Subgroups — #16854373

Order 3968.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521107	C₂	1	2	yes	maximal, complemented
3	#14521107	C₂	1	2	yes	maximal, complemented
4	#14521103	C₂	1	2	yes	maximal, complemented
5	#14521099	C₂	1	2	yes	maximal, complemented
6	#14521072	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521016	C₂	1	1	yes	maximal, characteristic
9	#14521007	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521004	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#408	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	Q₈×Dic₃₁	C₄	1	1	yes	characteristic
16	Q₈×Dic₃₁	C₄	1	1	yes	characteristic
17	D₄×Dic₃₁	C₄	1	2	yes	—
18	D₄×Dic₃₁	C₄	1	1	yes	characteristic
19	D₄×Dic₃₁	C₄	1	2	yes	—
20	D₄×Dic₃₁	C₄	1	1	yes	characteristic
21	#11758503	C₂²	1	2	yes	—
22	#11758503	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
25	C₂×#92474	C₂²	1	1	yes	characteristic
26	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
27	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
30	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
31	#11758403	C₂²	1	2	yes	—
32	#11758403	C₂²	1	2	yes	—
33	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
34	Dic₃₁⋊C₈	C₂²	1	2	yes	—
35	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₃₁⋊C₈	C₂²	1	2	yes	—
39	C₈×Dic₃₁	C₂²	1	2	yes	—
40	C₈×Dic₃₁	C₂²	1	2	yes	—
41	#41×C₃₁	—	2	2	no	—
42	#41×C₃₁	—	2	2	no	—
43	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
49	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
51	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
52	M₁₆×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
59	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
62	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
65	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
67	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₃₁	C₂³	1	2	yes	—
69	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
70	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
71	#92474	—	2	2	no	—
72	C₂×C₃₁⋊C₈	—	2	2	no	—
73	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
75	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	Q₈×C₃₁	—	2	2	no	—
79	D₄×C₃₁	—	2	4	no	—
80	D₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#41	1	2	yes	—
86	C₂×C₄×C₃₁	#41	1	2	yes	—
87	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₃₁	—	2	2	no	—
89	C₂×Dic₃₁	—	2	4	no	—
90	C₂×Dic₃₁	—	4	1	no	—
91	C₂×Dic₃₁	—	4	2	no	—
92	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
94	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
96	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	—	2	2	no	—
101	C₃₁⋊C₈	—	2	1	no	—
102	C₃₁⋊C₈	—	2	1	no	—
103	C₃₁⋊C₈	—	2	2	no	—
104	#2925	—	31	1	no	maximal, Sylow
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	—	2	1	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	#117	1	2	yes	—
111	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	#117	1	2	yes	—
114	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
115	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
116	Dic₃₁	—	2	1	no	—
117	Dic₃₁	—	2	2	no	—
118	Dic₃₁	—	2	1	no	—
119	Dic₃₁	—	2	1	no	—
120	Dic₃₁	—	2	2	no	—
121	Dic₃₁	#131	1	2	yes	—
122	C₄×#41	—	31	1	no	—
123	#433	—	31	2	no	—
124	#433	—	31	2	no	—
125	#432	—	31	2	no	—
126	#431	—	31	2	no	—
127	#423	—	31	1	no	—
128	C₂×C₄⋊C₈	—	31	1	no	—
129	#408	—	31	1	no	—
130	#408	D₃₁	1	1	yes	characteristic
131	#405	—	31	1	no	—
132	#405	—	31	1	no	—
133	C₂×C₃₁	—	2	2	no	—
134	C₂×C₃₁	—	4	2	no	—
135	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
136	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
137	C₂×C₃₁	#568	1	1	yes	characteristic
138	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
139	C₂×M₁₆	—	31	1	no	—
140	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
141	C₂²×C₈	—	31	1	no	—
142	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
143	C₄×Q₈	—	31	1	no	—
144	C₄×Q₈	—	31	1	no	—
145	C₄×D₄	—	31	2	no	—
146	C₄×D₄	—	31	1	no	—
147	C₄×D₄	—	31	2	no	—
148	C₄×D₄	—	31	1	no	—
149	#117	—	31	1	no	—
150	#117	—	31	2	no	—
151	C₂×C₄²	—	31	1	no	—
152	C₂×C₄²	—	31	2	no	—
153	C₄⋊C₈	—	31	1	no	—
154	C₄⋊C₈	—	31	1	no	—
155	C₄⋊C₈	—	31	1	no	—
156	C₄⋊C₈	—	31	1	no	—
157	C₄⋊C₈	—	31	2	no	—
158	C₄⋊C₈	—	31	2	no	—
159	C₂²⋊C₈	—	31	2	no	—
160	C₂²⋊C₈	—	31	2	no	—
161	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
162	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
163	#97	—	31	2	no	—
164	#97	—	31	2	no	—
165	C₄×C₈	—	31	2	no	—
166	C₄×C₈	—	31	2	no	—
167	C₃₁	#2925	1	1	yes	characteristic, complemented, Sylow
168	#41	—	2	2	no	—
169	#41	—	2	2	no	—
170	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
171	C₂×D₄	C₄×D₃₁	1	2	yes	—
172	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
173	C₂²×C₄	—	31	2	no	—
174	C₂²×C₄	—	31	1	no	—
175	C₂²×C₄	—	31	2	no	—
176	C₂²×C₄	—	31	1	no	—
177	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
178	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	62	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	31	1	no	—
183	C₂×C₈	—	31	2	no	—
184	C₂×C₈	—	31	1	no	—
185	C₂×C₈	—	62	2	no	—
186	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₃₁	1	2	yes	—
188	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
189	C₄⋊C₄	—	31	2	no	—
190	C₄⋊C₄	—	31	2	no	—
191	C₄⋊C₄	—	31	1	no	—
192	C₄⋊C₄	—	31	1	no	—
193	C₂²⋊C₄	—	31	2	no	—
194	C₂²⋊C₄	—	31	2	no	—
195	C₂²⋊C₄	—	31	2	no	—
196	C₄²	—	31	2	no	—
197	C₄²	—	31	1	no	—
198	C₄²	—	31	2	no	—
199	C₄²	—	31	1	no	—
200	C₄²	—	31	2	no	—
201	C₄²	—	31	1	no	—
202	C₄²	—	31	1	no	—
203	C₂³	C₂×C₄×D₃₁	1	2	yes	—
204	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	D₄	—	2	4	no	—
207	D₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	31	1	no	—
211	C₂×C₄	—	31	1	no	—
212	C₂×C₄	—	31	2	no	—
213	C₂×C₄	—	31	1	no	—
214	C₂×C₄	—	31	2	no	—
215	C₂×C₄	—	31	1	no	—
216	C₂×C₄	—	62	2	no	—
217	C₂×C₄	—	62	4	no	—
218	C₂×C₄	—	124	1	no	—
219	C₂×C₄	—	124	2	no	—
220	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
222	C₂×C₄	#92497	1	2	yes	—
223	C₂×C₄	#92497	1	2	yes	—
224	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	62	1	no	—
229	C₈	—	62	1	no	—
230	C₈	—	62	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	#11758454	1	2	yes	—
237	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	31	2	no	—
240	C₄	—	62	2	no	—
241	C₄	—	62	1	no	—
242	C₄	—	62	1	no	—
243	C₄	—	62	2	no	—
244	C₄	—	62	1	no	—
245	C₄	#11758454	1	2	yes	—
246	C₄	C₂×#92497	1	1	yes	characteristic
247	C₄	C₂×#92497	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×#11758454	1	1	yes	characteristic
251	C₂	#131×D₃₁	1	1	yes	characteristic
252	C₂	#14521814	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#408	C₂
prime radical	C₂×#41×C₃₁	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#408	D₃₁
hypocenter	C₃₁	#2925
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. #14521107	C₂	semidirect
3. #14521107	C₂	semidirect
4. #14521103	C₂	semidirect
5. #14521099	C₂	semidirect
6. #14521072	C₂	semidirect
7. C₂×Dic₃₁⋊C₈	C₂	semidirect
9. #14521007	C₂	semidirect
10. #14521004	C₂	semidirect
167. C₃₁	#2925	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₃₁	2
#14521107	2
#14521103	2
#14521099	2
#14521072	2
C₂×Dic₃₁⋊C₈	2
#14521016	2
#14521007	2
#14521004	2
C₃₁×#408	2
#2925	31