Subgroups — #16854274

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92474	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
3	#14521077	C₂	1	4	yes	maximal, complemented
4	#14521075	C₂	1	4	yes	maximal, complemented
5	#14521072	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₃₁	—	2	4	no	—
9	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₃₁	C₄	1	2	yes	—
13	#11758503	C₄	1	2	yes	—
14	#11758503	C₄	1	2	yes	—
15	#11758503	C₂²	1	2	yes	—
16	#11758503	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₃₁	C₄	1	2	yes	—
18	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
19	C₂×#92474	—	2	4	no	—
20	C₂×#92474	C₂²	1	2	yes	—
21	C₂×#92474	C₂²	1	2	yes	—
22	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
23	Dic₃₁⋊C₈	C₂²	1	16	yes	complemented
24	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	D₄	1	4	yes	—
29	C₂²×C₄×C₃₁	Q₈	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃₁	C₂×C₄	1	4	yes	—
34	M₁₆×C₃₁	—	2	4	no	—
35	M₁₆×C₃₁	—	2	4	no	—
36	C₂×C₈×C₃₁	—	2	4	no	—
37	C₂×C₈×C₃₁	C₂³	1	4	yes	—
38	C₂²⋊Dic₃₁	C₂×C₄	1	4	yes	—
39	C₂²⋊Dic₃₁	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
42	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
43	C₄×Dic₃₁	C₂³	1	4	yes	—
44	#92474	—	2	4	no	—
45	#92474	—	2	4	no	—
46	C₂×C₃₁⋊C₈	—	2	4	no	—
47	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
48	C₂³×C₃₁	—	4	2	no	—
49	C₂³×C₃₁	C₄⋊C₄	1	4	yes	—
50	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₃₁	—	4	4	no	—
67	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
68	C₈×C₃₁	—	2	4	no	—
69	C₃₁⋊C₈	—	2	4	no	—
70	#2905	—	31	1	no	maximal, Sylow
71	C₂²×C₃₁	—	2	2	no	—
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	—	2	2	no	—
74	C₂²×C₃₁	—	4	4	no	—
75	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
79	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
82	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
84	Dic₃₁	—	2	8	no	—
85	C₂²×M₁₆	—	31	1	no	—
86	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
87	C₂×#117	—	31	1	no	—
88	#424	—	31	4	no	—
89	#424	—	31	4	no	—
90	#423	—	31	4	no	—
91	C₂×C₃₁	—	2	2	no	—
92	C₂×C₃₁	—	2	2	no	—
93	C₂×C₃₁	—	2	2	no	—
94	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃₁	#568	1	2	yes	—
96	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
97	C₂×M₁₆	—	2	4	no	—
98	C₂×M₁₆	—	31	2	no	—
99	C₂×M₁₆	—	31	2	no	—
100	C₂×M₁₆	—	62	4	no	—
101	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
102	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
103	C₂²×C₈	—	31	2	no	—
104	C₂²×C₈	C₂×D₃₁	1	2	yes	—
105	#117	—	31	2	no	—
106	#117	—	31	2	no	—
107	#117	—	31	2	no	—
108	#117	—	31	2	no	—
109	C₂×C₄⋊C₄	—	31	2	no	—
110	C₂×C₂²⋊C₄	—	31	2	no	—
111	C₂×C₄²	—	31	2	no	—
112	C₄⋊C₈	—	31	16	no	—
113	C₃₁	#2905	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴	C₄×D₃₁	1	1	yes	characteristic
115	C₂²×C₄	—	31	4	no	—
116	C₂²×C₄	Dic₆₂	1	4	yes	—
117	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊D₃₁	1	4	yes	—
121	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	2	4	no	—
126	M₁₆	—	62	4	no	—
127	M₁₆	—	62	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	31	4	no	—
130	C₂×C₈	—	62	4	no	—
131	C₂×C₈	C₂²×D₃₁	1	4	yes	—
132	C₄⋊C₄	—	31	4	no	—
133	C₄⋊C₄	—	31	4	no	—
134	C₂²⋊C₄	—	31	4	no	—
135	C₂²⋊C₄	—	31	4	no	—
136	C₄²	—	31	4	no	—
137	C₄²	—	31	4	no	—
138	C₂³	—	4	2	no	—
139	C₂³	Dic₃₁⋊C₄	1	4	yes	—
140	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
142	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
143	C₂×C₄	—	31	8	no	—
144	C₂×C₄	—	124	4	no	—
145	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
146	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
147	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
148	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
149	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
150	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
157	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	62	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
165	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
166	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
167	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
168	C₄	—	62	8	no	—
169	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
170	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
171	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
172	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
173	C₄	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×Dic₃₁⋊C₄	1	1	yes	characteristic
178	C₂	#14521815	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂²×M₁₆×C₃₁	C₂
prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	#2905
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14521077	C₂	semidirect
4. #14521075	C₂	semidirect
5. #14521072	C₂	semidirect
23. Dic₃₁⋊C₈	C₂²	semidirect
113. C₃₁	#2905	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#92474	2
C₂×#11758503	2
#14521077	2
#14521075	2
#14521072	2
C₂²×M₁₆×C₃₁	2
#2905	31