Finite Group Databasebeta

Subgroups — #16847684

Order 3968.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14521854	C₂	1	1	yes	maximal, characteristic
2	#14521492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758388	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521324	C₂	1	1	yes	maximal, characteristic, complemented
5	#14520863	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520857	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#11758566	—	2	1	no	—
9	C₃₁×#131	C₂²	1	1	yes	characteristic
10	#11758528	—	2	1	no	—
11	#11758485	—	2	1	no	—
12	#11758485	—	2	2	no	—
13	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic, complemented
14	#11758481	—	2	1	no	—
15	#11758481	C₂²	1	1	yes	characteristic
16	C₂×D₂₄₈	—	2	1	no	—
17	C₂×D₂₄₈	C₂²	1	1	yes	characteristic
18	C₃₁×M₃₂	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
20	#11758412	—	2	1	no	—
21	#11758407	C₂²	1	1	yes	characteristic, complemented
22	Dic₂₄₈	—	2	1	no	—
23	#11758388	—	2	2	no	—
24	#11758388	—	2	1	no	—
25	#41×C₃₁	D₄	1	1	yes	characteristic
26	#92499	—	4	1	no	—
27	D₄×D₃₁	—	4	2	no	—
28	D₄×D₃₁	—	4	1	no	—
29	#92495	—	2	2	no	—
30	#92495	—	2	1	no	—
31	C₂×D₁₂₄	—	2	2	no	—
32	C₂×D₁₂₄	—	2	1	no	—
33	C₂×Dic₆₂	—	2	1	no	—
34	M₁₆×C₃₁	—	2	1	no	—
35	M₁₆×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₈×C₃₁	—	2	1	no	—
37	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	#92482	—	4	1	no	—
39	#92480	—	4	1	no	—
40	#92474	—	2	1	no	—
41	Dic₁₂₄	—	2	1	no	—
42	Dic₁₂₄	—	2	1	no	—
43	Dic₁₂₄	D₄	1	1	yes	characteristic
44	D₂₄₈	—	2	2	no	—
45	D₂₄₈	—	2	1	no	—
46	D₂₄₈	D₄	1	1	yes	characteristic
47	D₂₄₈	D₄	1	2	yes	—
48	#92470	—	2	2	no	—
49	#92470	—	2	1	no	—
50	C₁₆×C₃₁	—	2	1	no	—
51	C₁₆×C₃₁	—	2	1	no	—
52	C₂²×D₃₁	—	4	2	no	—
53	C₂²×D₃₁	—	4	1	no	—
54	Q₈×C₃₁	D₈	1	1	yes	characteristic
55	D₄×C₃₁	—	2	1	no	—
56	D₄×C₃₁	D₈	1	1	yes	characteristic
57	C₂×C₄×C₃₁	—	2	1	no	—
58	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₃₁	—	4	2	no	—
60	C₂²⋊D₃₁	—	4	1	no	—
61	C₂×Dic₃₁	—	4	1	no	—
62	D₁₂₄	—	2	2	no	—
63	D₁₂₄	—	2	1	no	—
64	D₁₂₄	—	4	2	no	—
65	D₁₂₄	—	4	1	no	—
66	C₄×D₃₁	—	4	2	no	—
67	C₄×D₃₁	—	4	1	no	—
68	Dic₆₂	—	2	1	no	—
69	Dic₆₂	—	4	1	no	—
70	Dic₆₂	—	4	1	no	—
71	C₈×C₃₁	—	2	1	no	—
72	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₃₁⋊C₈	—	4	1	no	—
75	#2194	—	31	1	no	maximal, Sylow
76	C₂²×C₃₁	—	2	1	no	—
77	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×D₃₁	—	4	2	no	—
79	C₂×D₃₁	—	4	1	no	—
80	C₂×D₃₁	—	16	1	no	—
81	C₂×D₃₁	—	16	2	no	—
82	C₄×C₃₁	—	2	1	no	—
83	C₄×C₃₁	#120	1	1	yes	characteristic
84	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
85	Dic₃₁	—	4	1	no	—
86	Dic₃₁	—	8	1	no	—
87	#576	—	31	1	no	—
88	#510	—	31	1	no	—
89	C₂×SD₃₂	—	31	1	no	—
90	#473	—	31	1	no	—
91	#361	—	31	1	no	—
92	#359	—	31	1	no	—
93	#350	D₃₁	1	1	yes	characteristic, complemented
94	C₂×C₃₁	—	2	1	no	—
95	C₂×C₃₁	—	4	1	no	—
96	C₂×C₃₁	#447	1	1	yes	characteristic
97	D₃₁	—	8	2	no	—
98	D₃₁	—	8	1	no	—
99	H(5,𝔽₂)	—	62	1	no	—
100	#137	—	62	1	no	—
101	AGL(1,ℤ/8)	—	62	2	no	—
102	AGL(1,ℤ/8)	—	62	1	no	—
103	#135	—	31	1	no	—
104	#135	—	62	1	no	—
105	C₂×Q₁₆	—	31	1	no	—
106	C₂×D₈	—	31	1	no	—
107	C₂×D₈	—	62	1	no	—
108	#131	C₂×D₃₁	1	1	yes	characteristic
109	Q₃₂	—	62	1	no	—
110	SD₃₂	—	62	2	no	—
111	SD₃₂	—	62	1	no	—
112	M₃₂	C₂×D₃₁	1	1	yes	characteristic, complemented
113	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic, complemented
114	#108	—	31	1	no	—
115	#101	—	62	1	no	—
116	C₃₁	#2194	1	1	yes	characteristic, complemented, Sylow
117	#41	—	62	2	no	—
118	#41	—	62	1	no	—
119	#41	—	124	1	no	—
120	#41	D₁₂₄	1	1	yes	characteristic
121	C₂×Q₈	—	62	1	no	—
122	C₂×D₄	—	62	2	no	—
123	C₂×D₄	—	62	1	no	—
124	C₂×D₄	—	124	2	no	—
125	C₂×D₄	—	124	1	no	—
126	Q₁₆	—	31	1	no	—
127	Q₁₆	—	62	1	no	—
128	Q₁₆	—	62	1	no	—
129	Q₁₆	—	124	1	no	—
130	SD₁₆	—	62	2	no	—
131	SD₁₆	—	62	1	no	—
132	SD₁₆	—	124	1	no	—
133	D₈	—	31	2	no	—
134	D₈	—	31	1	no	—
135	D₈	—	62	2	no	—
136	D₈	—	62	1	no	—
137	M₁₆	—	2	1	no	—
138	M₁₆	—	62	1	no	—
139	M₁₆	D₁₂₄	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
142	C₁₆	—	2	1	no	—
143	C₁₆	—	2	1	no	—
144	C₂³	—	124	2	no	—
145	C₂³	—	124	1	no	—
146	Q₈	—	62	1	no	—
147	Q₈	—	124	1	no	—
148	Q₈	—	124	1	no	—
149	Q₈	D₂₄₈	1	1	yes	characteristic
150	D₄	—	2	1	no	—
151	D₄	—	62	2	no	—
152	D₄	—	62	1	no	—
153	D₄	—	124	2	no	—
154	D₄	—	124	2	no	—
155	D₄	—	124	1	no	—
156	D₄	—	124	1	no	—
157	D₄	D₂₄₈	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	124	2	no	—
160	C₂×C₄	—	124	1	no	—
161	C₂×C₄	—	124	1	no	—
162	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	124	1	no	—
165	C₈	D₄×D₃₁	1	1	yes	characteristic
166	C₈	D₄×D₃₁	1	1	yes	characteristic
167	C₂²	—	2	1	no	—
168	C₂²	—	124	2	no	—
169	C₂²	—	124	1	no	—
170	C₂²	—	496	1	no	—
171	C₂²	—	496	2	no	—
172	C₂²	#11758485	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	124	1	no	—
175	C₄	—	248	1	no	—
176	C₄	#11758459	1	1	yes	characteristic
177	C₄	C₂×D₂₄₈	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	4	1	no	—
180	C₂	—	248	2	no	—
181	C₂	—	248	1	no	—
182	C₂	#14521185	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14521185
commutator, cosocle	C₂×C₈×C₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×#350	C₂
socle	C₂×C₃₁	#447
prime radical	#14521854	C₂
hypercenter	#350	D₃₁
hypocenter	C₃₁	#2194
LCS	G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
2. #14521492	C₂	semidirect
3. C₂×#11758388	C₂	semidirect
4. #14521324	C₂	semidirect
5. #14520863	C₂	semidirect
6. #14520857	C₂	semidirect
7. C₃₁×#350	C₂	semidirect
13. C₂×Dic₁₂₄	C₂²	semidirect
18. C₃₁×M₃₂	C₂²	semidirect
19. C₂×C₁₆×C₃₁	C₂²	semidirect
21. #11758407	C₂²	semidirect
93. #350	D₃₁	semidirect
112. M₃₂	C₂×D₃₁	semidirect
113. C₂×C₁₆	C₂×D₃₁	semidirect
116. C₃₁	#2194	semidirect

Maximal subgroups

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

subgroup	index
#14521854	2
#14521492	2
C₂×#11758388	2
#14521324	2
#14520863	2
#14520857	2
C₃₁×#350	2
#2194	31