Subgroups — #16854368

Order 3968.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521109	C₂	1	2	yes	maximal, complemented
3	#14521109	C₂	1	2	yes	maximal, complemented
4	#14521105	C₂	1	2	yes	maximal, complemented
5	#14521101	C₂	1	2	yes	maximal, complemented
6	#14521076	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521012	C₂	1	1	yes	maximal, characteristic
8	C₂×(C₂×D₃₁)⋊C₈	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×#424	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#117	C₂²	1	1	yes	characteristic
16	#11758503	C₂²	1	1	yes	characteristic
17	C₂×#92474	C₂²	1	1	yes	characteristic
18	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	#11758469	C₄	1	1	yes	characteristic
20	#11758469	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₃₁	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₃₁	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₃₁	C₄	1	2	yes	—
24	#11758454	C₄	1	2	yes	—
25	#11758449	C₂²	1	2	yes	—
26	C₄²×D₃₁	C₂²	1	2	yes	—
27	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
29	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
34	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
35	#11758403	C₂²	1	2	yes	complemented
36	#11758403	C₂²	1	2	yes	complemented
37	C₈×Dic₃₁	C₂²	1	2	yes	complemented
38	C₈×Dic₃₁	C₂²	1	2	yes	complemented
39	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	complemented
40	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	complemented
41	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₃₁	—	2	2	no	—
45	C₂×C₄×D₃₁	—	2	2	no	—
46	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₃₁	C₂³	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	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₄²×C₃₁	C₂³	1	2	yes	—
61	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
62	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
63	(C₂×D₃₁)⋊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	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
67	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
69	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
70	C₄×Dic₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
77	C₂²×D₃₁	—	2	2	no	—
78	C₂²×D₃₁	—	2	2	no	—
79	C₂²×D₃₁	—	4	1	no	—
80	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
91	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
92	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₃₁	—	2	4	no	—
94	C₄×D₃₁	—	2	2	no	—
95	C₄×D₃₁	—	2	2	no	—
96	C₄×D₃₁	—	2	1	no	—
97	C₄×D₃₁	—	2	1	no	—
98	C₄×D₃₁	—	2	1	no	—
99	C₄×D₃₁	—	2	1	no	—
100	C₄×D₃₁	—	4	2	no	—
101	C₈×C₃₁	—	2	1	no	—
102	C₈×C₃₁	—	2	1	no	—
103	C₈×C₃₁	—	2	2	no	—
104	C₃₁⋊C₈	—	2	1	no	—
105	C₃₁⋊C₈	—	2	1	no	—
106	C₃₁⋊C₈	—	2	2	no	—
107	#2924	—	31	1	no	maximal, Sylow
108	C₂²×C₃₁	—	2	1	no	—
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	#117	1	2	yes	—
111	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₃₁	—	2	1	no	—
113	C₂×D₃₁	—	2	1	no	—
114	C₂×D₃₁	—	2	1	no	—
115	C₂×D₃₁	—	2	1	no	—
116	C₂×D₃₁	—	2	1	no	—
117	C₂×D₃₁	—	4	4	no	—
118	C₂×D₃₁	#131	1	2	yes	—
119	C₄×C₃₁	—	2	2	no	—
120	C₄×C₃₁	—	2	2	no	—
121	C₄×C₃₁	#117	1	2	yes	—
122	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
123	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
124	Dic₃₁	—	2	1	no	—
125	Dic₃₁	—	2	1	no	—
126	Dic₃₁	—	2	2	no	—
127	Dic₃₁	—	2	2	no	—
128	C₂×#117	—	31	1	no	—
129	#433	—	31	2	no	—
130	#433	—	31	2	no	—
131	#432	—	31	2	no	—
132	#431	—	31	2	no	—
133	#424	—	31	1	no	—
134	#424	D₃₁	1	1	yes	characteristic, complemented
135	#407	—	31	1	no	—
136	C₂×C₂²⋊C₈	—	31	1	no	—
137	#405	—	31	1	no	—
138	#405	—	31	1	no	—
139	C₂×C₃₁	—	2	2	no	—
140	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
141	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
142	C₂×C₃₁	#568	1	1	yes	characteristic
143	D₃₁	—	2	2	no	—
144	D₃₁	—	4	1	no	—
145	C₂³×C₄	—	31	1	no	—
146	C₂×M₁₆	—	31	1	no	—
147	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
148	C₂²×C₈	—	31	1	no	—
149	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
150	#117	—	31	1	no	—
151	#117	—	31	1	no	—
152	#117	—	31	1	no	—
153	#117	—	31	2	no	—
154	#117	—	31	2	no	—
155	#117	C₂×D₃₁	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	31	1	no	—
157	C₂×C₄⋊C₄	—	31	1	no	—
158	C₂×C₂²⋊C₄	—	31	2	no	—
159	C₂×C₄²	—	31	2	no	—
160	C₄⋊C₈	—	31	2	no	—
161	C₄⋊C₈	—	31	2	no	—
162	C₄⋊C₈	C₂×D₃₁	1	2	yes	complemented
163	C₄⋊C₈	C₂×D₃₁	1	2	yes	complemented
164	C₂²⋊C₈	—	31	1	no	—
165	C₂²⋊C₈	—	31	1	no	—
166	C₂²⋊C₈	—	31	1	no	—
167	C₂²⋊C₈	—	31	1	no	—
168	C₂²⋊C₈	—	31	2	no	—
169	C₂²⋊C₈	—	31	2	no	—
170	#97	—	31	2	no	—
171	#97	—	31	2	no	—
172	C₄×C₈	—	31	2	no	—
173	C₄×C₈	—	31	2	no	—
174	C₃₁	#2924	1	1	yes	characteristic, complemented, Sylow
175	C₂⁴	—	31	1	no	—
176	C₂²×C₄	—	31	1	no	—
177	C₂²×C₄	—	31	2	no	—
178	C₂²×C₄	—	31	1	no	—
179	C₂²×C₄	—	31	1	no	—
180	C₂²×C₄	—	31	1	no	—
181	C₂²×C₄	—	31	1	no	—
182	C₂²×C₄	—	31	2	no	—
183	C₂²×C₄	—	62	2	no	—
184	C₂²×C₄	—	62	2	no	—
185	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	62	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	31	1	no	—
190	C₂×C₈	—	31	2	no	—
191	C₂×C₈	—	31	1	no	—
192	C₂×C₈	—	62	2	no	—
193	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₃₁	1	2	yes	—
195	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
196	C₄⋊C₄	—	31	2	no	—
197	C₄⋊C₄	—	31	2	no	—
198	C₄⋊C₄	—	31	1	no	—
199	C₄⋊C₄	—	31	1	no	—
200	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
201	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
202	C₂²⋊C₄	—	31	2	no	—
203	C₂²⋊C₄	—	31	2	no	—
204	C₂²⋊C₄	—	31	2	no	—
205	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
206	C₄²	—	31	2	no	—
207	C₄²	—	31	2	no	—
208	C₄²	—	31	2	no	—
209	C₄²	C₂²×D₃₁	1	2	yes	—
210	C₂³	—	31	1	no	—
211	C₂³	—	31	1	no	—
212	C₂³	—	62	2	no	—
213	C₂³	—	62	2	no	—
214	C₂³	—	124	1	no	—
215	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
216	C₂×C₄	—	31	1	no	—
217	C₂×C₄	—	31	2	no	—
218	C₂×C₄	—	31	2	no	—
219	C₂×C₄	—	31	1	no	—
220	C₂×C₄	—	62	4	no	—
221	C₂×C₄	—	62	2	no	—
222	C₂×C₄	—	62	2	no	—
223	C₂×C₄	—	62	2	no	—
224	C₂×C₄	—	62	1	no	—
225	C₂×C₄	—	62	1	no	—
226	C₂×C₄	—	62	1	no	—
227	C₂×C₄	—	62	1	no	—
228	C₂×C₄	—	124	2	no	—
229	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
232	C₂×C₄	#92497	1	2	yes	—
233	C₂×C₄	#92499	1	2	yes	—
234	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	62	1	no	—
239	C₈	—	62	1	no	—
240	C₈	—	62	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	31	2	no	—
244	C₂²	—	62	1	no	—
245	C₂²	—	62	1	no	—
246	C₂²	—	62	1	no	—
247	C₂²	—	62	1	no	—
248	C₂²	—	62	1	no	—
249	C₂²	—	124	4	no	—
250	C₂²	#11758469	1	2	yes	—
251	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	62	1	no	—
255	C₄	—	62	1	no	—
256	C₄	—	62	2	no	—
257	C₄	—	62	2	no	—
258	C₄	#11758469	1	2	yes	—
259	C₄	C₂×#92497	1	1	yes	characteristic
260	C₄	C₂×#92499	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	62	2	no	—
263	C₂	—	124	1	no	—
264	C₂	C₂×#11758469	1	1	yes	characteristic
265	C₂	#131×D₃₁	1	1	yes	characteristic
266	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×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#424	D₃₁
hypocenter	C₃₁	#2924
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. #14521109	C₂	semidirect
3. #14521109	C₂	semidirect
4. #14521105	C₂	semidirect
5. #14521101	C₂	semidirect
6. #14521076	C₂	semidirect
9. #14521007	C₂	semidirect
10. #14521004	C₂	semidirect
11. C₃₁×#424	C₂	semidirect
27. C₃₁×C₄⋊C₈	C₂²	semidirect
28. C₃₁×C₄⋊C₈	C₂²	semidirect
35. #11758403	C₂²	semidirect
36. #11758403	C₂²	semidirect
37. C₈×Dic₃₁	C₂²	semidirect
38. C₈×Dic₃₁	C₂²	semidirect
39. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
40. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
134. #424	D₃₁	semidirect
162. C₄⋊C₈	C₂×D₃₁	semidirect
163. C₄⋊C₈	C₂×D₃₁	semidirect
174. C₃₁	#2924	semidirect

Maximal subgroups

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

subgroup	index
#117×D₃₁	2
#14521109	2
#14521105	2
#14521101	2
#14521076	2
#14521012	2
C₂×(C₂×D₃₁)⋊C₈	2
#14521007	2
#14521004	2
C₃₁×#424	2
#2924	31