Subgroups — #16854874

Order 3968.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 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	Q₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521126	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521122	C₂	1	2	yes	maximal, complemented
5	C₈×D₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521114	C₂	1	2	yes	maximal, complemented
7	#14521111	C₂	1	2	yes	maximal, complemented
8	#14521100	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521074	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#446	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	2	yes	—
16	C₂×C₈×D₃₁	C₂²	1	2	yes	—
17	C₂×C₈×D₃₁	C₂²	1	2	yes	—
18	#11758475	C₄	1	2	yes	—
19	#11758472	C₄	1	1	yes	characteristic
20	#11758471	C₄	1	2	yes	—
21	#11758467	C₄	1	1	yes	characteristic
22	C₄×D₁₂₄	C₂²	1	2	yes	—
23	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
24	#11758449	C₂²	1	2	yes	—
25	C₄²×D₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
27	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃₁×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
34	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₃₁⋊C₈	C₂²	1	2	yes	—
37	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
39	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
41	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×D₁₂₄	C₂×C₄	1	2	yes	—
43	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂³	1	2	yes	—
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	—	4	2	no	—
71	C₈×D₃₁	—	4	2	no	—
72	C₈×D₃₁	—	4	2	no	—
73	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
74	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₃₁	C₂²×C₄	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	2	yes	—
85	D₁₂₄	—	2	4	no	—
86	D₁₂₄	—	2	2	no	—
87	C₄×D₃₁	—	2	2	no	—
88	C₄×D₃₁	—	2	2	no	—
89	C₄×D₃₁	—	2	1	no	—
90	C₄×D₃₁	—	2	2	no	—
91	C₄×D₃₁	—	2	1	no	—
92	C₄×D₃₁	—	2	2	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₃₁⋊C₈	—	2	2	no	—
97	C₃₁⋊C₈	—	2	1	no	—
98	C₃₁⋊C₈	—	2	1	no	—
99	#2994	—	31	1	no	maximal, Sylow
100	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₃₁	—	2	1	no	—
102	C₂×D₃₁	—	2	1	no	—
103	C₂×D₃₁	—	2	2	no	—
104	C₂×D₃₁	—	2	2	no	—
105	C₂×D₃₁	—	2	2	no	—
106	C₂×D₃₁	#131	1	2	yes	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	#131	1	2	yes	—
111	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₃₁	—	2	1	no	—
114	Dic₃₁	—	2	2	no	—
115	Dic₃₁	—	2	1	no	—
116	Dic₃₁	—	2	1	no	—
117	#529	—	31	1	no	—
118	#446	D₃₁	1	1	yes	characteristic, complemented
119	C₈×Q₈	—	31	1	no	—
120	#436	—	31	1	no	—
121	#435	—	31	2	no	—
122	C₈×D₄	—	31	2	no	—
123	C₈×D₄	—	31	1	no	—
124	#433	—	31	2	no	—
125	#431	—	31	1	no	—
126	#424	—	31	2	no	—
127	C₂×C₄⋊C₈	—	31	1	no	—
128	C₂×C₃₁	#520	1	1	yes	characteristic
129	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
130	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
131	D₃₁	—	2	2	no	—
132	D₃₁	—	4	2	no	—
133	C₂×M₁₆	—	31	2	no	—
134	C₂²×C₈	—	31	2	no	—
135	C₂²×C₈	—	31	2	no	—
136	#126	—	31	2	no	—
137	#125	—	31	1	no	—
138	#124	—	31	1	no	—
139	#123	—	31	2	no	—
140	C₂²⋊Q₈	—	31	1	no	—
141	#121	—	31	1	no	—
142	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
143	C₄×D₄	—	31	2	no	—
144	C₄×D₄	—	31	1	no	—
145	#117	—	31	2	no	—
146	C₂×C₄²	—	31	1	no	—
147	C₄⋊C₈	—	31	2	no	—
148	C₄⋊C₈	—	31	1	no	—
149	C₄⋊C₈	—	31	1	no	—
150	C₄⋊C₈	—	31	1	no	—
151	C₄⋊C₈	—	31	2	no	—
152	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
154	C₂²⋊C₈	—	31	1	no	—
155	C₂²⋊C₈	—	31	1	no	—
156	C₂²⋊C₈	—	31	2	no	—
157	C₂²⋊C₈	—	31	2	no	—
158	#97	C₂×D₃₁	1	2	yes	—
159	C₄×C₈	—	31	1	no	—
160	C₄×C₈	—	31	2	no	—
161	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
162	C₃₁	#2994	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
164	C₂×D₄	—	31	2	no	—
165	C₂×D₄	—	31	1	no	—
166	C₂²×C₄	—	31	1	no	—
167	C₂²×C₄	—	31	1	no	—
168	C₂²×C₄	—	31	1	no	—
169	C₂²×C₄	—	31	2	no	—
170	M₁₆	—	124	2	no	—
171	C₂×C₈	—	31	1	no	—
172	C₂×C₈	—	31	2	no	—
173	C₂×C₈	—	31	1	no	—
174	C₂×C₈	—	124	2	no	—
175	C₂×C₈	—	124	2	no	—
176	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₃₁	1	2	yes	—
179	C₄⋊C₄	—	31	2	no	—
180	C₄⋊C₄	—	31	2	no	—
181	C₄⋊C₄	—	31	1	no	—
182	C₄⋊C₄	—	31	2	no	—
183	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
184	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
185	C₂²⋊C₄	—	31	2	no	—
186	C₂²⋊C₄	—	31	2	no	—
187	C₂²⋊C₄	—	31	2	no	—
188	C₂²⋊C₄	—	31	2	no	—
189	C₂²⋊C₄	—	31	2	no	—
190	C₄²	—	31	1	no	—
191	C₄²	—	31	1	no	—
192	C₄²	—	31	1	no	—
193	C₄²	C₂²×D₃₁	1	1	yes	characteristic
194	C₄²	C₂²×D₃₁	1	2	yes	—
195	C₂³	—	31	1	no	—
196	C₂³	—	31	2	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	62	4	no	—
199	D₄	—	62	2	no	—
200	C₂×C₄	—	31	1	no	—
201	C₂×C₄	—	31	1	no	—
202	C₂×C₄	—	31	1	no	—
203	C₂×C₄	—	31	2	no	—
204	C₂×C₄	—	62	2	no	—
205	C₂×C₄	—	62	2	no	—
206	C₂×C₄	—	62	1	no	—
207	C₂×C₄	—	62	2	no	—
208	C₂×C₄	—	62	1	no	—
209	C₂×C₄	—	62	2	no	—
210	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	62	1	no	—
219	C₈	—	62	2	no	—
220	C₈	—	62	1	no	—
221	C₂²	—	31	2	no	—
222	C₂²	—	62	1	no	—
223	C₂²	—	62	1	no	—
224	C₂²	—	62	2	no	—
225	C₂²	—	62	2	no	—
226	C₂²	—	62	2	no	—
227	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	62	1	no	—
232	C₄	—	62	2	no	—
233	C₄	—	62	1	no	—
234	C₄	—	62	1	no	—
235	C₄	#11758484	1	2	yes	—
236	C₄	#11758564	1	1	yes	characteristic
237	C₄	#11758566	1	1	yes	characteristic
238	C₂	—	62	2	no	—
239	C₂	—	124	2	no	—
240	C₂	#14521531	1	1	yes	characteristic
241	C₂	#131×D₃₁	1	1	yes	characteristic
242	C₂	C₂×#11758484	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#446	C₂
prime radical	#11758472	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#446	D₃₁
hypocenter	C₃₁	#2994
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃₁⋊C₈	C₂	semidirect
3. #14521126	C₂	semidirect
4. #14521122	C₂	semidirect
5. C₈×D₁₂₄	C₂	semidirect
6. #14521114	C₂	semidirect
7. #14521111	C₂	semidirect
8. #14521100	C₂	semidirect
9. #14521074	C₂	semidirect
10. C₄⋊C₈×D₃₁	C₂	semidirect
11. C₃₁×#446	C₂	semidirect
118. #446	D₃₁	semidirect
162. C₃₁	#2994	semidirect

Maximal subgroups

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

subgroup	index
#14521586	2
Q₈×C₃₁⋊C₈	2
#14521126	2
#14521122	2
C₈×D₁₂₄	2
#14521114	2
#14521111	2
#14521100	2
#14521074	2
C₄⋊C₈×D₃₁	2
C₃₁×#446	2
#2994	31