Subgroups — #16854837

Order 3968.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#92495	C₂	1	1	yes	maximal, characteristic
2	#14521176	C₂	1	2	yes	maximal, complemented
3	#14521118	C₂	1	4	yes	maximal, complemented
4	#14521114	C₂	1	2	yes	maximal, complemented
5	#14521105	C₂	1	2	yes	maximal, complemented
6	#14521102	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#92495	C₄	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	2	yes	—
13	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
14	#11758503	C₄	1	1	yes	characteristic
15	C₂×C₈×D₃₁	C₂²	1	4	yes	—
16	C₄×D₁₂₄	C₄	1	1	yes	characteristic
17	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
18	#11758449	C₄	1	2	yes	—
19	C₄²×D₃₁	C₂²	1	2	yes	—
20	C₄×Dic₆₂	C₄	1	1	yes	characteristic
21	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
25	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	—
26	#11758403	C₂²	1	4	yes	—
27	Dic₃₁⋊C₈	C₂²	1	4	yes	—
28	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
30	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
31	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
35	#92495	—	2	4	no	—
36	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₃₁	C₂³	1	2	yes	—
40	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₃₁	—	4	2	no	—
42	C₂×C₈×C₃₁	C₂³	1	4	yes	—
43	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
48	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
49	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
52	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
54	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
55	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
56	C₄×Dic₃₁	C₂³	1	2	yes	—
57	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
58	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
59	C₈×D₃₁	—	4	4	no	—
60	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	1	no	—
66	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₃₁	—	2	4	no	—
73	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
74	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
76	D₁₂₄	M₁₆	1	4	yes	complemented
77	C₄×D₃₁	—	2	4	no	—
78	C₄×D₃₁	—	2	4	no	—
79	C₄×D₃₁	—	2	2	no	—
80	C₄×D₃₁	—	2	2	no	—
81	Dic₆₂	—	2	2	no	—
82	C₈×C₃₁	—	2	4	no	—
83	C₃₁⋊C₈	—	2	2	no	—
84	C₃₁⋊C₈	—	2	2	no	—
85	#2990	—	31	1	no	maximal, Sylow
86	C₂²×C₃₁	—	2	1	no	—
87	C₂²×C₃₁	—	2	1	no	—
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₃₁	—	2	2	no	—
91	C₂×D₃₁	—	2	2	no	—
92	C₂×D₃₁	C₂×M₁₆	1	4	yes	—
93	C₄×C₃₁	—	2	1	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
97	C₄×C₃₁	#131	1	2	yes	—
98	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
99	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
100	Dic₃₁	—	2	2	no	—
101	Dic₃₁	—	2	2	no	—
102	Dic₃₁	—	2	2	no	—
103	C₄×#41	—	31	1	no	—
104	#446	—	31	2	no	—
105	#435	—	31	4	no	—
106	C₈×D₄	—	31	2	no	—
107	#432	—	31	2	no	—
108	#431	—	31	1	no	—
109	#423	D₃₁	1	1	yes	characteristic, complemented
110	C₂×C₄⋊C₈	—	31	2	no	—
111	C₂×C₃₁	—	4	1	no	—
112	C₂×C₃₁	#520	1	1	yes	characteristic
113	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
114	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
115	D₃₁	—	2	4	no	—
116	C₂×#41	—	31	1	no	—
117	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
118	C₂²×C₈	—	31	4	no	—
119	C₄×Q₈	—	31	1	no	—
120	C₄×Q₈	—	31	1	no	—
121	C₄×D₄	—	31	2	no	—
122	C₄×D₄	—	31	2	no	—
123	C₄×D₄	—	31	1	no	—
124	C₄×D₄	—	31	1	no	—
125	#117	—	31	1	no	—
126	#117	—	31	2	no	—
127	C₂×C₄²	—	31	2	no	—
128	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
129	C₄⋊C₈	—	31	4	no	—
130	C₄⋊C₈	—	31	1	no	—
131	C₄⋊C₈	—	31	1	no	—
132	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
133	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
134	C₂²⋊C₈	—	31	2	no	—
135	C₂²⋊C₈	—	31	4	no	—
136	#97	—	31	4	no	—
137	C₄×C₈	—	31	2	no	—
138	C₃₁	#2990	1	1	yes	characteristic, complemented, Sylow
139	#41	—	62	4	no	—
140	C₂×Q₈	—	31	1	no	—
141	C₂×D₄	—	31	2	no	—
142	C₂×D₄	—	31	1	no	—
143	C₂²×C₄	—	31	2	no	—
144	C₂²×C₄	—	31	2	no	—
145	C₂²×C₄	—	31	2	no	—
146	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	31	2	no	—
151	C₂×C₈	—	31	2	no	—
152	C₂×C₈	—	124	4	no	—
153	C₂×C₈	C₂²×D₃₁	1	4	yes	—
154	C₄⋊C₄	—	31	2	no	—
155	C₄⋊C₄	—	31	2	no	—
156	C₄⋊C₄	—	31	1	no	—
157	C₄⋊C₄	—	31	1	no	—
158	C₂²⋊C₄	—	31	2	no	—
159	C₂²⋊C₄	—	31	2	no	—
160	C₂²⋊C₄	—	31	2	no	—
161	C₄²	—	31	2	no	—
162	C₄²	—	31	2	no	—
163	C₄²	—	31	2	no	—
164	C₄²	C₄×D₃₁	1	1	yes	characteristic
165	C₄²	C₄×D₃₁	1	1	yes	characteristic
166	C₄²	C₂²×D₃₁	1	1	yes	characteristic
167	C₄²	C₂²×D₃₁	1	1	yes	characteristic
168	C₂³	—	31	2	no	—
169	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
170	Q₈	—	62	2	no	—
171	D₄	—	31	4	no	—
172	D₄	—	62	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	31	2	no	—
178	C₂×C₄	—	31	2	no	—
179	C₂×C₄	—	31	2	no	—
180	C₂×C₄	—	62	4	no	—
181	C₂×C₄	—	62	4	no	—
182	C₂×C₄	—	62	2	no	—
183	C₂×C₄	—	62	2	no	—
184	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	62	2	no	—
192	C₈	—	62	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	31	4	no	—
197	C₂²	—	62	2	no	—
198	C₂²	—	62	2	no	—
199	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	62	2	no	—
204	C₄	—	62	2	no	—
205	C₄	—	62	2	no	—
206	C₄	M₁₆×D₃₁	1	2	yes	—
207	C₄	#11758484	1	2	yes	—
208	C₄	#11758561	1	1	yes	characteristic
209	C₄	#11758564	1	1	yes	characteristic
210	C₂	—	4	1	no	—
211	C₂	—	62	4	no	—
212	C₂	#14521530	1	1	yes	characteristic
213	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
214	C₂	C₂×#11758484	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#423	C₂
prime radical	C₂×#92495	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#423	D₃₁
hypocenter	C₃₁	#2990
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #14521176	C₂	semidirect
3. #14521118	C₂	semidirect
4. #14521114	C₂	semidirect
5. #14521105	C₂	semidirect
6. #14521102	C₂	semidirect
7. C₄⋊C₈×D₃₁	C₂	semidirect
8. C₃₁×#423	C₂	semidirect
76. D₁₂₄	M₁₆	semidirect
109. #423	D₃₁	semidirect
138. C₃₁	#2990	semidirect

Maximal subgroups

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

subgroup	index
C₄×#92495	2
#14521176	2
#14521118	2
#14521114	2
#14521105	2
#14521102	2
C₄⋊C₈×D₃₁	2
C₃₁×#423	2
#2990	31