Subgroups — #16854657

Order 3968.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521602	C₂	1	1	yes	maximal, characteristic
2	#14521178	C₂	1	2	yes	maximal, complemented
3	#14521178	C₂	1	2	yes	maximal, complemented
4	#14521176	C₂	1	2	yes	maximal, complemented
5	#14521173	C₂	1	2	yes	maximal, complemented
6	#14521077	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#14521005	C₂	1	1	yes	maximal, characteristic
9	#14521004	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#424	C₂	1	1	yes	maximal, characteristic
12	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#117	C₂²	1	1	yes	characteristic
15	#11758503	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×#92474	C₂²	1	1	yes	characteristic
18	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	#11758467	C₄	1	1	yes	characteristic
20	#11758467	C₄	1	1	yes	characteristic
21	#11758465	C₄	1	1	yes	characteristic
22	#11758465	C₄	1	1	yes	characteristic
23	#11758455	C₄	1	2	yes	—
24	#11758455	C₄	1	2	yes	—
25	C₄×Dic₆₂	C₂²	1	2	yes	—
26	C₄×Dic₆₂	C₂²	1	2	yes	—
27	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
28	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
29	#11758403	C₂²	1	2	yes	—
30	#11758403	C₂²	1	2	yes	—
31	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
32	Dic₃₁⋊C₈	C₂²	1	2	yes	—
33	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
34	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₃₁⋊C₈	C₂²	1	2	yes	—
37	C₈×Dic₃₁	C₂²	1	2	yes	—
38	C₈×Dic₃₁	C₂²	1	2	yes	—
39	#11758391	C₂²	1	2	yes	—
40	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
45	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
46	M₁₆×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
53	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₄²×C₃₁	C₂³	1	2	yes	—
55	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
60	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
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	C₄×Dic₃₁	C₂³	1	2	yes	—
68	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
69	#92474	Q₈	1	4	yes	complemented
70	C₂×C₃₁⋊C₈	—	2	2	no	—
71	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
74	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₃₁	#41	1	2	yes	—
80	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₃₁	—	2	2	no	—
82	C₂×Dic₃₁	—	4	1	no	—
83	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
87	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
88	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
89	Dic₆₂	—	2	4	no	—
90	Dic₆₂	—	4	2	no	—
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	2	no	—
94	C₃₁⋊C₈	—	2	1	no	—
95	C₃₁⋊C₈	—	2	1	no	—
96	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
97	#2962	—	31	1	no	maximal, Sylow
98	C₂²×C₃₁	—	2	1	no	—
99	C₂²×C₃₁	—	2	1	no	—
100	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
101	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	C₄×Q₈	1	2	yes	—
105	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
107	Dic₃₁	—	2	2	no	—
108	Dic₃₁	—	2	1	no	—
109	Dic₃₁	—	2	1	no	—
110	Dic₃₁	—	2	2	no	—
111	Dic₃₁	—	2	1	no	—
112	Dic₃₁	#131	1	2	yes	—
113	#533	—	31	1	no	—
114	#446	—	31	2	no	—
115	#446	—	31	2	no	—
116	#446	—	31	2	no	—
117	C₈×Q₈	—	31	2	no	—
118	#424	—	31	1	no	—
119	#424	D₃₁	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	31	1	no	—
121	#405	—	31	1	no	—
122	#405	—	31	1	no	—
123	C₄×M₁₆	—	31	1	no	—
124	C₂×C₃₁	—	2	2	no	—
125	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
126	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
127	C₂×C₃₁	#568	1	1	yes	characteristic
128	C₂×M₁₆	—	31	1	no	—
129	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
130	C₂²×C₈	—	31	1	no	—
131	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
132	C₄⋊Q₈	—	31	1	no	—
133	C₄⋊Q₈	—	31	1	no	—
134	#125	—	31	1	no	—
135	#125	—	31	1	no	—
136	C₂²⋊Q₈	—	31	2	no	—
137	C₂²⋊Q₈	—	31	2	no	—
138	C₄×Q₈	—	31	2	no	—
139	C₄×Q₈	—	31	2	no	—
140	#117	—	31	1	no	—
141	#117	C₂×D₃₁	1	1	yes	characteristic
142	C₂×C₄²	—	31	1	no	—
143	C₄⋊C₈	—	31	1	no	—
144	C₄⋊C₈	—	31	1	no	—
145	C₄⋊C₈	—	31	1	no	—
146	C₄⋊C₈	—	31	1	no	—
147	C₄⋊C₈	—	31	2	no	—
148	C₄⋊C₈	—	31	2	no	—
149	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
150	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
151	#97	—	31	2	no	—
152	#97	—	31	2	no	—
153	#97	—	31	2	no	—
154	C₄×C₈	—	31	2	no	—
155	C₄×C₈	—	31	2	no	—
156	C₄×C₈	—	31	2	no	—
157	C₃₁	#2962	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	31	2	no	—
159	C₂×Q₈	—	31	2	no	—
160	C₂²×C₄	—	31	1	no	—
161	C₂²×C₄	—	31	1	no	—
162	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
163	M₁₆	—	2	2	no	—
164	M₁₆	—	31	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	31	2	no	—
167	C₂×C₈	—	31	1	no	—
168	C₂×C₈	—	31	1	no	—
169	C₂×C₈	—	62	2	no	—
170	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₃₁	1	2	yes	—
172	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
173	C₄⋊C₄	—	31	2	no	—
174	C₄⋊C₄	—	31	2	no	—
175	C₄⋊C₄	—	31	2	no	—
176	C₄⋊C₄	—	31	2	no	—
177	C₄⋊C₄	—	31	1	no	—
178	C₄⋊C₄	—	31	1	no	—
179	C₄⋊C₄	—	31	2	no	—
180	C₄⋊C₄	—	31	2	no	—
181	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
182	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
183	C₂²⋊C₄	—	31	2	no	—
184	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
185	C₄²	—	31	1	no	—
186	C₄²	—	31	1	no	—
187	C₄²	—	31	1	no	—
188	C₄²	—	31	2	no	—
189	C₄²	—	31	1	no	—
190	C₄²	C₂²×D₃₁	1	2	yes	—
191	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
192	Q₈	—	62	4	no	—
193	Q₈	—	124	2	no	—
194	C₂×C₄	—	31	1	no	—
195	C₂×C₄	—	31	1	no	—
196	C₂×C₄	—	31	1	no	—
197	C₂×C₄	—	31	1	no	—
198	C₂×C₄	—	31	2	no	—
199	C₂×C₄	—	31	2	no	—
200	C₂×C₄	—	62	2	no	—
201	C₂×C₄	—	124	1	no	—
202	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
205	C₂×C₄	#92497	1	2	yes	—
206	C₂×C₄	Q₈×D₃₁	1	2	yes	—
207	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	31	4	no	—
212	C₈	—	62	1	no	—
213	C₈	—	62	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	#11758464	1	2	yes	—
217	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	31	2	no	—
221	C₄	—	62	1	no	—
222	C₄	—	62	2	no	—
223	C₄	—	62	1	no	—
224	C₄	—	62	1	no	—
225	C₄	—	62	2	no	—
226	C₄	#11758464	1	2	yes	—
227	C₄	C₂×#92497	1	1	yes	characteristic
228	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×#11758464	1	1	yes	characteristic
231	C₂	#131×D₃₁	1	1	yes	characteristic
232	C₂	#14521814	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#424	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#424	D₃₁
hypocenter	C₃₁	#2962
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #14521178	C₂	semidirect
3. #14521178	C₂	semidirect
4. #14521176	C₂	semidirect
5. #14521173	C₂	semidirect
69. #92474	Q₈	semidirect
157. C₃₁	#2962	semidirect

Maximal subgroups

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

subgroup	index
#14521602	2
#14521178	2
#14521176	2
#14521173	2
#14521077	2
C₂×Dic₃₁⋊C₈	2
#14521005	2
#14521004	2
M₁₆×Dic₃₁	2
C₃₁×#424	2
#2962	31