Finite Group Databasebeta

Subgroups — #34034400

Order 18688.

214 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#17989696	C₂	1	1	yes	maximal, characteristic
2	#22627162	C₂	1	1	yes	maximal, characteristic
3	#22627022	C₂	1	2	yes	maximal, complemented
4	#22627021	C₂	1	2	yes	maximal, complemented
5	#22626143	C₂	1	1	yes	maximal, characteristic
6	#17989696	C₄	1	2	yes	complemented
7	#17989696	C₂²	1	2	yes	—
8	#17989696	C₂²	1	1	yes	characteristic
9	#17989696	C₂²	1	1	yes	characteristic
10	#17989695	C₄	1	2	yes	complemented
11	#17989694	C₄	1	2	yes	complemented
12	C₂×#14556681	C₄	1	2	yes	complemented
13	C₂×#14556664	C₂²	1	1	yes	characteristic
14	#17989550	C₄	1	1	yes	characteristic
15	#17989474	C₂²	1	2	yes	—
16	C₁₆×#64408	C₄	1	2	yes	—
17	M₃₂×D₇₃	C₄	1	1	yes	characteristic
18	#14556681	C₂×C₄	1	2	yes	—
19	#14556681	C₂×C₄	1	2	yes	—
20	#14556681	C₂×C₄	1	4	yes	—
21	D₇₃⋊M₁₆	C₈	1	1	yes	characteristic
22	#14556670	C₈	1	1	yes	characteristic
23	#14556664	C₂×C₄	1	1	yes	characteristic
24	#14556664	C₂×C₄	1	1	yes	characteristic
25	#14556664	C₂×C₄	1	1	yes	characteristic
26	#14556664	C₂³	1	1	yes	characteristic
27	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
29	#14556645	C₈	1	2	yes	—
30	C₈×#64408	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
32	#14556531	C₈	1	1	yes	characteristic
33	#14556530	C₈	1	2	yes	—
34	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
35	M₃₂×C₇₃	C₈	1	1	yes	characteristic
36	#11924038	—	2	1	no	—
37	#11924037	—	2	1	no	—
38	#11924036	C₂×C₈	1	2	yes	—
39	C₄⋊#64408	—	2	1	no	—
40	C₂²⋊#64408	—	2	1	no	—
41	C₄×#64408	C₂×C₈	1	2	yes	—
42	#11924031	C₄²	1	1	yes	characteristic
43	#11924031	C₄²	1	1	yes	characteristic
44	#11924031	C₂²×C₄	1	1	yes	characteristic
45	#11924031	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×#10596831	C₄²	1	1	yes	characteristic
48	C₈×D₇₃	C₄²	1	1	yes	characteristic
49	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
50	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
51	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
52	#11924003	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
54	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
55	C₂²×D₇₃	M₃₂	1	1	yes	characteristic
56	C₂×#64408	—	2	2	no	—
57	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
58	C₂×Dic₇₃	M₃₂	1	1	yes	characteristic
59	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
60	C₄×D₇₃	M₃₂	1	1	yes	characteristic
61	C₄×D₇₃	M₃₂	1	1	yes	characteristic
62	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
63	#10596833	—	2	2	no	—
64	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
65	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	#10596831	C₄×C₈	1	1	yes	characteristic
67	#10596831	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₇₃	#346	1	1	yes	characteristic
69	C₂×D₇₃	—	4	1	no	—
70	C₂×D₇₃	#346	1	1	yes	characteristic
71	C₂×D₇₃	C₂×M₃₂	1	1	yes	characteristic
72	#64408	—	4	2	no	—
73	C₄×C₇₃	#346	1	1	yes	characteristic
74	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
75	Dic₇₃	#346	1	1	yes	characteristic
76	Dic₇₃	C₂×M₃₂	1	1	yes	characteristic
77	#13551	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	—	2	1	no	—
79	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
80	D₇₃	—	2	1	no	—
81	D₇₃	—	2	1	no	—
82	C₂×M₆₄	—	73	1	no	—
83	C₂×M₆₄	—	73	1	no	—
84	#2108	—	73	1	no	—
85	#1398	—	73	2	no	—
86	#1398	—	73	2	no	—
87	C₇₃	#13551	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₃₂	—	73	1	no	—
89	C₂²×C₁₆	—	73	1	no	—
90	#405	—	73	1	no	—
91	M₆₄	—	73	2	no	—
92	M₆₄	—	73	2	no	—
93	M₆₄	—	73	2	no	—
94	M₆₄	—	73	1	no	—
95	M₆₄	—	73	1	no	—
96	C₂×C₃₂	—	73	2	no	—
97	C₂×C₃₂	—	73	2	no	—
98	#346	—	73	2	no	—
99	C₄×C₁₆	—	73	2	no	—
100	C₂×M₁₆	—	73	1	no	—
101	C₂²×C₈	—	73	1	no	—
102	#117	—	73	1	no	—
103	M₃₂	—	73	1	no	—
104	M₃₂	—	73	2	no	—
105	M₃₂	#10596842	1	1	yes	characteristic
106	C₂×C₁₆	—	73	1	no	—
107	C₂×C₁₆	—	73	1	no	—
108	C₂×C₁₆	—	73	1	no	—
109	C₂×C₁₆	—	73	1	no	—
110	C₂×C₁₆	—	73	1	no	—
111	C₂×C₁₆	—	73	1	no	—
112	C₂×C₁₆	—	73	2	no	—
113	#97	—	73	2	no	—
114	C₄×C₈	—	73	2	no	—
115	C₃₂	—	73	2	no	—
116	C₃₂	—	73	2	no	—
117	C₃₂	—	73	4	no	—
118	C₂²×C₄	—	73	1	no	—
119	M₁₆	—	146	1	no	—
120	M₁₆	—	146	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
128	C₄⋊C₄	—	146	1	no	—
129	C₂²⋊C₄	—	146	1	no	—
130	C₄²	—	73	2	no	—
131	C₁₆	—	73	1	no	—
132	C₁₆	—	73	1	no	—
133	C₁₆	—	73	1	no	—
134	C₁₆	—	73	1	no	—
135	C₁₆	—	73	2	no	—
136	C₁₆	C₂×#10596842	1	2	yes	—
137	C₂³	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
145	C₈	—	73	1	no	—
146	C₈	—	73	1	no	—
147	C₈	—	146	2	no	—
148	C₈	C₄×#10596842	1	1	yes	characteristic
149	C₈	C₂²×#10596842	1	1	yes	characteristic
150	C₂²	—	292	1	no	—
151	C₂²	—	73	1	no	—
152	C₂²	—	73	1	no	—
153	C₂²	#17989682	1	1	yes	characteristic
154	C₄	—	292	2	no	—
155	C₄	—	73	1	no	—
156	C₄	—	73	1	no	—
157	C₄	#17989682	1	1	yes	characteristic
158	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	146	1	no	—
161	C₂	—	146	1	no	—
162	C₂	C₂×#17989682	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#10596842
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₈	C₂²×#10596842
Fitting	M₃₂×C₇₃	C₈
socle	C₂×C₇₃	C₂×#346
prime radical	C₂²×D₇₃	M₃₂
cosocle	#14556664	C₂³
hypercenter	M₃₂	#10596842
hypocenter	C₇₃	#13551
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
3. #22627022	C₂	semidirect
4. #22627021	C₂	semidirect
6. #17989696	C₄	semidirect
10. #17989695	C₄	semidirect
11. #17989694	C₄	semidirect
12. C₂×#14556681	C₄	semidirect
87. C₇₃	#13551	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989696	2
#22627162	2
#22627022	2
#22627021	2
#22626143	2
#13551	73