Finite Group Databasebeta

Subgroups — #34009727

Order 18688.

290 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 156 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	#22626288	C₂	1	2	yes	maximal
2	#22624844	C₂	1	2	yes	maximal, complemented
3	#22624843	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11924031	C₄	1	4	yes	—
6	C₂²⋊#11924031	C₄	1	4	yes	—
7	C₄×#11924031	C₂²	1	4	yes	—
8	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₈×#10596831	C₄	1	1	yes	characteristic
11	C₈×#10596831	C₄	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	C₈×#10596831	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924031	C₂×C₄	1	8	yes	—
16	C₂²×#10596831	C₈	1	2	yes	—
17	C₂²×#10596831	C₂×C₄	1	2	yes	—
18	C₄×#10596831	C₈	1	2	yes	—
19	C₄×#10596831	C₂×C₄	1	2	yes	—
20	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	C₈	1	2	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	C₈	1	2	yes	—
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924031	—	2	8	no	—
30	C₂×#10596831	—	2	4	no	—
31	C₂×#10596831	—	2	4	no	—
32	C₂×#10596831	C₄²	1	2	yes	—
33	C₂×#10596831	C₂×C₈	1	2	yes	—
34	C₂×#10596831	C₂×C₈	1	2	yes	—
35	C₂×#10596831	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	—	2	4	no	—
40	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
42	C₄²×C₇₃	C₄²	1	1	yes	characteristic
43	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
53	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₇₃	—	2	2	no	—
57	C₈×C₇₃	M₃₂	1	4	yes	—
58	#10596831	—	2	2	no	—
59	#10596831	—	2	2	no	—
60	#10596831	—	2	2	no	—
61	#10596831	#131	1	4	yes	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	#346	1	2	yes	—
65	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	#346	1	2	yes	—
69	C₄×C₇₃	#405	1	2	yes	—
70	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
71	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
72	#11411	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	2	2	no	—
74	C₂×C₇₃	#1450	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
76	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
77	#2162	—	73	2	no	—
78	C₂×C₈²	—	73	1	no	—
79	#1312	—	73	2	no	—
80	#1311	—	73	2	no	—
81	C₇₃	#11411	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄×C₈	—	73	1	no	—
83	C₂×C₄×C₈	—	73	1	no	—
84	C₂×C₄×C₈	#64408	1	1	yes	characteristic
85	C₄⋊C₁₆	—	73	4	no	—
86	C₂²⋊C₁₆	—	73	4	no	—
87	C₄×C₁₆	—	73	4	no	—
88	C₈²	—	73	1	no	—
89	C₈²	—	73	1	no	—
90	C₈²	—	73	1	no	—
91	C₈²	—	73	1	no	—
92	C₂²×C₈	—	73	2	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	#10596833	1	2	yes	—
95	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
96	C₂×C₁₆	—	73	8	no	—
97	C₄×C₈	—	73	2	no	—
98	C₄×C₈	—	73	2	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	#10596833	1	2	yes	—
104	C₄×C₈	C₂×#64408	1	1	yes	characteristic
105	C₄×C₈	C₂×#64408	1	1	yes	characteristic
106	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
107	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
108	C₂²×C₄	#11924036	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	73	2	no	—
113	C₂×C₈	—	73	2	no	—
114	C₂×C₈	—	146	4	no	—
115	C₂×C₈	—	146	4	no	—
116	C₂×C₈	C₂×#10596833	1	2	yes	—
117	C₂×C₈	C₂×#10596833	1	2	yes	—
118	C₄²	C₄×#64408	1	1	yes	characteristic
119	C₄²	C₂×#10596833	1	1	yes	characteristic
120	C₄²	#11924036	1	1	yes	characteristic
121	C₄²	C₂²×#64408	1	1	yes	characteristic
122	C₁₆	—	146	8	no	—
123	C₂³	C₄×#10596833	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
127	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
128	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
129	C₂×C₄	#14556666	1	4	yes	—
130	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
131	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
132	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	73	4	no	—
135	C₈	—	146	2	no	—
136	C₈	—	146	2	no	—
137	C₈	—	146	2	no	—
138	C₈	#14556665	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#17989476	1	2	yes	—
142	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	#17989476	1	2	yes	—
146	C₄	#17989550	1	2	yes	—
147	C₄	C₂×#14556665	1	2	yes	—
148	C₄	C₂×#14556666	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#22625134	1	1	yes	characteristic
151	C₂	C₂×#17989476	1	1	yes	characteristic
152	C₂	C₄×#14556665	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₇₃	C₄
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#11411
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22624844	C₂	semidirect
3. #22624843	C₂	semidirect
81. C₇₃	#11411	semidirect

Maximal subgroups

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

subgroup	index
#22626288	2
#22624844	2
#22624843	2
C₂×C₈×#10596831	2
#11411	73