	no	—
28	C₂×D₈×C₇₃	—	2	1	no	—
29	D₁₆×C₇₃	—	2	1	no	—
30	D₁₆×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	C₂²×Dic₇₃	—	8	1	no	—
36	C₂×#10596831	D₈	1	1	yes	characteristic
37	C₄⋊Dic₇₃	—	4	1	no	—
38	C₄⋊Dic₇₃	—	4	1	no	—
39	#11924007	—	8	1	no	—
40	C₂²⋊Dic₇₃	—	8	1	no	—
41	C₄×Dic₇₃	D₈	1	1	yes	characteristic
42	#11924003	—	2	1	no	—
43	C₂×D₄×C₇₃	—	4	1	no	—
44	D₈×C₇₃	—	2	2	no	—
45	D₈×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	C₂³×C₇₃	—	8	1	no	—
51	D₄×C₇₃	—	4	2	no	—
52	D₄×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	—	16	2	no	—
58	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
59	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
60	Dic₁₄₆	—	16	2	no	—
61	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₇₃	#137	1	1	yes	characteristic
63	#10596831	—	2	1	no	—
64	C₂²×C₇₃	—	8	2	no	—
65	C₂²×C₇₃	—	16	1	no	—
66	C₂²×C₇₃	—	16	1	no	—
67	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
68	C₄×C₇₃	—	16	1	no	—
69	C₄×C₇₃	#474	1	1	yes	characteristic
70	C₄×C₇₃	#510	1	1	yes	characteristic
71	Dic₇₃	—	2	1	no	—
72	Dic₇₃	—	16	1	no	—
73	Dic₇₃	—	16	1	no	—
74	Dic₇₃	SD₆₄	1	2	yes	—
75	#13705	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	16	2	no	—
77	C₂×C₇₃	#2224	1	1	yes	characteristic
78	C₂×C₇₃	C₂×SD₆₄	1	1	yes	characteristic
79	C₂×C₇₃	#2263	1	1	yes	characteristic
80	#2252	—	73	1	no	—
81	C₄×D₁₆	—	73	1	no	—
82	#1424	—	73	1	no	—
83	#1424	—	73	1	no	—
84	C₄⋊C₃₂	—	73	1	no	—
85	#1415	—	73	1	no	—
86	#1415	D₇₃	1	1	yes	characteristic
87	C₇₃	#13705	1	1	yes	characteristic, complemented, Sylow
88	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
89	#500	—	146	1	no	—
90	C₄×D₈	—	146	1	no	—
91	C₂×C₃₂	—	73	1	no	—
92	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
93	#366	—	73	1	no	—
94	#366	—	73	1	no	—
95	#366	—	146	1	no	—
96	#366	C₂×D₇₃	1	1	yes	characteristic
97	#357	—	146	1	no	—
98	C₄×C₁₆	—	73	1	no	—
99	C₂×D₈	—	2	1	no	—
100	C₄⋊Q₈	—	292	1	no	—
101	C₄×D₄	—	292	1	no	—
102	D₁₆	—	2	1	no	—
103	D₁₆	Dic₁₄₆	1	2	yes	complemented
104	C₂×C₁₆	—	73	1	no	—
105	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
106	#107	—	2	1	no	—
107	#107	—	292	1	no	—
108	#107	—	146	1	no	—
109	#107	—	146	1	no	—
110	#102	—	292	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₃₂	—	2	1	no	—
113	C₃₂	—	146	1	no	—
114	C₂×Q₈	—	584	1	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂²×C₄	—	584	1	no	—
117	D₈	—	2	2	no	—
118	D₈	—	4	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	292	1	no	—
123	C₄⋊C₄	—	292	1	no	—
124	C₄⋊C₄	—	584	1	no	—
125	C₂²⋊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	C₂³	—	8	1	no	—
131	Q₈	—	1168	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	584	1	no	—
136	C₂×C₄	—	584	1	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	1168	2	no	—
140	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
141	C₈	—	146	1	no	—
142	C₈	#14556568	1	1	yes	characteristic
143	C₈	#14556620	1	1	yes	characteristic
144	C₂²	—	8	2	no	—
145	C₂²	—	16	1	no	—
146	C₂²	—	16	1	no	—
147	C₂²	D₁₆×D₇₃	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	73	2	no	—
150	C₄	—	1168	1	no	—
151	C₄	—	1168	1	no	—
152	C₄	—	146	1	no	—
153	C₄	#17988865	1	1	yes	characteristic
154	C₄	#17989026	1	1	yes	characteristic
155	C₂	—	16	2	no	—
156	C₂	#22612066	1	1	yes	characteristic
157	C₂	SD₆₄×D₇₃	1	1	yes	characteristic
158	C₂	#22612242	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₃
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1415	C₂
socle	C₂²×C₇₃	C₂×D₁₆
prime radical	C₂×D₁₆×C₇₃	C₂²
hypercenter	#1415	D₇₃
hypocenter	C₇₃	#13705
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

Complements

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

N	H ≅ G/N	type
1. #22612205	C₂	semidirect
3. #22608133	C₂	semidirect
4. #22608132	C₂	semidirect
5. #22608126	C₂	semidirect
30. D₁₆×C₇₃	Q₈	semidirect
87. C₇₃	#13705	semidirect
103. D₁₆	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
D₁₆×Dic₇₃	2
#22608133	2
#22608132	2
#22608126	2
#22608109	2
C₇₃×#1415	2
#13705	73