Subgroups — #16858912

Order 3968.

1540 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92495	C₂	1	1	yes	maximal, characteristic
2	#14521530	C₂	1	4	yes	maximal, complemented
3	#14521529	C₂	1	4	yes	maximal, complemented
4	#14521526	C₂	1	2	yes	maximal, complemented
5	#14521523	C₂	1	2	yes	maximal, complemented
6	#14521522	C₂	1	4	yes	maximal, complemented
7	C₄×#92495	C₂	1	4	yes	maximal, complemented
8	#14521517	C₂	1	4	yes	maximal, complemented
9	C₂×#11758503	C₂	1	1	yes	maximal, characteristic, complemented
10	#117×D₃₁	C₂	1	4	yes	maximal, complemented
11	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₃₁	C₂²	1	2	yes	—
14	C₂×#92495	C₄	1	16	yes	complemented
15	C₂×#92495	C₂²	1	2	yes	—
16	C₂×#92495	C₂²	1	2	yes	—
17	C₂×#92495	C₂²	1	2	yes	—
18	C₂×#92495	C₂²	1	2	yes	—
19	C₂²×D₁₂₄	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
21	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
22	C₃₁×#117	C₂²	1	2	yes	—
23	C₃₁×#117	C₂²	1	2	yes	—
24	C₃₁×#117	C₂²	1	2	yes	complemented
25	C₃₁×#117	C₂²	1	2	yes	complemented
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊Dic₃₁	C₂²	1	2	yes	—
30	C₄×C₂²⋊D₃₁	C₂²	1	16	yes	complemented
31	#11758503	C₂²	1	2	yes	—
32	#11758503	C₂²	1	2	yes	—
33	#11758503	C₂²	1	2	yes	complemented
34	#11758503	C₂²	1	2	yes	complemented
35	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	complemented
36	C₂×C₄×Dic₃₁	C₂²	1	2	yes	complemented
37	#11758470	C₂²	1	8	yes	complemented
38	#11758469	C₂²	1	8	yes	complemented
39	C₄⋊C₄×D₃₁	C₂²	1	8	yes	complemented
40	#11758464	C₂²	1	8	yes	complemented
41	(C₂²⋊D₃₁)⋊C₄	C₂²	1	16	yes	complemented
42	C₂²⋊C₄×D₃₁	C₂²	1	8	yes	—
43	#11758454	C₂²	1	8	yes	—
44	C₄×D₁₂₄	C₂²	1	8	yes	complemented
45	#11758449	C₂²	1	8	yes	complemented
46	C₄²×D₃₁	C₂²	1	8	yes	complemented
47	C₄×Dic₆₂	C₂²	1	8	yes	complemented
48	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×D₃₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
51	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₃₁	C₂×C₄	1	16	yes	complemented
59	C₂×C₂²⋊D₃₁	C₂³	1	4	yes	—
60	C₂×C₂²⋊D₃₁	C₂³	1	4	yes	—
61	C₂²×Dic₃₁	C₂³	1	4	yes	—
62	C₂²×Dic₃₁	C₂³	1	2	yes	—
63	#92495	C₂×C₄	1	16	yes	complemented
64	#92495	C₂×C₄	1	16	yes	complemented
65	#92495	C₂×C₄	1	16	yes	complemented
66	#92495	C₂×C₄	1	16	yes	complemented
67	C₂×D₁₂₄	C₂×C₄	1	8	yes	complemented
68	C₂×D₁₂₄	C₂³	1	2	yes	—
69	C₂×D₁₂₄	C₂³	1	2	yes	—
70	C₂×C₄×D₃₁	C₂×C₄	1	16	yes	complemented
71	C₂×C₄×D₃₁	C₂³	1	16	yes	—
72	C₂×C₄×D₃₁	C₂³	1	4	yes	—
73	C₂×C₄×D₃₁	C₂³	1	4	yes	—
74	C₂×Dic₆₂	C₂×C₄	1	8	yes	complemented
75	C₂×Dic₆₂	C₂³	1	2	yes	—
76	C₂×Dic₆₂	C₂³	1	2	yes	—
77	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
78	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₄²×C₃₁	C₂³	1	4	yes	—
82	C₄²×C₃₁	C₂³	1	4	yes	—
83	C₂²⋊Dic₃₁	C₂³	1	4	yes	—
84	C₂²⋊Dic₃₁	C₂³	1	4	yes	—
85	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	—
86	C₄⋊Dic₃₁	C₂³	1	4	yes	—
87	C₄⋊Dic₃₁	C₂³	1	4	yes	—
88	Dic₃₁⋊C₄	C₂³	1	16	yes	—
89	C₄×Dic₃₁	C₂³	1	16	yes	—
90	C₄×Dic₃₁	C₂³	1	4	yes	—
91	C₄×Dic₃₁	C₂³	1	4	yes	—
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
94	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₂²×D₃₁	—	4	4	no	—
98	C₂²×D₃₁	C₂²×C₄	1	8	yes	—
99	C₂²×D₃₁	C₂⁴	1	4	yes	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
103	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
107	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
108	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₃₁	C₂²×C₄	1	16	yes	—
113	C₂²⋊D₃₁	C₂²×C₄	1	16	yes	—
114	C₂×Dic₃₁	—	4	4	no	—
115	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
116	C₂×Dic₃₁	C₂⁴	1	8	yes	—
117	C₂×Dic₃₁	C₂⁴	1	4	yes	—
118	D₁₂₄	C₂²×C₄	1	8	yes	—
119	D₁₂₄	C₂²×C₄	1	8	yes	—
120	C₄×D₃₁	—	4	16	no	—
121	C₄×D₃₁	C₂²×C₄	1	16	yes	—
122	C₄×D₃₁	C₂²×C₄	1	16	yes	—
123	Dic₆₂	C₂²×C₄	1	8	yes	—
124	Dic₆₂	C₂²×C₄	1	8	yes	—
125	#3428	—	31	1	no	maximal, Sylow
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	4	4	no	—
130	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
131	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
132	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
133	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
134	C₂×D₃₁	—	2	8	no	—
135	C₂×D₃₁	—	4	8	no	—
136	C₂×D₃₁	C₂³×C₄	1	8	yes	—
137	C₄×C₃₁	—	2	8	no	—
138	C₄×C₃₁	C₂³×C₄	1	2	yes	—
139	C₄×C₃₁	C₂³×C₄	1	2	yes	—
140	C₄×C₃₁	C₂³×C₄	1	2	yes	—
141	C₄×C₃₁	C₂³×C₄	1	2	yes	—
142	Dic₃₁	—	2	8	no	—
143	Dic₃₁	C₂³×C₄	1	8	yes	—
144	C₂²×#41	—	31	1	no	—
145	#520	—	31	4	no	—
146	#520	—	31	4	no	—
147	#519	—	31	2	no	—
148	#518	—	31	2	no	—
149	#518	—	31	4	no	—
150	C₄×#41	—	31	4	no	—
151	C₄×#41	—	31	4	no	—
152	C₂×#117	—	31	1	no	—
153	C₂×#117	—	31	4	no	—
154	C₂×#117	D₃₁	1	1	yes	characteristic, complemented
155	C₂×C₃₁	—	2	2	no	—
156	C₂×C₃₁	—	2	2	no	—
157	C₂×C₃₁	—	2	2	no	—
158	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
159	C₂×C₃₁	#585	1	2	yes	—
160	D₃₁	—	2	8	no	—
161	C₂×#41	—	31	16	no	—
162	C₂×#41	—	31	2	no	—
163	C₂×#41	—	31	2	no	—
164	C₂×#41	—	31	2	no	—
165	C₂×#41	—	31	2	no	—
166	C₂²×Q₈	—	31	1	no	—
167	C₂²×D₄	—	31	2	no	—
168	C₂²×D₄	—	31	1	no	—
169	C₂³×C₄	—	31	2	no	—
170	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
171	C₄×Q₈	—	31	8	no	—
172	C₄×Q₈	—	31	8	no	—
173	C₄×D₄	—	31	16	no	—
174	C₄×D₄	—	31	8	no	—
175	C₄×D₄	—	31	16	no	—
176	C₄×D₄	—	31	8	no	—
177	#117	—	31	2	no	—
178	#117	—	31	2	no	—
179	#117	—	31	2	no	—
180	#117	—	31	2	no	—
181	#117	—	31	8	no	—
182	#117	—	31	8	no	—
183	#117	—	31	8	no	—
184	#117	C₂×D₃₁	1	2	yes	—
185	#117	C₂×D₃₁	1	2	yes	—
186	#117	C₂×D₃₁	1	2	yes	complemented
187	#117	C₂×D₃₁	1	2	yes	complemented
188	C₂×C₄⋊C₄	—	31	2	no	—
189	C₂×C₄⋊C₄	—	31	8	no	—
190	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	complemented
191	C₂×C₂²⋊C₄	—	31	2	no	—
192	C₂×C₂²⋊C₄	—	31	8	no	—
193	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
194	C₂×C₄²	—	31	2	no	—
195	C₂×C₄²	—	31	8	no	—
196	C₂×C₄²	C₂×D₃₁	1	2	yes	complemented
197	C₃₁	#3428	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	31	2	no	—
199	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
200	#41	—	31	16	no	—
201	#41	—	31	16	no	—
202	#41	—	31	16	no	—
203	#41	—	31	16	no	—
204	C₂×Q₈	—	31	8	no	—
205	C₂×Q₈	—	31	2	no	—
206	C₂×Q₈	—	31	2	no	—
207	C₂×D₄	—	31	16	no	—
208	C₂×D₄	—	31	4	no	—
209	C₂×D₄	—	31	4	no	—
210	C₂×D₄	—	31	8	no	—
211	C₂×D₄	—	31	2	no	—
212	C₂×D₄	—	31	2	no	—
213	C₂²×C₄	—	31	2	no	—
214	C₂²×C₄	—	31	16	no	—
215	C₂²×C₄	—	31	4	no	—
216	C₂²×C₄	—	31	4	no	—
217	C₂²×C₄	—	31	4	no	—
218	C₂²×C₄	—	31	16	no	—
219	C₂²×C₄	C₄×D₃₁	1	8	yes	complemented
220	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
221	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
227	C₄⋊C₄	—	31	16	no	—
228	C₄⋊C₄	—	31	4	no	—
229	C₄⋊C₄	—	31	4	no	—
230	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
231	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
232	C₂²⋊C₄	—	31	4	no	—
233	C₂²⋊C₄	—	31	4	no	—
234	C₂²⋊C₄	—	31	16	no	—
235	C₂²⋊C₄	C₂²×D₃₁	1	4	yes	—
236	C₂²⋊C₄	C₂²×D₃₁	1	4	yes	—
237	C₄²	—	31	16	no	—
238	C₄²	—	31	4	no	—
239	C₄²	—	31	4	no	—
240	C₄²	C₂²×D₃₁	1	4	yes	—
241	C₄²	C₂²×D₃₁	1	4	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	—	31	4	no	—
244	C₂³	—	31	8	no	—
245	C₂³	—	124	4	no	—
246	C₂³	C₂×C₄×D₃₁	1	4	yes	—
247	C₂³	C₂³×D₃₁	1	1	yes	characteristic
248	C₂³	C₂³×D₃₁	1	1	yes	characteristic
249	C₂³	C₂³×D₃₁	1	1	yes	characteristic
250	Q₈	—	31	8	no	—
251	Q₈	—	31	8	no	—
252	D₄	—	31	16	no	—
253	D₄	—	31	16	no	—
254	D₄	—	31	8	no	—
255	D₄	—	31	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	31	8	no	—
258	C₂×C₄	—	31	4	no	—
259	C₂×C₄	—	31	16	no	—
260	C₂×C₄	—	31	16	no	—
261	C₂×C₄	—	31	8	no	—
262	C₂×C₄	—	124	16	no	—
263	C₂×C₄	—	124	4	no	—
264	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
265	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
266	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
267	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
268	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
269	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
270	C₂×C₄	C₂³×D₃₁	1	8	yes	—
271	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	31	8	no	—
280	C₂²	—	62	8	no	—
281	C₂²	—	124	8	no	—
282	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
283	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
284	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
285	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	31	8	no	—
288	C₄	—	62	8	no	—
289	C₄	C₂²×C₄×D₃₁	1	2	yes	—
290	C₄	C₂²×C₄×D₃₁	1	2	yes	—
291	C₄	C₂²×C₄×D₃₁	1	2	yes	—
292	C₄	C₂²×C₄×D₃₁	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	62	8	no	—
297	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
298	C₂	#14521885	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle, cosocle	C₂²×C₃₁	C₂⁵
prime radical	C₂²×#92495	C₂
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	#3428
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #14521530	C₂	semidirect
3. #14521529	C₂	semidirect
4. #14521526	C₂	semidirect
5. #14521523	C₂	semidirect
6. #14521522	C₂	semidirect
7. C₄×#92495	C₂	semidirect
8. #14521517	C₂	semidirect
9. C₂×#11758503	C₂	semidirect
10. #117×D₃₁	C₂	semidirect
11. C₂×C₃₁×#117	C₂	semidirect
14. C₂×#92495	C₄	semidirect
24. C₃₁×#117	C₂²	semidirect
25. C₃₁×#117	C₂²	semidirect
26. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
28. C₂×C₄²×C₃₁	C₂²	semidirect
30. C₄×C₂²⋊D₃₁	C₂²	semidirect
33. #11758503	C₂²	semidirect
34. #11758503	C₂²	semidirect
35. C₂×C₄⋊Dic₃₁	C₂²	semidirect
36. C₂×C₄×Dic₃₁	C₂²	semidirect
37. #11758470	C₂²	semidirect
38. #11758469	C₂²	semidirect
39. C₄⋊C₄×D₃₁	C₂²	semidirect
40. #11758464	C₂²	semidirect
41. (C₂²⋊D₃₁)⋊C₄	C₂²	semidirect
44. C₄×D₁₂₄	C₂²	semidirect
45. #11758449	C₂²	semidirect
46. C₄²×D₃₁	C₂²	semidirect
47. C₄×Dic₆₂	C₂²	semidirect
50. C₂²×C₄×C₃₁	C₂×C₄	semidirect
58. C₂×C₂²⋊D₃₁	C₂×C₄	semidirect
63. #92495	C₂×C₄	semidirect
64. #92495	C₂×C₄	semidirect
65. #92495	C₂×C₄	semidirect
66. #92495	C₂×C₄	semidirect
67. C₂×D₁₂₄	C₂×C₄	semidirect
70. C₂×C₄×D₃₁	C₂×C₄	semidirect
74. C₂×Dic₆₂	C₂×C₄	semidirect
154. C₂×#117	D₃₁	semidirect
186. #117	C₂×D₃₁	semidirect
187. #117	C₂×D₃₁	semidirect
190. C₂×C₄⋊C₄	C₂×D₃₁	semidirect
196. C₂×C₄²	C₂×D₃₁	semidirect
197. C₃₁	#3428	semidirect
219. C₂²×C₄	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#92495	2
#14521530	2
#14521529	2
#14521526	2
#14521523	2
#14521522	2
C₄×#92495	2
#14521517	2
C₂×#11758503	2
#117×D₃₁	2
C₂×C₃₁×#117	2
#3428	31