Finite Group Databasebeta

Subgroups — #16847683

Order 3968.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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	#14521851	C₂	1	1	yes	maximal, characteristic
2	#14521489	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758388	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521325	C₂	1	1	yes	maximal, characteristic, complemented
5	#14520861	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520832	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#11758566	—	2	1	no	—
9	C₃₁×#135	C₂²	1	1	yes	characteristic
10	#11758526	—	2	1	no	—
11	#11758496	—	2	1	no	—
12	#11758491	—	2	2	no	—
13	D₈×D₃₁	—	2	1	no	—
14	#11758486	—	2	1	no	—
15	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₂₄₈	C₂²	1	1	yes	characteristic
17	#11758478	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
20	#11758417	—	2	1	no	—
21	#11758415	—	2	1	no	—
22	#11758412	—	2	1	no	—
23	#11758400	C₂²	1	1	yes	characteristic, complemented
24	#11758388	—	2	2	no	—
25	#41×C₃₁	—	2	1	no	—
26	#92499	—	2	2	no	—
27	D₄×D₃₁	—	2	2	no	—
28	D₄×D₃₁	—	4	2	no	—
29	#92495	—	4	1	no	—
30	#92495	D₄	1	1	yes	characteristic
31	C₂×D₁₂₄	—	2	1	no	—
32	C₂×D₁₂₄	—	4	1	no	—
33	C₂×Dic₆₂	—	2	1	no	—
34	Q₁₆×C₃₁	D₄	1	1	yes	characteristic
35	SD₁₆×C₃₁	—	2	1	no	—
36	D₈×C₃₁	D₄	1	1	yes	characteristic
37	M₁₆×C₃₁	—	2	1	no	—
38	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	#92481	—	2	2	no	—
40	#92479	—	2	2	no	—
41	#92474	D₄	1	1	yes	characteristic
42	Dic₁₂₄	—	2	1	no	—
43	Dic₁₂₄	—	2	1	no	—
44	Dic₁₂₄	—	4	1	no	—
45	D₂₄₈	—	2	1	no	—
46	D₂₄₈	D₄	1	2	yes	—
47	#92470	—	4	1	no	—
48	#92469	—	2	1	no	—
49	C₈×D₃₁	—	2	1	no	—
50	C₁₆×C₃₁	—	2	1	no	—
51	C₃₁⋊C₁₆	—	2	1	no	—
52	C₂²×D₃₁	—	4	2	no	—
53	C₂²×D₃₁	—	4	1	no	—
54	Q₈×C₃₁	—	2	1	no	—
55	D₄×C₃₁	—	2	1	no	—
56	D₄×C₃₁	—	4	1	no	—
57	C₂×C₄×C₃₁	—	4	1	no	—
58	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₃₁	—	2	1	no	—
60	C₂²⋊D₃₁	—	4	2	no	—
61	C₂×Dic₃₁	—	4	1	no	—
62	D₁₂₄	—	2	2	no	—
63	D₁₂₄	—	4	2	no	—
64	D₁₂₄	—	4	1	no	—
65	D₁₂₄	D₈	1	1	yes	characteristic
66	C₄×D₃₁	—	2	1	no	—
67	C₄×D₃₁	—	4	2	no	—
68	Dic₆₂	—	4	1	no	—
69	Dic₆₂	—	4	1	no	—
70	Dic₆₂	D₈	1	1	yes	characteristic
71	C₈×C₃₁	—	4	1	no	—
72	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₃₁⋊C₈	—	2	1	no	—
75	#2194	—	31	1	no	maximal, Sylow
76	C₂²×C₃₁	—	4	1	no	—
77	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×D₃₁	—	2	1	no	—
79	C₂×D₃₁	—	4	2	no	—
80	C₂×D₃₁	—	16	2	no	—
81	C₂×D₃₁	—	16	1	no	—
82	C₄×C₃₁	—	4	1	no	—
83	C₄×C₃₁	#120	1	1	yes	characteristic
84	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
85	Dic₃₁	—	2	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	D₃₁	1	1	yes	characteristic, complemented
93	#350	—	31	1	no	—
94	C₂×C₃₁	—	2	1	no	—
95	C₂×C₃₁	—	8	1	no	—
96	C₂×C₃₁	#447	1	1	yes	characteristic
97	D₃₁	—	4	1	no	—
98	D₃₁	—	8	2	no	—
99	H(5,𝔽₂)	—	62	1	no	—
100	#137	—	62	1	no	—
101	AGL(1,ℤ/8)	—	62	1	no	—
102	AGL(1,ℤ/8)	—	62	2	no	—
103	#135	—	62	1	no	—
104	#135	C₂×D₃₁	1	1	yes	characteristic
105	C₂×Q₁₆	—	31	1	no	—
106	C₂×D₈	—	31	1	no	—
107	C₂×D₈	—	62	1	no	—
108	#131	—	31	1	no	—
109	Q₃₂	—	62	1	no	—
110	SD₃₂	—	62	2	no	—
111	SD₃₂	—	62	1	no	—
112	M₃₂	—	31	1	no	—
113	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic, complemented
114	#108	C₂×D₃₁	1	1	yes	characteristic, complemented
115	#101	—	62	1	no	—
116	C₃₁	#2194	1	1	yes	characteristic, complemented, Sylow
117	#41	—	2	1	no	—
118	#41	—	31	1	no	—
119	#41	—	62	2	no	—
120	#41	—	124	1	no	—
121	C₂×Q₈	—	62	1	no	—
122	C₂×D₄	—	62	2	no	—
123	C₂×D₄	—	62	1	no	—
124	C₂×D₄	—	124	1	no	—
125	C₂×D₄	—	124	2	no	—
126	Q₁₆	—	62	1	no	—
127	Q₁₆	—	62	1	no	—
128	Q₁₆	—	124	1	no	—
129	Q₁₆	D₁₂₄	1	1	yes	characteristic
130	SD₁₆	—	2	1	no	—
131	SD₁₆	—	62	2	no	—
132	SD₁₆	—	124	1	no	—
133	D₈	—	31	2	no	—
134	D₈	—	62	2	no	—
135	D₈	—	62	1	no	—
136	D₈	D₁₂₄	1	1	yes	characteristic
137	M₁₆	—	2	1	no	—
138	M₁₆	—	31	1	no	—
139	M₁₆	—	62	1	no	—
140	C₂×C₈	—	62	1	no	—
141	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
142	C₁₆	—	2	1	no	—
143	C₁₆	—	62	1	no	—
144	C₂³	—	124	2	no	—
145	C₂³	—	124	1	no	—
146	Q₈	—	2	1	no	—
147	Q₈	—	31	1	no	—
148	Q₈	—	124	1	no	—
149	Q₈	—	124	1	no	—
150	D₄	—	2	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	31	1	no	—
153	D₄	—	62	1	no	—
154	D₄	—	62	2	no	—
155	D₄	—	124	2	no	—
156	D₄	—	124	2	no	—
157	D₄	—	124	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	62	1	no	—
160	C₂×C₄	—	124	2	no	—
161	C₂×C₄	—	124	1	no	—
162	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
163	C₈	—	4	1	no	—
164	C₈	—	62	1	no	—
165	C₈	C₂×D₁₂₄	1	1	yes	characteristic
166	C₈	D₄×D₃₁	1	1	yes	characteristic
167	C₂²	—	4	1	no	—
168	C₂²	—	62	1	no	—
169	C₂²	—	124	2	no	—
170	C₂²	—	496	1	no	—
171	C₂²	—	496	2	no	—
172	C₂²	#11758491	1	1	yes	characteristic
173	C₄	—	4	1	no	—
174	C₄	—	62	1	no	—
175	C₄	—	248	1	no	—
176	C₄	#11758459	1	1	yes	characteristic
177	C₄	D₈×D₃₁	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	8	1	no	—
180	C₂	—	124	1	no	—
181	C₂	—	248	2	no	—
182	C₂	#14521184	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14521184
commutator, cosocle	C₂×C₈×C₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×#359	C₂
socle	C₂×C₃₁	#447
prime radical	#14521851	C₂
hypercenter	#359	D₃₁
hypocenter	C₃₁	#2194
LCS	G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
2. #14521489	C₂	semidirect
3. C₂×#11758388	C₂	semidirect
4. #14521325	C₂	semidirect
5. #14520861	C₂	semidirect
6. #14520832	C₂	semidirect
7. C₃₁×#359	C₂	semidirect
15. C₂×Dic₁₂₄	C₂²	semidirect
18. C₂×C₁₆×C₃₁	C₂²	semidirect
19. C₃₁×#108	C₂²	semidirect
23. #11758400	C₂²	semidirect
92. #359	D₃₁	semidirect
113. C₂×C₁₆	C₂×D₃₁	semidirect
114. #108	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
#14521851	2
#14521489	2
C₂×#11758388	2
#14521325	2
#14520861	2
#14520832	2
C₃₁×#359	2
#2194	31