Finite Group Databasebeta

Subgroups — #34034407

Order 18688.

298 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 109 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	1	yes	maximal, characteristic, complemented
2	C₂²×#14556681	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14556681	C₂	1	4	yes	maximal, complemented
4	C₂×#17989645	C₂	1	1	yes	maximal, characteristic
5	#17989695	—	2	2	no	—
6	C₂×#14556681	—	2	2	no	—
7	C₂×#14556681	C₂²	1	2	yes	—
8	C₂×#14556681	C₂²	1	2	yes	—
9	#17989645	—	2	2	no	—
10	#17989645	—	2	2	no	—
11	C₂×#14556665	C₄	1	2	yes	—
12	C₂×#14556665	C₄	1	1	yes	characteristic
13	C₂²×#11924031	C₂²	1	2	yes	—
14	C₂²×#11924031	C₂²	1	1	yes	characteristic
15	C₂×#14556598	C₄	1	1	yes	characteristic
16	#14556681	—	2	2	no	—
17	#14556681	—	2	2	no	—
18	#14556665	—	2	4	no	—
19	#14556665	—	2	2	no	—
20	C₂×#11924031	—	2	2	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	C₂×C₄	1	2	yes	—
23	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924031	D₄	1	2	yes	—
25	C₂×#11924031	D₄	1	2	yes	—
26	C₂×#11924031	C₂³	1	1	yes	characteristic
27	#14556598	—	2	2	no	—
28	#14556598	—	2	2	no	—
29	C₂×#11924012	C₈	1	2	yes	—
30	C₂×#11924012	C₈	1	1	yes	characteristic
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₂×#41×C₇₃	C₈	1	1	yes	characteristic
34	#11924031	—	2	2	no	—
35	#11924031	C₂²⋊C₄	1	2	yes	—
36	#11924031	C₂×D₄	1	1	yes	characteristic
37	#11924031	C₂×D₄	1	1	yes	characteristic
38	#11924012	—	2	2	no	—
39	#11924012	—	2	4	no	—
40	C₂×#10596831	—	2	2	no	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
43	C₂×#10596831	C₂×C₈	1	2	yes	—
44	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
45	C₂×#10596831	M₁₆	1	2	yes	—
46	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
47	#41×C₇₃	—	2	2	no	—
48	#41×C₇₃	—	2	2	no	—
49	C₂×Q₈×C₇₃	C₁₆	1	1	yes	characteristic
50	C₂×D₄×C₇₃	C₁₆	1	2	yes	—
51	C₂×D₄×C₇₃	C₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
54	C₂³×C₇₃	C₂×C₁₆	1	2	yes	—
55	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
56	Q₈×C₇₃	—	2	2	no	—
57	D₄×C₇₃	—	2	4	no	—
58	D₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
65	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	#10596831	—	2	2	no	—
67	#10596831	C₂²⋊C₈	1	2	yes	—
68	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	#10596831	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₇₃	—	2	2	no	—
71	C₂²×C₇₃	—	2	2	no	—
72	C₂²×C₇₃	—	2	4	no	—
73	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
74	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
77	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
79	#13554	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	—	4	2	no	—
82	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
83	C₂×C₇₃	#2258	1	2	yes	—
84	C₂×#504	—	73	1	no	—
85	C₂×M₆₄	—	73	1	no	—
86	C₂²×C₃₂	—	73	1	no	—
87	C₂²⋊C₃₂	—	73	4	no	—
88	C₇₃	#13554	1	1	yes	characteristic, complemented, Sylow
89	C₂×#131	—	73	1	no	—
90	#504	—	146	2	no	—
91	#504	—	146	2	no	—
92	C₂×M₃₂	—	73	2	no	—
93	C₂×M₃₂	—	73	1	no	—
94	C₂²×C₁₆	—	73	1	no	—
95	C₂²×C₁₆	—	73	2	no	—
96	M₆₄	—	146	2	no	—
97	C₂×C₃₂	—	73	2	no	—
98	C₂×C₃₂	—	73	2	no	—
99	C₂×C₃₂	—	146	2	no	—
100	C₂×#41	#10596842	1	1	yes	characteristic
101	#131	—	146	2	no	—
102	#131	—	146	2	no	—
103	C₂×M₁₆	—	73	2	no	—
104	C₂×M₁₆	—	73	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	2	no	—
107	M₃₂	—	146	4	no	—
108	M₃₂	—	146	2	no	—
109	C₂×C₁₆	—	73	1	no	—
110	C₂×C₁₆	—	73	2	no	—
111	C₂×C₁₆	—	73	2	no	—
112	C₂×C₁₆	—	73	1	no	—
113	C₂×C₁₆	—	73	2	no	—
114	C₂×C₁₆	—	146	2	no	—
115	C₂×C₁₆	—	146	2	no	—
116	C₃₂	—	146	2	no	—
117	C₃₂	—	146	2	no	—
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	C₂×Q₈	#11924040	1	1	yes	characteristic
121	C₂×D₄	#11924040	1	2	yes	—
122	C₂×D₄	#11924040	1	1	yes	characteristic
123	C₂²×C₄	C₂×#10596842	1	2	yes	—
124	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
125	M₁₆	—	146	4	no	—
126	M₁₆	—	146	2	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	2	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	2	no	—
132	C₂×C₈	—	146	2	no	—
133	C₂×C₈	—	146	2	no	—
134	C₁₆	—	73	2	no	—
135	C₁₆	—	73	1	no	—
136	C₁₆	—	73	1	no	—
137	C₁₆	—	146	2	no	—
138	C₂³	C₂×#11924040	1	2	yes	—
139	C₂³	C₂×#11924040	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	D₄	—	2	4	no	—
142	D₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₂²⋊#10596842	1	2	yes	—
146	C₂×C₄	C₂×#11924040	1	2	yes	—
147	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
148	C₂×C₄	#14556687	1	2	yes	—
149	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
150	C₈	—	73	1	no	—
151	C₈	—	73	1	no	—
152	C₈	—	73	2	no	—
153	C₈	—	146	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	C₂²⋊#11924040	1	2	yes	—
158	C₂²	C₂²×#11924040	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	C₂²⋊#11924040	1	2	yes	—
161	C₄	C₂×C₂²⋊#10596842	1	1	yes	characteristic
162	C₄	C₂×#14556687	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	C₂×C₂²⋊#11924040	1	1	yes	characteristic
166	C₂	#22627164	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting, prime radical	C₂×#41×C₇₃	C₈
cosocle	C₂×#11924031	C₂³
hypercenter	C₂×#41	#10596842
hypocenter	C₇₃	#13554
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×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. C₂²×#14556681	C₂	semidirect
3. C₂²⋊#14556681	C₂	semidirect
88. C₇₃	#13554	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
C₂²×#14556681	2
C₂²⋊#14556681	2
C₂×#17989645	2
#13554	73