Subgroups — #16854140

Order 3968.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758484	C₂	1	2	yes	maximal
2	C₄×#92495	C₂	1	1	yes	maximal, characteristic
3	#14521007	C₂	1	2	yes	maximal, complemented
4	#14521003	C₂	1	4	yes	maximal, complemented
5	#14521001	C₂	1	2	yes	maximal, complemented
6	C₂×#11758391	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#92495	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₃₁	C₄	1	4	yes	—
13	#11758503	C₂²	1	1	yes	characteristic
14	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
15	#11758484	C₄	1	16	yes	complemented
16	C₂×#92469	C₂²	1	4	yes	—
17	C₂×C₈×D₃₁	C₂²	1	4	yes	—
18	C₄×D₁₂₄	C₄	1	2	yes	—
19	#11758449	C₂²	1	2	yes	—
20	C₄²×D₃₁	C₂²	1	2	yes	—
21	C₄×Dic₆₂	C₄	1	2	yes	—
22	C₃₁×#97	C₂²	1	2	yes	—
23	C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	#11758403	C₂²	1	4	yes	—
25	C₈×Dic₃₁	C₂²	1	4	yes	—
26	#11758391	C₂²	1	1	yes	characteristic
27	#11758391	C₂²	1	1	yes	characteristic
28	#11758391	C₂²	1	2	yes	—
29	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
32	#92495	C₂×C₄	1	8	yes	—
33	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₃₁	C₂³	1	2	yes	—
36	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
37	M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₃₁	C₂³	1	4	yes	—
39	C₄²×C₃₁	C₂×C₄	1	2	yes	—
40	C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
43	(C₂×D₃₁)⋊C₄	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
45	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
46	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
47	C₄×Dic₃₁	C₂³	1	2	yes	—
48	C₂×C₃₁⋊C₈	C₂×C₄	1	8	yes	complemented
49	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
50	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
51	#92469	C₂×C₄	1	16	yes	complemented
52	C₈×D₃₁	C₂×C₄	1	16	yes	complemented
53	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₁	—	4	2	no	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₃₁	C₄²	1	8	yes	—
62	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
63	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
64	D₁₂₄	C₄²	1	4	yes	—
65	C₄×D₃₁	—	4	4	no	—
66	C₄×D₃₁	C₂²×C₄	1	8	yes	—
67	Dic₆₂	C₄²	1	4	yes	—
68	C₈×C₃₁	C₂²×C₄	1	8	yes	—
69	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
70	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
71	#2875	—	31	1	no	maximal, Sylow
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	C₂×C₄²	1	2	yes	—
74	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
75	C₂×D₃₁	—	2	4	no	—
76	C₂×D₃₁	C₂×C₄²	1	4	yes	—
77	C₄×C₃₁	—	2	2	no	—
78	C₄×C₃₁	—	2	2	no	—
79	C₄×C₃₁	C₂×C₄²	1	2	yes	—
80	C₄×C₃₁	C₂³×C₄	1	2	yes	—
81	Dic₃₁	—	2	4	no	—
82	Dic₃₁	C₂×C₄²	1	4	yes	—
83	C₂×#131	—	31	2	no	—
84	C₄×#41	—	31	1	no	—
85	#405	—	31	2	no	—
86	#405	—	31	4	no	—
87	C₄×M₁₆	—	31	2	no	—
88	C₄×M₁₆	D₃₁	1	1	yes	characteristic, complemented
89	C₂×#97	—	31	1	no	—
90	C₂×#97	—	31	2	no	—
91	C₂×C₃₁	—	2	2	no	—
92	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
93	C₂×C₃₁	#568	1	2	yes	—
94	D₃₁	—	2	4	no	—
95	C₂×#41	—	31	1	no	—
96	#131	—	31	16	no	—
97	C₂×M₁₆	—	31	4	no	—
98	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
99	C₂²×C₈	—	31	2	no	—
100	C₂²×C₈	—	31	4	no	—
101	C₄×Q₈	—	31	2	no	—
102	C₄×D₄	—	31	4	no	—
103	C₄×D₄	—	31	2	no	—
104	#117	—	31	1	no	—
105	#117	—	31	2	no	—
106	C₂×C₄²	—	31	2	no	—
107	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
108	#97	—	31	4	no	—
109	#97	—	31	1	no	—
110	#97	—	31	1	no	—
111	#97	—	31	2	no	—
112	#97	C₂×D₃₁	1	2	yes	—
113	C₄×C₈	—	31	4	no	—
114	C₄×C₈	C₂×D₃₁	1	2	yes	—
115	C₃₁	#2875	1	1	yes	characteristic, complemented, Sylow
116	#41	—	31	8	no	—
117	C₂×Q₈	—	31	1	no	—
118	C₂×D₄	—	31	2	no	—
119	C₂×D₄	—	31	1	no	—
120	C₂²×C₄	—	31	2	no	—
121	C₂²×C₄	—	31	4	no	—
122	C₂²×C₄	C₄×D₃₁	1	2	yes	—
123	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
124	M₁₆	—	31	16	no	—
125	M₁₆	C₄×D₃₁	1	8	yes	complemented
126	C₂×C₈	—	31	8	no	—
127	C₂×C₈	—	31	2	no	—
128	C₂×C₈	—	31	2	no	—
129	C₂×C₈	—	31	16	no	—
130	C₂×C₈	C₂²×D₃₁	1	4	yes	—
131	C₄⋊C₄	—	31	4	no	—
132	C₄⋊C₄	—	31	2	no	—
133	C₂²⋊C₄	—	31	2	no	—
134	C₂²⋊C₄	—	31	4	no	—
135	C₄²	—	31	4	no	—
136	C₄²	—	31	2	no	—
137	C₄²	C₄×D₃₁	1	2	yes	—
138	C₄²	C₂²×D₃₁	1	1	yes	characteristic
139	C₄²	C₂²×D₃₁	1	1	yes	characteristic
140	C₂³	—	31	2	no	—
141	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
142	Q₈	—	31	4	no	—
143	D₄	—	31	8	no	—
144	D₄	—	31	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	31	2	no	—
147	C₂×C₄	—	31	8	no	—
148	C₂×C₄	—	31	4	no	—
149	C₂×C₄	—	124	4	no	—
150	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
154	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
155	C₈	—	31	4	no	—
156	C₈	—	31	4	no	—
157	C₈	C₂×C₄×D₃₁	1	8	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	31	4	no	—
160	C₂²	—	62	4	no	—
161	C₂²	C₄²×D₃₁	1	2	yes	—
162	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	31	4	no	—
166	C₄	—	62	4	no	—
167	C₄	C₄²×D₃₁	1	2	yes	—
168	C₄	C₂²×C₄×D₃₁	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	62	4	no	—
171	C₂	C₂×C₄²×D₃₁	1	1	yes	characteristic
172	C₂	#14521815	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Fitting	C₄×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂×#92495	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×M₁₆	D₃₁
hypocenter	C₃₁	#2875
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14521007	C₂	semidirect
4. #14521003	C₂	semidirect
5. #14521001	C₂	semidirect
6. C₂×#11758391	C₂	semidirect
7. #97×D₃₁	C₂	semidirect
8. C₄×M₁₆×C₃₁	C₂	semidirect
15. #11758484	C₄	semidirect
37. M₁₆×C₃₁	C₂×C₄	semidirect
48. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
51. #92469	C₂×C₄	semidirect
52. C₈×D₃₁	C₂×C₄	semidirect
88. C₄×M₁₆	D₃₁	semidirect
115. C₃₁	#2875	semidirect
125. M₁₆	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758484	2
C₄×#92495	2
#14521007	2
#14521003	2
#14521001	2
C₂×#11758391	2
#97×D₃₁	2
C₄×M₁₆×C₃₁	2
#2875	31