Subgroups — #16854705

Order 3968.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521583	C₂	1	1	yes	maximal, characteristic
2	#14521177	C₂	1	2	yes	maximal, complemented
3	#14521122	C₂	1	2	yes	maximal, complemented
4	#14521120	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
6	#14521111	C₂	1	2	yes	maximal, complemented
7	#14521102	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521006	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	#11758507	C₄	1	1	yes	characteristic
13	#11758506	C₄	1	2	yes	—
14	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
15	#11758501	C₄	1	1	yes	characteristic
16	C₂×#92469	C₂²	1	2	yes	—
17	C₂×C₈×D₃₁	C₂²	1	2	yes	—
18	#11758453	C₄	1	2	yes	—
19	#11758452	C₄	1	1	yes	characteristic
20	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
21	#11758449	C₂²	1	2	yes	—
22	#11758447	C₄	1	1	yes	characteristic
23	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
24	C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₃₁×#97	C₂²	1	2	yes	—
26	C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
29	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
31	#11758403	C₂²	1	2	yes	—
32	Dic₃₁⋊C₈	C₂²	1	2	yes	—
33	Dic₃₁⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₃₁	C₂²	1	2	yes	—
35	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
36	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
37	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
41	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₃₁	C₂³	1	2	yes	—
43	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂×C₈×C₃₁	C₂³	1	2	yes	—
47	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
52	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
58	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₃₁	C₂³	1	2	yes	—
63	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
64	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
65	#92469	—	2	4	no	—
66	C₈×D₃₁	—	2	4	no	—
67	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	4	1	no	—
72	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₃₁	—	2	4	no	—
79	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
80	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
81	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
82	D₁₂₄	—	4	1	no	—
83	C₄×D₃₁	—	2	2	no	—
84	C₄×D₃₁	—	2	2	no	—
85	Dic₆₂	—	4	1	no	—
86	C₈×C₃₁	—	2	2	no	—
87	C₈×C₃₁	#41	1	4	yes	—
88	C₃₁⋊C₈	—	2	2	no	—
89	C₃₁⋊C₈	—	2	2	no	—
90	#2968	—	31	1	no	maximal, Sylow
91	C₂²×C₃₁	—	2	1	no	—
92	C₂²×C₃₁	—	2	1	no	—
93	C₂²×C₃₁	#131	1	2	yes	—
94	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₃₁	—	2	2	no	—
96	C₂×D₃₁	—	2	2	no	—
97	C₂×D₃₁	—	2	2	no	—
98	C₄×C₃₁	—	2	1	no	—
99	C₄×C₃₁	—	2	1	no	—
100	C₄×C₃₁	—	2	1	no	—
101	C₄×C₃₁	—	2	1	no	—
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	C₂×#41	1	2	yes	—
104	Dic₃₁	—	2	2	no	—
105	Dic₃₁	—	2	2	no	—
106	Dic₃₁	—	2	2	no	—
107	#529	—	31	1	no	—
108	#446	—	31	2	no	—
109	#435	—	31	2	no	—
110	#435	—	31	2	no	—
111	C₈×D₄	—	31	2	no	—
112	#433	—	31	2	no	—
113	#431	—	31	1	no	—
114	#405	—	31	2	no	—
115	C₂×#97	D₃₁	1	1	yes	characteristic, complemented
116	C₂×C₃₁	—	2	2	no	—
117	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
118	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
119	C₂×C₃₁	#568	1	1	yes	characteristic
120	D₃₁	—	4	2	no	—
121	C₂×M₁₆	—	31	2	no	—
122	C₂²×C₈	—	31	2	no	—
123	C₂²×C₈	C₂×D₃₁	1	2	yes	—
124	#126	—	31	2	no	—
125	#125	—	31	1	no	—
126	#124	—	31	1	no	—
127	#123	—	31	2	no	—
128	C₂²⋊Q₈	—	31	1	no	—
129	#121	—	31	1	no	—
130	C₄×Q₈	—	31	1	no	—
131	C₄×D₄	—	31	2	no	—
132	C₄×D₄	—	31	1	no	—
133	#117	—	31	2	no	—
134	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
135	C₄⋊C₈	—	31	2	no	—
136	C₄⋊C₈	—	31	2	no	—
137	C₄⋊C₈	—	31	2	no	—
138	C₂²⋊C₈	—	31	1	no	—
139	C₂²⋊C₈	—	31	1	no	—
140	C₂²⋊C₈	—	31	2	no	—
141	C₂²⋊C₈	—	31	2	no	—
142	#97	—	31	2	no	—
143	#97	C₂×D₃₁	1	1	yes	characteristic
144	#97	C₂×D₃₁	1	1	yes	characteristic
145	#97	C₂×D₃₁	1	2	yes	—
146	C₄×C₈	—	31	2	no	—
147	C₄×C₈	—	31	2	no	—
148	C₃₁	#2968	1	1	yes	characteristic, complemented, Sylow
149	C₂×Q₈	—	31	1	no	—
150	C₂×D₄	—	31	2	no	—
151	C₂×D₄	—	31	1	no	—
152	C₂²×C₄	—	31	2	no	—
153	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
156	M₁₆	—	62	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	31	2	no	—
159	C₂×C₈	—	31	2	no	—
160	C₂×C₈	—	62	4	no	—
161	C₂×C₈	C₂²×D₃₁	1	2	yes	—
162	C₂×C₈	C₂²×D₃₁	1	2	yes	—
163	C₄⋊C₄	—	31	2	no	—
164	C₄⋊C₄	—	31	2	no	—
165	C₄⋊C₄	—	31	2	no	—
166	C₄⋊C₄	—	31	2	no	—
167	C₄⋊C₄	—	31	1	no	—
168	C₄⋊C₄	—	31	1	no	—
169	C₂²⋊C₄	—	31	2	no	—
170	C₂²⋊C₄	—	31	2	no	—
171	C₂²⋊C₄	—	31	2	no	—
172	C₂²⋊C₄	—	31	2	no	—
173	C₂²⋊C₄	—	31	2	no	—
174	C₄²	—	31	2	no	—
175	C₄²	C₄×D₃₁	1	1	yes	characteristic
176	C₄²	C₄×D₃₁	1	1	yes	characteristic
177	C₄²	C₂²×D₃₁	1	1	yes	characteristic
178	C₄²	C₂²×D₃₁	1	1	yes	characteristic
179	C₂³	—	31	2	no	—
180	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
181	Q₈	—	124	1	no	—
182	D₄	—	62	4	no	—
183	D₄	—	124	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	31	2	no	—
188	C₂×C₄	—	31	2	no	—
189	C₂×C₄	—	31	2	no	—
190	C₂×C₄	—	62	2	no	—
191	C₂×C₄	—	62	2	no	—
192	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	62	2	no	—
200	C₈	—	62	2	no	—
201	C₈	#92495	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	62	2	no	—
205	C₂²	—	62	2	no	—
206	C₂²	—	62	2	no	—
207	C₂²	#11758484	1	2	yes	—
208	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	62	2	no	—
215	C₄	—	62	2	no	—
216	C₄	—	62	2	no	—
217	C₄	C₂×#92495	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	124	2	no	—
220	C₂	C₄×#92495	1	1	yes	characteristic
221	C₂	C₂×#11758484	1	1	yes	characteristic
222	C₂	#14521815	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×#97	C₂
prime radical	#11758507	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#97	D₃₁
hypocenter	C₃₁	#2968
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #14521177	C₂	semidirect
3. #14521122	C₂	semidirect
4. #14521120	C₂	semidirect
5. C₈×C₂²⋊D₃₁	C₂	semidirect
6. #14521111	C₂	semidirect
7. #14521102	C₂	semidirect
8. #14521006	C₂	semidirect
9. C₂×C₃₁×#97	C₂	semidirect
115. C₂×#97	D₃₁	semidirect
148. C₃₁	#2968	semidirect

Maximal subgroups

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

subgroup	index
#14521583	2
#14521177	2
#14521122	2
#14521120	2
C₈×C₂²⋊D₃₁	2
#14521111	2
#14521102	2
#14521006	2
C₂×C₃₁×#97	2
#2968	31