Finite Group Databasebeta

Subgroups — #34034413

Order 18688.

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

normal subgroups only

This group has 290 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×#17989695	C₂	1	2	yes	maximal, complemented
2	#22627025	C₂	1	4	yes	maximal, complemented
3	C₂×#17989645	C₂	1	1	yes	maximal, characteristic
4	#17989695	—	2	2	no	—
5	#17989695	C₂²	1	4	yes	—
6	C₂×#14556681	—	2	2	no	—
7	#17989645	—	2	2	no	—
8	#17989645	—	2	2	no	—
9	C₂×#14556665	C₄	1	1	yes	characteristic
10	C₂×#14556665	C₂²	1	2	yes	—
11	C₂²×#11924031	C₄	1	2	yes	—
12	C₂²×#11924031	C₂²	1	1	yes	characteristic
13	C₂×#14556598	C₄	1	1	yes	characteristic
14	#14556681	—	2	4	no	—
15	#14556665	—	2	4	no	—
16	#14556665	—	2	2	no	—
17	C₂×#11924031	—	2	2	no	—
18	C₂×#11924031	—	2	2	no	—
19	C₂×#11924031	C₂×C₄	1	2	yes	—
20	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924031	D₄	1	2	yes	—
22	C₂×#11924031	D₄	1	2	yes	—
23	C₂×#11924031	C₂³	1	1	yes	characteristic
24	#14556598	—	2	2	no	—
25	#14556598	—	2	2	no	—
26	C₂×#11924012	C₈	1	2	yes	—
27	C₂×#11924012	C₈	1	1	yes	characteristic
28	C₂²×#10596831	C₂×C₄	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×#41×C₇₃	C₈	1	1	yes	characteristic
31	#11924031	—	2	2	no	—
32	#11924031	C₂²⋊C₄	1	1	yes	characteristic
33	#11924031	C₂²⋊C₄	1	1	yes	characteristic
34	#11924031	C₂×D₄	1	2	yes	—
35	#11924012	—	2	2	no	—
36	#11924012	—	2	4	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
40	C₂×#10596831	C₂×C₈	1	2	yes	—
41	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
42	C₂×#10596831	M₁₆	1	2	yes	—
43	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
44	#41×C₇₃	—	2	2	no	—
45	#41×C₇₃	—	2	2	no	—
46	C₂×Q₈×C₇₃	C₁₆	1	1	yes	characteristic
47	C₂×D₄×C₇₃	C₁₆	1	2	yes	—
48	C₂×D₄×C₇₃	C₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂³×C₇₃	C₂×C₁₆	1	2	yes	—
52	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
53	Q₈×C₇₃	—	2	2	no	—
54	D₄×C₇₃	—	2	4	no	—
55	D₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
62	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
63	#10596831	—	2	2	no	—
64	#10596831	C₂²⋊C₈	1	2	yes	—
65	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	#10596831	C₂×M₁₆	1	1	yes	characteristic
67	C₂²×C₇₃	—	2	2	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	—	4	2	no	—
70	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
71	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
74	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
76	#13556	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	—	2	2	no	—
79	C₂×C₇₃	—	4	2	no	—
80	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
81	C₂×#504	—	73	1	no	—
82	C₂×M₆₄	—	73	2	no	—
83	#1400	—	73	4	no	—
84	C₇₃	#13556	1	1	yes	characteristic, complemented, Sylow
85	C₂×#131	—	73	1	no	—
86	#504	—	146	2	no	—
87	#504	—	146	2	no	—
88	C₂×M₃₂	—	73	1	no	—
89	C₂×M₃₂	—	73	2	no	—
90	C₂²×C₁₆	—	73	1	no	—
91	C₂²×C₁₆	—	73	2	no	—
92	M₆₄	—	73	4	no	—
93	M₆₄	—	146	2	no	—
94	C₂×C₃₂	—	146	2	no	—
95	C₂×#41	#10596842	1	1	yes	characteristic
96	#131	—	146	2	no	—
97	#131	—	146	2	no	—
98	C₂×M₁₆	—	73	2	no	—
99	C₂×M₁₆	—	73	1	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂²×C₈	—	73	2	no	—
102	M₃₂	—	146	4	no	—
103	M₃₂	—	146	2	no	—
104	C₂×C₁₆	—	73	2	no	—
105	C₂×C₁₆	—	73	1	no	—
106	C₂×C₁₆	—	73	2	no	—
107	C₂×C₁₆	—	73	1	no	—
108	C₂×C₁₆	—	73	2	no	—
109	C₂×C₁₆	—	146	2	no	—
110	C₂×C₁₆	—	146	2	no	—
111	C₃₂	—	146	4	no	—
112	#41	—	2	2	no	—
113	#41	—	2	2	no	—
114	C₂×Q₈	#11924040	1	1	yes	characteristic
115	C₂×D₄	#11924040	1	2	yes	—
116	C₂×D₄	#11924040	1	1	yes	characteristic
117	C₂²×C₄	C₂×#10596842	1	2	yes	—
118	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
119	M₁₆	—	146	4	no	—
120	M₁₆	—	146	2	no	—
121	C₂×C₈	—	73	2	no	—
122	C₂×C₈	—	73	2	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	—	146	2	no	—
128	C₁₆	—	73	1	no	—
129	C₁₆	—	73	1	no	—
130	C₁₆	—	73	2	no	—
131	C₁₆	—	146	2	no	—
132	C₂³	C₂×#11924040	1	2	yes	—
133	C₂³	C₂×#11924040	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₂²⋊#10596842	1	2	yes	—
140	C₂×C₄	C₂×#11924040	1	2	yes	—
141	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
142	C₂×C₄	#14556687	1	2	yes	—
143	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
144	C₈	—	73	1	no	—
145	C₈	—	73	1	no	—
146	C₈	—	73	2	no	—
147	C₈	—	146	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	2	no	—
151	C₂²	C₂²⋊#11924040	1	2	yes	—
152	C₂²	C₂²×#11924040	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	C₂²⋊#11924040	1	2	yes	—
155	C₄	C₂×C₂²⋊#10596842	1	1	yes	characteristic
156	C₄	C₂×#14556687	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₂×C₂²⋊#11924040	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#10596842
commutator	C₂²×C₇₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting, prime radical	C₂×#41×C₇₃	C₈
socle	C₂×C₇₃	C₂×C₂²⋊C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	C₂×#41	#10596842
hypocenter	C₇₃	#13556
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#17989695	C₂	semidirect
2. #22627025	C₂	semidirect
84. C₇₃	#13556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989695	2
#22627025	2
C₂×#17989645	2
#13556	73