Subgroups — #30674616

Order 16768.

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×D₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#22553374	C₂	1	2	yes	maximal, complemented
3	#22553371	C₂	1	2	yes	maximal, complemented
4	#22553367	C₂	1	2	yes	maximal, complemented
5	#22553363	C₂	1	2	yes	maximal, complemented
6	#22553340	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553338	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553275	C₂	1	1	yes	maximal, characteristic
9	C₂×(C₂×D₁₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic
10	#22553268	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×#14523497	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	#16862205	C₂²	1	1	yes	characteristic
16	#16862204	C₄	1	1	yes	characteristic
17	#16862204	C₄	1	1	yes	characteristic
18	#16862203	C₂²	1	2	yes	—
19	#16862202	C₄	1	2	yes	—
20	C₄⋊C₄×D₁₃₁	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₁₃₁	C₄	1	1	yes	characteristic
22	C₂²⋊C₄×D₁₃₁	C₄	1	2	yes	—
23	C₄²×D₁₃₁	C₂²	1	2	yes	—
24	Dic₁₃₁⋊C₈	C₂²	1	2	yes	complemented
25	Dic₁₃₁⋊C₈	C₂²	1	2	yes	complemented
26	Dic₁₃₁⋊C₈	C₂²	1	2	yes	complemented
27	Dic₁₃₁⋊C₈	C₂²	1	2	yes	complemented
28	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
34	#16862157	C₂²	1	2	yes	complemented
35	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
38	#117×C₁₃₁	C₂²	1	1	yes	characteristic
39	#97×C₁₃₁	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₃₁	C₂²	1	2	yes	complemented
41	C₂³×D₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃₁	—	2	2	no	—
44	C₂×C₄×D₁₃₁	—	2	2	no	—
45	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
51	#14523497	—	2	2	no	—
52	C₂×C₁₃₁⋊C₈	—	2	2	no	—
53	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
56	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
58	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
61	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
62	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₃₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₃₁	—	2	2	no	—
68	C₂×C₈×C₁₃₁	—	2	2	no	—
69	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
71	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
75	C₄²×C₁₃₁	C₂³	1	2	yes	—
76	C₂²×D₁₃₁	—	2	2	no	—
77	C₂²×D₁₃₁	—	2	2	no	—
78	C₂²×D₁₃₁	—	4	1	no	—
79	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃₁	—	2	2	no	—
82	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₄×D₁₃₁	—	2	4	no	—
87	C₄×D₁₃₁	—	2	2	no	—
88	C₄×D₁₃₁	—	2	2	no	—
89	C₄×D₁₃₁	—	2	1	no	—
90	C₄×D₁₃₁	—	2	1	no	—
91	C₄×D₁₃₁	—	2	1	no	—
92	C₄×D₁₃₁	—	2	1	no	—
93	C₄×D₁₃₁	—	4	2	no	—
94	C₁₃₁⋊C₈	—	2	1	no	—
95	C₁₃₁⋊C₈	—	2	1	no	—
96	C₁₃₁⋊C₈	—	2	2	no	—
97	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	#41	1	2	yes	—
102	C₂×C₄×C₁₃₁	#41	1	2	yes	—
103	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
104	C₈×C₁₃₁	—	2	1	no	—
105	C₈×C₁₃₁	—	2	1	no	—
106	C₈×C₁₃₁	—	2	2	no	—
107	C₂²×C₁₃₁	—	2	1	no	—
108	C₂²×C₁₃₁	—	2	1	no	—
109	C₂²×C₁₃₁	#117	1	2	yes	—
110	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×D₁₃₁	—	2	1	no	—
112	C₂×D₁₃₁	—	2	1	no	—
113	C₂×D₁₃₁	—	2	1	no	—
114	C₂×D₁₃₁	—	2	1	no	—
115	C₂×D₁₃₁	—	2	1	no	—
116	C₂×D₁₃₁	—	4	4	no	—
117	C₂×D₁₃₁	#131	1	2	yes	—
118	C₄×C₁₃₁	—	2	2	no	—
119	C₄×C₁₃₁	—	2	2	no	—
120	C₄×C₁₃₁	#117	1	2	yes	—
121	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
123	Dic₁₃₁	—	2	1	no	—
124	Dic₁₃₁	—	2	1	no	—
125	Dic₁₃₁	—	2	2	no	—
126	Dic₁₃₁	—	2	2	no	—
127	C₂×C₁₃₁	—	2	2	no	—
128	C₂×C₁₃₁	C₂×#117	1	1	yes	characteristic
129	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₃₁	#568	1	1	yes	characteristic
131	D₁₃₁	—	2	2	no	—
132	D₁₃₁	—	4	1	no	—
133	C₁₃₁	#2924	1	1	yes	characteristic, complemented, Sylow
134	#2924	—	131	1	no	maximal, Sylow
135	C₂×#117	—	131	1	no	—
136	#433	—	131	2	no	—
137	#433	—	131	2	no	—
138	#432	—	131	2	no	—
139	#431	—	131	2	no	—
140	#424	—	131	1	no	—
141	#424	—	131	1	no	—
142	#407	—	131	1	no	—
143	C₂×C₂²⋊C₈	—	131	1	no	—
144	#405	—	131	1	no	—
145	#405	D₁₃₁	1	1	yes	characteristic, complemented
146	C₂³×C₄	—	131	1	no	—
147	C₂×M₁₆	—	131	1	no	—
148	C₂×M₁₆	C₂×D₁₃₁	1	1	yes	characteristic
149	C₂²×C₈	—	131	1	no	—
150	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
151	#117	—	131	1	no	—
152	#117	—	131	1	no	—
153	#117	—	131	1	no	—
154	#117	—	131	2	no	—
155	#117	—	131	2	no	—
156	#117	C₂×D₁₃₁	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	131	1	no	—
158	C₂×C₄⋊C₄	—	131	1	no	—
159	C₂×C₂²⋊C₄	—	131	2	no	—
160	C₂×C₄²	—	131	2	no	—
161	C₄⋊C₈	—	131	2	no	—
162	C₄⋊C₈	—	131	2	no	—
163	C₄⋊C₈	—	131	2	no	—
164	C₄⋊C₈	—	131	2	no	—
165	C₂²⋊C₈	—	131	1	no	—
166	C₂²⋊C₈	—	131	1	no	—
167	C₂²⋊C₈	—	131	1	no	—
168	C₂²⋊C₈	—	131	1	no	—
169	C₂²⋊C₈	—	131	2	no	—
170	C₂²⋊C₈	—	131	2	no	—
171	#97	—	131	2	no	—
172	#97	C₂×D₁₃₁	1	2	yes	complemented
173	C₄×C₈	—	131	2	no	—
174	C₄×C₈	C₂×D₁₃₁	1	2	yes	complemented
175	C₂⁴	—	131	1	no	—
176	C₂²×C₄	—	262	2	no	—
177	C₂²×C₄	—	262	2	no	—
178	C₂²×C₄	—	131	1	no	—
179	C₂²×C₄	—	131	2	no	—
180	C₂²×C₄	—	131	1	no	—
181	C₂²×C₄	—	131	1	no	—
182	C₂²×C₄	—	131	1	no	—
183	C₂²×C₄	—	131	1	no	—
184	C₂²×C₄	—	131	2	no	—
185	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	262	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	262	2	no	—
190	C₂×C₈	—	131	1	no	—
191	C₂×C₈	—	131	2	no	—
192	C₂×C₈	—	131	1	no	—
193	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
195	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
196	C₄⋊C₄	—	131	2	no	—
197	C₄⋊C₄	—	131	2	no	—
198	C₄⋊C₄	—	131	1	no	—
199	C₄⋊C₄	—	131	1	no	—
200	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
201	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
202	C₂²⋊C₄	—	131	2	no	—
203	C₂²⋊C₄	—	131	2	no	—
204	C₂²⋊C₄	—	131	2	no	—
205	C₂²⋊C₄	C₄×D₁₃₁	1	2	yes	—
206	C₄²	—	131	2	no	—
207	C₄²	—	131	2	no	—
208	C₄²	—	131	2	no	—
209	C₄²	C₂²×D₁₃₁	1	2	yes	—
210	C₂³	—	262	2	no	—
211	C₂³	—	262	2	no	—
212	C₂³	—	524	1	no	—
213	C₂³	—	131	1	no	—
214	C₂³	—	131	1	no	—
215	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
216	C₂×C₄	—	262	4	no	—
217	C₂×C₄	—	262	2	no	—
218	C₂×C₄	—	262	1	no	—
219	C₂×C₄	—	262	1	no	—
220	C₂×C₄	—	262	1	no	—
221	C₂×C₄	—	262	1	no	—
222	C₂×C₄	—	262	2	no	—
223	C₂×C₄	—	262	2	no	—
224	C₂×C₄	—	524	2	no	—
225	C₂×C₄	—	131	1	no	—
226	C₂×C₄	—	131	2	no	—
227	C₂×C₄	—	131	2	no	—
228	C₂×C₄	—	131	1	no	—
229	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
232	C₂×C₄	#14523513	1	2	yes	—
233	C₂×C₄	#14523513	1	2	yes	—
234	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	262	1	no	—
239	C₈	—	262	1	no	—
240	C₈	—	262	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	262	1	no	—
244	C₂²	—	262	1	no	—
245	C₂²	—	262	1	no	—
246	C₂²	—	262	1	no	—
247	C₂²	—	262	1	no	—
248	C₂²	—	524	4	no	—
249	C₂²	—	131	2	no	—
250	C₂²	#16862203	1	2	yes	—
251	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	262	1	no	—
255	C₄	—	262	1	no	—
256	C₄	—	262	2	no	—
257	C₄	—	262	2	no	—
258	C₄	#16862203	1	2	yes	—
259	C₄	C₂×#14523513	1	1	yes	characteristic
260	C₄	C₂×#14523513	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	262	2	no	—
263	C₂	—	524	1	no	—
264	C₂	C₂×#16862203	1	1	yes	characteristic
265	C₂	#131×D₁₃₁	1	1	yes	characteristic
266	C₂	#22554077	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#405×C₁₃₁	C₂
prime radical	C₂³×D₁₃₁	C₂×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#405	D₁₃₁
hypocenter	C₁₃₁	#2924
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22553374	C₂	semidirect
3. #22553371	C₂	semidirect
4. #22553367	C₂	semidirect
5. #22553363	C₂	semidirect
6. #22553340	C₂	semidirect
7. #22553338	C₂	semidirect
10. #22553268	C₂	semidirect
11. #405×C₁₃₁	C₂	semidirect
24. Dic₁₃₁⋊C₈	C₂²	semidirect
25. Dic₁₃₁⋊C₈	C₂²	semidirect
26. Dic₁₃₁⋊C₈	C₂²	semidirect
27. Dic₁₃₁⋊C₈	C₂²	semidirect
34. #16862157	C₂²	semidirect
35. C₄×C₁₃₁⋊C₈	C₂²	semidirect
39. #97×C₁₃₁	C₂²	semidirect
40. C₄×C₈×C₁₃₁	C₂²	semidirect
133. C₁₃₁	#2924	semidirect
145. #405	D₁₃₁	semidirect
172. #97	C₂×D₁₃₁	semidirect
174. C₄×C₈	C₂×D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#117×D₁₃₁	2
#22553374	2
#22553371	2
#22553367	2
#22553363	2
#22553340	2
#22553338	2
#22553275	2
C₂×(C₂×D₁₃₁)⋊C₈	2
#22553268	2
#405×C₁₃₁	2
#2924	131