Finite Group Databasebeta

Subgroups — #34032903

Order 18688.

200 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22627125	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989691	C₂	1	1	yes	maximal, characteristic
3	M₃₂×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556688	—	2	1	no	—
5	#17989706	—	2	1	no	—
6	#17989692	C₄	1	2	yes	—
7	#17989691	—	2	1	no	—
8	#17989691	—	2	1	no	—
9	#17989691	C₄	1	2	yes	—
10	#17989643	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
12	#17989474	—	2	1	no	—
13	C₁₆×#64408	—	2	1	no	—
14	M₃₂×D₇₃	C₄	1	1	yes	characteristic
15	#14556688	—	2	2	no	—
16	#14556688	—	2	2	no	—
17	#14556688	—	4	1	no	—
18	#14556687	—	4	1	no	—
19	#14556686	—	4	1	no	—
20	C₂×#11924040	—	4	1	no	—
21	C₂×#11924036	D₄	1	1	yes	characteristic
22	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
23	#14556666	C₈	1	2	yes	—
24	#14556665	—	2	1	no	—
25	#14556664	—	2	1	no	—
26	C₈×#64408	C₂×C₄	1	1	yes	characteristic
27	C₈×#64408	C₂×C₄	1	1	yes	characteristic
28	C₈×#64408	D₄	1	1	yes	characteristic
29	C₈×#64408	D₄	1	1	yes	characteristic
30	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
31	#14556531	C₈	1	1	yes	characteristic
32	#14556530	—	2	1	no	—
33	C₁₆×D₇₃	—	2	1	no	—
34	M₃₂×C₇₃	C₈	1	1	yes	characteristic
35	#11924040	—	4	2	no	—
36	#11924040	—	4	2	no	—
37	C₂²×#64408	—	2	1	no	—
38	#11924036	C₂²⋊C₄	1	1	yes	characteristic
39	#11924036	C₂²⋊C₄	1	1	yes	characteristic
40	#11924036	SD₁₆	1	1	yes	characteristic
41	#11924036	SD₁₆	1	1	yes	characteristic
42	C₂×#10596833	D₈	1	1	yes	characteristic
43	C₂×#10596833	Q₁₆	1	1	yes	characteristic
44	C₄×#64408	—	2	1	no	—
45	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
46	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
47	#11924031	—	2	2	no	—
48	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#10596831	M₁₆	1	1	yes	characteristic
50	C₈×D₇₃	C₄²	1	1	yes	characteristic
51	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
53	C₈×D₇₃	M₁₆	1	1	yes	characteristic
54	#11924003	—	2	1	no	—
55	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₇₃	—	2	1	no	—
57	C₂²×D₇₃	#108	1	1	yes	characteristic
58	C₂×#64408	—	2	1	no	—
59	C₂×#64408	—	2	1	no	—
60	C₂×#64408	—	4	2	no	—
61	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×Dic₇₃	#107	1	1	yes	characteristic
63	C₄×D₇₃	#95	1	1	yes	characteristic
64	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
65	C₄×D₇₃	#106	1	1	yes	characteristic
66	C₄×D₇₃	#108	1	1	yes	characteristic
67	#10596833	#102	1	1	yes	characteristic
68	#10596833	#102	1	1	yes	characteristic
69	#10596833	#103	1	1	yes	characteristic
70	#10596833	#103	1	1	yes	characteristic
71	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
72	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
73	#10596831	#97	1	1	yes	characteristic
74	#10596831	C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₇₃	#335	1	1	yes	characteristic
76	C₂×D₇₃	—	4	1	no	—
77	C₂×D₇₃	#334	1	1	yes	characteristic
78	C₂×D₇₃	#341	1	1	yes	characteristic
79	#64408	—	2	2	no	—
80	#64408	—	4	1	no	—
81	C₄×C₇₃	#334	1	1	yes	characteristic
82	C₄×C₇₃	#336	1	1	yes	characteristic
83	Dic₇₃	#335	1	1	yes	characteristic
84	Dic₇₃	#340	1	1	yes	characteristic
85	#7162	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	#1277	1	1	yes	characteristic
88	D₇₃	—	2	1	no	—
89	D₇₃	—	2	1	no	—
90	C₂×#364	—	73	1	no	—
91	C₂×#364	—	73	1	no	—
92	C₄×M₃₂	—	73	1	no	—
93	C₇₃	#7162	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	—	73	1	no	—
95	C₂×M₃₂	—	73	1	no	—
96	C₂×M₃₂	—	146	1	no	—
97	C₂×M₃₂	—	146	1	no	—
98	C₂×C₄×C₈	—	73	1	no	—
99	#364	—	73	2	no	—
100	#364	—	73	2	no	—
101	#364	—	146	1	no	—
102	#364	—	146	1	no	—
103	#346	—	146	1	no	—
104	C₄×C₁₆	—	146	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂×C₄²	—	73	1	no	—
108	M₃₂	—	292	1	no	—
109	M₃₂	—	292	1	no	—
110	M₃₂	—	73	2	no	—
111	M₃₂	—	73	1	no	—
112	M₃₂	—	146	1	no	—
113	M₃₂	—	146	2	no	—
114	M₃₂	—	146	2	no	—
115	M₃₂	—	146	1	no	—
116	M₃₂	#10596842	1	1	yes	characteristic
117	C₂×C₁₆	—	292	1	no	—
118	C₂×C₁₆	—	292	1	no	—
119	C₂×C₁₆	—	146	1	no	—
120	C₂×C₁₆	—	146	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	73	1	no	—
123	C₄×C₈	—	73	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₂²×C₄	—	73	1	no	—
126	C₂²×C₄	—	146	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
139	C₄²	—	73	1	no	—
140	C₄²	—	73	1	no	—
141	C₄²	—	146	1	no	—
142	C₁₆	—	2	1	no	—
143	C₁₆	—	292	2	no	—
144	C₁₆	—	292	2	no	—
145	C₁₆	—	146	2	no	—
146	C₁₆	—	146	1	no	—
147	C₂³	—	73	1	no	—
148	C₂×C₄	—	292	2	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	1	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	1	no	—
154	C₂×C₄	—	146	1	no	—
155	C₂×C₄	—	146	1	no	—
156	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
157	C₈	—	73	1	no	—
158	C₈	—	73	1	no	—
159	C₈	—	73	1	no	—
160	C₈	—	73	1	no	—
161	C₈	—	73	1	no	—
162	C₈	—	73	1	no	—
163	C₈	C₄×#10596842	1	1	yes	characteristic
164	C₈	C₂²⋊#10596842	1	1	yes	characteristic
165	C₂²	—	292	1	no	—
166	C₂²	—	73	1	no	—
167	C₂²	—	73	1	no	—
168	C₂²	#17989677	1	1	yes	characteristic
169	C₄	—	292	1	no	—
170	C₄	—	73	1	no	—
171	C₄	—	73	1	no	—
172	C₄	—	146	2	no	—
173	C₄	#17989676	1	1	yes	characteristic
174	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	146	1	no	—
177	C₂	—	146	1	no	—
178	C₂	#22626937	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22626937
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₈	C₂×#10596842
Fitting	M₃₂×C₇₃	C₈
socle	C₂×C₇₃	#1277
prime radical	C₂²×D₇₃	#108
cosocle	C₂×C₈×#64408	C₂²
hypercenter	M₃₂	#10596842
hypocenter	C₇₃	#7162
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
93. C₇₃	#7162	semidirect

Maximal subgroups

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

subgroup	index
#22627125	2
C₂×#17989691	2
M₃₂×#64408	2
#7162	73