Subgroups — #30674625

Order 16768.

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	#22553780	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	#22553335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553280	C₂	1	1	yes	maximal, characteristic
9	#22553278	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	C₂×#14523513	C₂²	1	1	yes	characteristic
13	C₂×#14523497	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	#16862211	C₄	1	1	yes	characteristic
16	#16862211	C₄	1	1	yes	characteristic
17	#16862208	C₄	1	1	yes	characteristic
18	#16862208	C₄	1	1	yes	characteristic
19	(C₂²⋊D₁₃₁)⋊C₄	C₄	1	2	yes	—
20	(C₂²⋊D₁₃₁)⋊C₄	C₄	1	2	yes	—
21	#16862203	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
23	C₄²×D₁₃₁	C₂²	1	2	yes	—
24	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
26	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
29	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
34	#16862157	C₂²	1	2	yes	—
35	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
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	—
40	C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
41	#14523513	—	2	2	no	—
42	#14523513	—	2	2	no	—
43	C₂×Dic₂₆₂	C₂×C₄	1	1	yes	characteristic
44	C₂×D₅₂₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	2	yes	—
46	C₂²×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
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	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
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	C₂²×C₄	1	2	yes	—
77	Dic₂₆₂	—	2	2	no	—
78	D₅₂₄	—	2	2	no	—
79	C₂²⋊D₁₃₁	—	2	4	no	—
80	C₂×Dic₁₃₁	—	2	2	no	—
81	C₂×Dic₁₃₁	—	4	1	no	—
82	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₄×D₁₃₁	—	2	2	no	—
88	C₄×D₁₃₁	—	2	2	no	—
89	C₄×D₁₃₁	—	2	4	no	—
90	C₄×D₁₃₁	—	4	2	no	—
91	C₁₃₁⋊C₈	—	2	1	no	—
92	C₁₃₁⋊C₈	—	2	1	no	—
93	C₁₃₁⋊C₈	—	2	2	no	—
94	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	#41	1	2	yes	—
99	C₂×C₄×C₁₃₁	#41	1	2	yes	—
100	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
101	C₈×C₁₃₁	—	2	1	no	—
102	C₈×C₁₃₁	—	2	1	no	—
103	C₈×C₁₃₁	—	2	2	no	—
104	C₂²×C₁₃₁	—	2	1	no	—
105	C₂²×C₁₃₁	—	2	1	no	—
106	C₂²×C₁₃₁	#117	1	2	yes	—
107	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₁₃₁	—	2	2	no	—
109	C₂×D₁₃₁	—	2	2	no	—
110	C₂×D₁₃₁	—	2	2	no	—
111	C₄×C₁₃₁	—	2	2	no	—
112	C₄×C₁₃₁	—	2	2	no	—
113	C₄×C₁₃₁	#117	1	2	yes	—
114	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
115	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
116	Dic₁₃₁	—	2	2	no	—
117	Dic₁₃₁	—	2	1	no	—
118	Dic₁₃₁	—	2	1	no	—
119	Dic₁₃₁	—	2	1	no	—
120	Dic₁₃₁	#131	1	2	yes	—
121	C₂×C₁₃₁	—	2	2	no	—
122	C₂×C₁₃₁	C₂×#117	1	1	yes	characteristic
123	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
124	C₂×C₁₃₁	#568	1	1	yes	characteristic
125	D₁₃₁	—	4	2	no	—
126	C₁₃₁	#2925	1	1	yes	characteristic, complemented, Sylow
127	#2925	—	131	1	no	maximal, Sylow
128	C₄×#41	—	131	1	no	—
129	#433	—	131	2	no	—
130	#433	—	131	2	no	—
131	#432	—	131	2	no	—
132	#431	—	131	2	no	—
133	#423	—	131	1	no	—
134	C₂×C₄⋊C₈	—	131	1	no	—
135	#408	—	131	1	no	—
136	#408	—	131	1	no	—
137	#405	—	131	1	no	—
138	#405	D₁₃₁	1	1	yes	characteristic, complemented
139	C₂×#41	—	131	1	no	—
140	C₂×M₁₆	—	131	1	no	—
141	C₂×M₁₆	C₂×D₁₃₁	1	1	yes	characteristic
142	C₂²×C₈	—	131	1	no	—
143	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
144	C₄×Q₈	—	131	1	no	—
145	C₄×Q₈	—	131	1	no	—
146	C₄×D₄	—	131	1	no	—
147	C₄×D₄	—	131	1	no	—
148	C₄×D₄	—	131	2	no	—
149	C₄×D₄	—	131	2	no	—
150	#117	—	131	2	no	—
151	#117	C₂×D₁₃₁	1	1	yes	characteristic
152	C₂×C₄²	—	131	1	no	—
153	C₂×C₄²	—	131	2	no	—
154	C₄⋊C₈	—	131	1	no	—
155	C₄⋊C₈	—	131	1	no	—
156	C₄⋊C₈	—	131	1	no	—
157	C₄⋊C₈	—	131	1	no	—
158	C₄⋊C₈	—	131	2	no	—
159	C₄⋊C₈	—	131	2	no	—
160	C₂²⋊C₈	—	131	2	no	—
161	C₂²⋊C₈	—	131	2	no	—
162	C₂²⋊C₈	—	131	2	no	—
163	C₂²⋊C₈	—	131	2	no	—
164	#97	—	131	2	no	—
165	#97	C₂×D₁₃₁	1	2	yes	—
166	C₄×C₈	—	131	2	no	—
167	C₄×C₈	C₂×D₁₃₁	1	2	yes	—
168	#41	—	262	2	no	—
169	#41	—	262	2	no	—
170	C₂×Q₈	—	131	1	no	—
171	C₂×D₄	—	131	2	no	—
172	C₂×D₄	—	131	1	no	—
173	C₂²×C₄	—	131	1	no	—
174	C₂²×C₄	—	131	2	no	—
175	C₂²×C₄	—	131	2	no	—
176	C₂²×C₄	—	131	1	no	—
177	C₂²×C₄	—	131	2	no	—
178	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	262	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	262	2	no	—
183	C₂×C₈	—	131	1	no	—
184	C₂×C₈	—	131	2	no	—
185	C₂×C₈	—	131	1	no	—
186	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
188	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
189	C₄⋊C₄	—	131	2	no	—
190	C₄⋊C₄	—	131	2	no	—
191	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
192	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
193	C₂²⋊C₄	—	131	2	no	—
194	C₂²⋊C₄	—	131	2	no	—
195	C₂²⋊C₄	C₄×D₁₃₁	1	2	yes	—
196	C₄²	—	131	1	no	—
197	C₄²	—	131	2	no	—
198	C₄²	—	131	1	no	—
199	C₄²	—	131	2	no	—
200	C₄²	—	131	1	no	—
201	C₄²	—	131	1	no	—
202	C₄²	C₂²×D₁₃₁	1	2	yes	—
203	C₂³	—	131	2	no	—
204	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
205	Q₈	—	262	2	no	—
206	D₄	—	262	4	no	—
207	D₄	—	262	2	no	—
208	C₂×C₄	—	262	4	no	—
209	C₂×C₄	—	262	2	no	—
210	C₂×C₄	—	262	2	no	—
211	C₂×C₄	—	262	2	no	—
212	C₂×C₄	—	524	2	no	—
213	C₂×C₄	—	524	1	no	—
214	C₂×C₄	—	131	1	no	—
215	C₂×C₄	—	131	1	no	—
216	C₂×C₄	—	131	1	no	—
217	C₂×C₄	—	131	2	no	—
218	C₂×C₄	—	131	1	no	—
219	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
222	C₂×C₄	#14523513	1	2	yes	—
223	C₂×C₄	#14523513	1	2	yes	—
224	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	262	1	no	—
229	C₈	—	262	1	no	—
230	C₈	—	262	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	262	2	no	—
234	C₂²	—	262	2	no	—
235	C₂²	—	262	2	no	—
236	C₂²	#16862203	1	2	yes	—
237	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	262	2	no	—
241	C₄	—	262	1	no	—
242	C₄	—	262	1	no	—
243	C₄	—	262	1	no	—
244	C₄	—	131	2	no	—
245	C₄	#16862203	1	2	yes	—
246	C₄	C₂×#14523513	1	1	yes	characteristic
247	C₄	C₂×#14523513	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	524	2	no	—
250	C₂	C₂×#16862203	1	1	yes	characteristic
251	C₂	#131×D₁₃₁	1	1	yes	characteristic
252	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	C₂×#14523513	C₂²
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#405	D₁₃₁
hypocenter	C₁₃₁	#2925
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. #22553335	C₂	semidirect
7. C₂×Dic₁₃₁⋊C₈	C₂	semidirect
10. #22553268	C₂	semidirect
11. #405×C₁₃₁	C₂	semidirect
126. C₁₃₁	#2925	semidirect
138. #405	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553780	2
#22553374	2
#22553371	2
#22553367	2
#22553363	2
#22553335	2
C₂×Dic₁₃₁⋊C₈	2
#22553280	2
#22553278	2
#22553268	2
#405×C₁₃₁	2
#2925	131