Subgroups — #16854803

Order 3968.

322 conjugacy classes of subgroups.

The subgroup lattice has 314 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	#14521179	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521177	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521177	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521176	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521173	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521110	C₂	1	1	yes	maximal, characteristic
11	#14521107	C₂	1	1	yes	maximal, characteristic
12	#14521107	C₂	1	1	yes	maximal, characteristic
13	#14521103	C₂	1	1	yes	maximal, characteristic
14	#14521099	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#433	C₂	1	1	yes	maximal, characteristic
16	C₃₁×#117	C₂²	1	1	yes	characteristic
17	#11758503	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₃₁	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	1	yes	characteristic
24	#11758455	C₄	1	1	yes	characteristic
25	#11758455	C₄	1	1	yes	characteristic
26	#11758455	C₄	1	1	yes	characteristic
27	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
28	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
29	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
30	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
39	#11758403	C₂²	1	1	yes	characteristic
40	#11758403	C₂²	1	1	yes	characteristic
41	#11758403	C₂²	1	1	yes	characteristic
42	#11758403	C₂²	1	1	yes	characteristic
43	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
46	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
52	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
53	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
54	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
55	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
57	#11758391	C₂²	1	1	yes	characteristic
58	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
63	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
83	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
84	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
85	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
86	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
87	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
88	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
89	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
90	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
95	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
96	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
97	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
99	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
100	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
102	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₃₁	—	2	2	no	—
112	C₂×Dic₃₁	—	4	1	no	—
113	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
121	Dic₆₂	—	2	2	no	—
122	Dic₆₂	—	2	2	no	—
123	Dic₆₂	—	4	1	no	—
124	Dic₆₂	—	4	1	no	—
125	C₈×C₃₁	—	2	1	no	—
126	C₈×C₃₁	—	2	1	no	—
127	C₈×C₃₁	—	2	1	no	—
128	C₈×C₃₁	—	2	1	no	—
129	C₃₁⋊C₈	—	2	1	no	—
130	C₃₁⋊C₈	—	2	1	no	—
131	C₃₁⋊C₈	—	2	1	no	—
132	C₃₁⋊C₈	—	2	1	no	—
133	#2985	—	31	1	no	maximal, Sylow
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₃₁	—	2	1	no	—
146	Dic₃₁	—	2	1	no	—
147	Dic₃₁	—	2	1	no	—
148	Dic₃₁	—	2	1	no	—
149	Dic₃₁	—	2	1	no	—
150	Dic₃₁	—	2	1	no	—
151	Dic₃₁	—	2	1	no	—
152	Dic₃₁	#131	1	2	yes	—
153	#533	—	31	1	no	—
154	#446	—	31	1	no	—
155	#446	—	31	1	no	—
156	#446	—	31	1	no	—
157	#446	—	31	1	no	—
158	#446	—	31	1	no	—
159	#446	—	31	1	no	—
160	C₈×Q₈	—	31	1	no	—
161	C₈×Q₈	—	31	1	no	—
162	#433	—	31	1	no	—
163	#433	—	31	1	no	—
164	#433	—	31	1	no	—
165	#433	D₃₁	1	1	yes	characteristic
166	#432	—	31	1	no	—
167	#431	—	31	1	no	—
168	C₂×C₃₁	—	4	1	no	—
169	C₂×C₃₁	#519	1	1	yes	characteristic
170	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
171	C₂×C₃₁	#568	1	1	yes	characteristic
172	C₄⋊Q₈	—	31	1	no	—
173	C₄⋊Q₈	—	31	1	no	—
174	#125	—	31	1	no	—
175	#125	—	31	1	no	—
176	C₂²⋊Q₈	—	31	1	no	—
177	C₂²⋊Q₈	—	31	1	no	—
178	C₂²⋊Q₈	—	31	1	no	—
179	C₂²⋊Q₈	—	31	1	no	—
180	C₄×Q₈	—	31	1	no	—
181	C₄×Q₈	—	31	1	no	—
182	C₄×Q₈	—	31	1	no	—
183	C₄×Q₈	—	31	1	no	—
184	#117	—	31	1	no	—
185	#117	C₂×D₃₁	1	1	yes	characteristic
186	C₂×C₄²	—	31	1	no	—
187	C₄⋊C₈	—	31	1	no	—
188	C₄⋊C₈	—	31	1	no	—
189	C₄⋊C₈	—	31	1	no	—
190	C₄⋊C₈	—	31	1	no	—
191	C₄⋊C₈	—	31	1	no	—
192	C₄⋊C₈	—	31	1	no	—
193	C₄⋊C₈	—	31	1	no	—
194	C₄⋊C₈	—	31	1	no	—
195	C₄⋊C₈	—	31	1	no	—
196	C₄⋊C₈	—	31	1	no	—
197	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
198	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
199	C₂²⋊C₈	—	31	1	no	—
200	C₂²⋊C₈	—	31	1	no	—
201	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
202	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
203	#97	—	31	1	no	—
204	#97	—	31	1	no	—
205	#97	—	31	1	no	—
206	#97	—	31	1	no	—
207	#97	—	31	1	no	—
208	#97	C₂×D₃₁	1	1	yes	characteristic
209	C₄×C₈	—	31	1	no	—
210	C₄×C₈	—	31	1	no	—
211	C₄×C₈	—	31	1	no	—
212	C₄×C₈	—	31	1	no	—
213	C₄×C₈	—	31	1	no	—
214	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
215	C₃₁	#2985	1	1	yes	characteristic, complemented, Sylow
216	C₂×Q₈	—	31	1	no	—
217	C₂×Q₈	—	31	1	no	—
218	C₂×Q₈	—	31	1	no	—
219	C₂×Q₈	—	31	1	no	—
220	C₂²×C₄	—	31	1	no	—
221	C₂²×C₄	—	31	1	no	—
222	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
223	C₂×C₈	—	31	1	no	—
224	C₂×C₈	—	31	1	no	—
225	C₂×C₈	—	31	1	no	—
226	C₂×C₈	—	31	1	no	—
227	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
231	C₄⋊C₄	—	31	1	no	—
232	C₄⋊C₄	—	31	1	no	—
233	C₄⋊C₄	—	31	1	no	—
234	C₄⋊C₄	—	31	1	no	—
235	C₄⋊C₄	—	31	1	no	—
236	C₄⋊C₄	—	31	1	no	—
237	C₄⋊C₄	—	31	1	no	—
238	C₄⋊C₄	—	31	1	no	—
239	C₄⋊C₄	—	31	1	no	—
240	C₄⋊C₄	—	31	1	no	—
241	C₄⋊C₄	—	31	1	no	—
242	C₄⋊C₄	—	31	1	no	—
243	C₄⋊C₄	—	31	1	no	—
244	C₄⋊C₄	—	31	1	no	—
245	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
246	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
247	C₂²⋊C₄	—	31	1	no	—
248	C₂²⋊C₄	—	31	1	no	—
249	C₂²⋊C₄	C₄×D₃₁	1	1	yes	characteristic
250	C₂²⋊C₄	C₄×D₃₁	1	1	yes	characteristic
251	C₄²	—	31	1	no	—
252	C₄²	—	31	1	no	—
253	C₄²	—	31	1	no	—
254	C₄²	—	31	1	no	—
255	C₄²	—	31	1	no	—
256	C₄²	—	31	1	no	—
257	C₄²	C₂²×D₃₁	1	1	yes	characteristic
258	C₄²	C₂²×D₃₁	1	1	yes	characteristic
259	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
260	Q₈	—	62	2	no	—
261	Q₈	—	62	2	no	—
262	Q₈	—	124	1	no	—
263	Q₈	—	124	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	31	1	no	—
267	C₂×C₄	—	31	1	no	—
268	C₂×C₄	—	31	1	no	—
269	C₂×C₄	—	31	1	no	—
270	C₂×C₄	—	31	1	no	—
271	C₂×C₄	—	31	1	no	—
272	C₂×C₄	—	31	1	no	—
273	C₂×C₄	—	31	1	no	—
274	C₂×C₄	—	62	2	no	—
275	C₂×C₄	—	124	1	no	—
276	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	62	1	no	—
287	C₈	—	62	1	no	—
288	C₈	—	62	1	no	—
289	C₈	—	62	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	31	2	no	—
300	C₄	—	62	1	no	—
301	C₄	—	62	1	no	—
302	C₄	—	62	1	no	—
303	C₄	—	62	1	no	—
304	C₄	—	62	1	no	—
305	C₄	—	62	1	no	—
306	C₄	—	62	1	no	—
307	C₄	#11758567	1	1	yes	characteristic
308	C₄	#11758567	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	#14521526	1	1	yes	characteristic
311	C₂	#131×D₃₁	1	1	yes	characteristic
312	C₂	#14521814	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#433	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#433	D₃₁
hypocenter	C₃₁	#2985
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #14521179	C₂	semidirect
3. #14521178	C₂	semidirect
4. #14521178	C₂	semidirect
5. #14521177	C₂	semidirect
6. #14521177	C₂	semidirect
7. #14521176	C₂	semidirect
8. C₈×Dic₆₂	C₂	semidirect
9. #14521173	C₂	semidirect
215. C₃₁	#2985	semidirect

Maximal subgroups

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

subgroup	index
#14521602	2
#14521179	2
#14521178	2
#14521177	2
#14521176	2
C₈×Dic₆₂	2
#14521173	2
#14521110	2
#14521107	2
#14521103	2
#14521099	2
C₃₁×#433	2
#2985	31