Subgroups — #16854714

Order 3968.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521586	C₂	1	1	yes	maximal, characteristic
2	#14521180	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521128	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521128	C₂	1	2	yes	maximal, complemented
5	#14521119	C₂	1	2	yes	maximal, complemented
6	#14521114	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521109	C₂	1	2	yes	maximal, complemented
8	#14521101	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521003	C₂	1	2	yes	maximal, complemented
10	C₄×#92469	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758524	C₄	1	1	yes	characteristic
14	#11758523	C₄	1	1	yes	characteristic
15	C₂×#92469	C₂²	1	1	yes	characteristic
16	C₂×#92469	C₂²	1	1	yes	characteristic
17	C₂×#92469	C₂²	1	2	yes	—
18	C₂×C₈×D₃₁	C₂²	1	2	yes	—
19	#11758475	C₄	1	2	yes	—
20	#11758472	C₄	1	1	yes	characteristic
21	#11758471	C₄	1	2	yes	—
22	#11758467	C₄	1	1	yes	characteristic
23	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
24	C₄×D₁₂₄	C₂²	1	2	yes	—
25	#11758449	C₂²	1	2	yes	—
26	C₄²×D₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₃₁	C₂²	1	2	yes	—
31	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
34	#11758403	C₂²	1	1	yes	characteristic
35	#11758403	C₂²	1	1	yes	characteristic
36	C₈×Dic₃₁	C₂²	1	2	yes	—
37	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
39	#11758391	C₂²	1	2	yes	—
40	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
43	C₂×D₁₂₄	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₃₁	C₂³	1	2	yes	—
48	C₂×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₄²×C₃₁	C₂³	1	2	yes	—
55	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
62	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
69	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	#92469	—	2	2	no	—
71	#92469	—	2	4	no	—
72	#92469	—	2	2	no	—
73	C₈×D₃₁	—	2	4	no	—
74	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
75	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
86	D₁₂₄	—	2	2	no	—
87	D₁₂₄	—	4	2	no	—
88	C₄×D₃₁	—	2	2	no	—
89	C₄×D₃₁	—	2	2	no	—
90	C₄×D₃₁	—	2	1	no	—
91	C₄×D₃₁	—	2	2	no	—
92	C₄×D₃₁	—	2	1	no	—
93	C₄×D₃₁	—	4	1	no	—
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₈×C₃₁	#41	1	2	yes	—
97	C₃₁⋊C₈	—	2	1	no	—
98	C₃₁⋊C₈	—	2	2	no	—
99	C₃₁⋊C₈	#41	1	2	yes	—
100	#2969	—	31	1	no	maximal, Sylow
101	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₃₁	—	2	2	no	—
103	C₂×D₃₁	—	2	1	no	—
104	C₂×D₃₁	—	2	1	no	—
105	C₂×D₃₁	—	2	2	no	—
106	C₂×D₃₁	—	2	1	no	—
107	C₂×D₃₁	—	2	2	no	—
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	#131	1	2	yes	—
112	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
113	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
114	Dic₃₁	—	2	2	no	—
115	Dic₃₁	—	2	1	no	—
116	Dic₃₁	—	2	1	no	—
117	Dic₃₁	—	2	1	no	—
118	#529	—	31	1	no	—
119	#446	—	31	1	no	—
120	C₈×Q₈	D₃₁	1	1	yes	characteristic, complemented
121	#436	—	31	1	no	—
122	#436	—	31	2	no	—
123	#435	—	31	2	no	—
124	C₈×D₄	—	31	1	no	—
125	#433	—	31	2	no	—
126	#431	—	31	1	no	—
127	#405	—	31	2	no	—
128	C₄×M₁₆	—	31	1	no	—
129	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
130	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
131	C₂×C₃₁	#568	1	1	yes	characteristic
132	D₃₁	—	4	1	no	—
133	D₃₁	—	4	2	no	—
134	C₂×M₁₆	—	31	1	no	—
135	C₂×M₁₆	—	31	1	no	—
136	C₂×M₁₆	—	31	2	no	—
137	C₂²×C₈	—	31	2	no	—
138	#126	—	31	2	no	—
139	#125	—	31	1	no	—
140	#124	—	31	1	no	—
141	#123	—	31	2	no	—
142	C₂²⋊Q₈	—	31	1	no	—
143	#121	—	31	1	no	—
144	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
145	C₄×D₄	—	31	1	no	—
146	C₄×D₄	—	31	2	no	—
147	#117	—	31	2	no	—
148	C₂×C₄²	—	31	1	no	—
149	C₄⋊C₈	—	31	1	no	—
150	C₄⋊C₈	—	31	2	no	—
151	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
153	C₂²⋊C₈	—	31	2	no	—
154	C₂²⋊C₈	—	31	2	no	—
155	C₂²⋊C₈	—	31	2	no	—
156	#97	—	31	1	no	—
157	#97	—	31	1	no	—
158	#97	—	31	2	no	—
159	C₄×C₈	—	31	2	no	—
160	C₄×C₈	—	31	1	no	—
161	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
162	C₄×C₈	C₂×D₃₁	1	2	yes	—
163	C₃₁	#2969	1	1	yes	characteristic, complemented, Sylow
164	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
165	C₂×D₄	—	31	1	no	—
166	C₂×D₄	—	31	2	no	—
167	C₂²×C₄	—	31	1	no	—
168	C₂²×C₄	—	31	1	no	—
169	C₂²×C₄	—	31	2	no	—
170	C₂²×C₄	—	31	1	no	—
171	M₁₆	—	62	2	no	—
172	M₁₆	—	62	2	no	—
173	M₁₆	—	62	4	no	—
174	C₂×C₈	—	31	1	no	—
175	C₂×C₈	—	31	1	no	—
176	C₂×C₈	—	31	2	no	—
177	C₂×C₈	—	62	4	no	—
178	C₂×C₈	C₂²×D₃₁	1	2	yes	—
179	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
181	C₄⋊C₄	—	31	2	no	—
182	C₄⋊C₄	—	31	2	no	—
183	C₄⋊C₄	—	31	1	no	—
184	C₄⋊C₄	—	31	2	no	—
185	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
186	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
187	C₂²⋊C₄	—	31	2	no	—
188	C₂²⋊C₄	—	31	2	no	—
189	C₂²⋊C₄	—	31	2	no	—
190	C₂²⋊C₄	—	31	2	no	—
191	C₂²⋊C₄	—	31	2	no	—
192	C₄²	—	31	1	no	—
193	C₄²	—	31	1	no	—
194	C₄²	—	31	1	no	—
195	C₄²	C₂²×D₃₁	1	1	yes	characteristic
196	C₄²	C₂²×D₃₁	1	2	yes	—
197	C₂³	—	31	2	no	—
198	C₂³	—	31	1	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	62	2	no	—
201	D₄	—	124	2	no	—
202	C₂×C₄	—	31	1	no	—
203	C₂×C₄	—	31	1	no	—
204	C₂×C₄	—	31	1	no	—
205	C₂×C₄	—	31	2	no	—
206	C₂×C₄	—	62	2	no	—
207	C₂×C₄	—	62	1	no	—
208	C₂×C₄	—	62	2	no	—
209	C₂×C₄	—	62	2	no	—
210	C₂×C₄	—	62	1	no	—
211	C₂×C₄	—	124	1	no	—
212	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	31	2	no	—
220	C₈	—	62	1	no	—
221	C₈	—	62	2	no	—
222	C₈	#92499	1	2	yes	—
223	C₂²	—	62	2	no	—
224	C₂²	—	62	1	no	—
225	C₂²	—	62	1	no	—
226	C₂²	—	62	2	no	—
227	C₂²	—	62	1	no	—
228	C₂²	—	62	2	no	—
229	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	62	1	no	—
234	C₄	—	62	2	no	—
235	C₄	—	62	1	no	—
236	C₄	—	62	1	no	—
237	C₄	#11758478	1	2	yes	—
238	C₄	C₂×#92499	1	1	yes	characteristic
239	C₄	#41×D₃₁	1	1	yes	characteristic
240	C₂	—	124	1	no	—
241	C₂	—	124	2	no	—
242	C₂	C₄×#92499	1	1	yes	characteristic
243	C₂	C₂×#11758478	1	1	yes	characteristic
244	C₂	#14521814	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₈×Q₈×C₃₁	C₂
prime radical	#11758472	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₈×Q₈	D₃₁
hypocenter	C₃₁	#2969
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #14521180	C₂	semidirect
3. #14521128	C₂	semidirect
4. #14521128	C₂	semidirect
5. #14521119	C₂	semidirect
6. #14521114	C₂	semidirect
7. #14521109	C₂	semidirect
8. #14521101	C₂	semidirect
9. #14521003	C₂	semidirect
10. C₄×#92469	C₂	semidirect
11. C₈×Q₈×C₃₁	C₂	semidirect
120. C₈×Q₈	D₃₁	semidirect
163. C₃₁	#2969	semidirect

Maximal subgroups

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

subgroup	index
#14521586	2
#14521180	2
#14521128	2
#14521119	2
#14521114	2
#14521109	2
#14521101	2
#14521003	2
C₄×#92469	2
C₈×Q₈×C₃₁	2
#2969	31