Subgroups — #15761777

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	#13115772	C₂	1	2	yes	maximal
3	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718048	C₂	1	2	yes	maximal
5	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88058	C₄	1	1	yes	characteristic
12	C₂×#88056	C₄	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₄×Dic₅₂	—	2	2	no	—
15	C₄×Dic₅₂	—	2	2	no	—
16	C₈²×C₁₃	—	2	2	no	—
17	#11718142	—	2	4	no	—
18	#11718050	—	2	2	no	—
19	#11718048	—	2	4	no	—
20	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂×#6101	—	2	2	no	—
28	C₂×C₄×Dic₁₃	—	2	2	no	—
29	C₂²×#1055	—	2	2	no	—
30	C₂×Dic₅₂	—	2	2	no	—
31	C₂×Dic₅₂	—	2	2	no	—
32	C₂×Dic₅₂	C₈	1	4	yes	complemented
33	C₄×Dic₂₆	—	2	2	no	—
34	C₄×Dic₂₆	—	2	2	no	—
35	C₄×Dic₂₆	—	2	2	no	—
36	C₄×Dic₂₆	—	2	2	no	—
37	C₄×C₈×C₁₃	—	2	1	no	—
38	C₄×C₈×C₁₃	—	2	1	no	—
39	C₄×C₈×C₁₃	—	2	1	no	—
40	C₄×C₈×C₁₃	D₄	1	2	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	C₄×C₈×C₁₃	D₄	1	2	yes	—
43	#88058	—	2	2	no	—
44	#88056	—	2	4	no	—
45	C₄⋊#1055	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	#9656	—	13	1	no	maximal, Sylow
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	2	2	no	—
52	C₂²×Dic₁₃	—	2	2	no	—
53	C₂×Dic₂₆	—	2	2	no	—
54	C₂×Dic₂₆	—	2	2	no	—
55	C₂×Dic₂₆	—	2	2	no	—
56	C₂×Dic₂₆	—	2	2	no	—
57	C₂×Dic₂₆	—	2	2	no	—
58	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₃	#41	1	1	yes	characteristic
74	C₄⋊Dic₁₃	—	2	2	no	—
75	C₄⋊Dic₁₃	—	2	2	no	—
76	#6101	—	4	2	no	—
77	#6101	—	4	2	no	—
78	C₄×Dic₁₃	—	4	2	no	—
79	C₄×Dic₁₃	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	C₂×#1055	—	4	4	no	—
82	Dic₅₂	—	2	8	no	—
83	C₂×C₄×Q₁₆	—	13	1	no	—
84	#1984	—	13	2	no	—
85	C₂×#335	—	13	1	no	—
86	C₂×Q₈⋊C₈	—	13	2	no	—
87	C₂×C₈²	D₁₃	1	1	yes	characteristic
88	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₁₃	#135	1	2	yes	—
102	C₂×Dic₁₃	—	2	4	no	—
103	C₂×Dic₁₃	—	4	2	no	—
104	C₂×Dic₁₃	—	4	2	no	—
105	C₂×Dic₁₃	—	4	2	no	—
106	C₂×Dic₁₃	—	4	4	no	—
107	Dic₂₆	—	2	4	no	—
108	Dic₂₆	—	2	4	no	—
109	Dic₂₆	—	4	2	no	—
110	Dic₂₆	—	4	2	no	—
111	C₈×C₁₃	—	2	4	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	—	2	4	no	—
114	C₈×C₁₃	C₂²⋊C₈	1	4	yes	—
115	#1055	—	8	2	no	—
116	C₂²×Q₁₆	—	13	1	no	—
117	C₂×C₄×Q₈	—	13	2	no	—
118	C₄×Q₁₆	—	26	2	no	—
119	C₄×Q₁₆	—	26	2	no	—
120	C₂×#107	—	13	1	no	—
121	C₂×C₄⋊C₈	—	13	2	no	—
122	C₂×#103	—	13	2	no	—
123	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
126	#336	—	26	4	no	—
127	#335	—	26	2	no	—
128	Q₈⋊C₈	—	26	4	no	—
129	C₈²	—	2	2	no	—
130	C₂²×C₁₃	#390	1	1	yes	characteristic
131	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
132	C₂²×C₁₃	#436	1	1	yes	characteristic
133	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
134	C₂²×C₁₃	#443	1	2	yes	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₁₃	#408	1	1	yes	characteristic
139	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
140	C₄×C₁₃	#436	1	1	yes	characteristic
141	C₄×C₁₃	#467	1	2	yes	—
142	C₄×C₁₃	#494	1	1	yes	characteristic
143	C₄×C₁₃	#495	1	1	yes	characteristic
144	Dic₁₃	—	4	4	no	—
145	Dic₁₃	—	8	2	no	—
146	C₂²×Q₈	—	13	2	no	—
147	C₂×Q₁₆	—	13	4	no	—
148	C₂×Q₁₆	—	26	2	no	—
149	C₂×Q₁₆	—	26	2	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	26	2	no	—
152	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₈	D₅₂	1	2	yes	—
155	C₄×Q₈	—	26	2	no	—
156	C₄×Q₈	—	26	2	no	—
157	C₄×Q₈	—	26	2	no	—
158	C₄×Q₈	—	26	2	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄⋊C₄	—	26	2	no	—
161	C₂×C₄²	—	26	2	no	—
162	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
163	#107	—	26	2	no	—
164	C₄⋊C₈	—	26	2	no	—
165	C₄⋊C₈	—	26	2	no	—
166	#103	—	26	4	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	D₅₂	1	2	yes	—
171	C₄×C₈	D₅₂	1	2	yes	—
172	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
173	C₂×C₁₃	C₈×Q₁₆	1	2	yes	—
174	C₂×C₁₃	#1591	1	2	yes	—
175	C₂×C₁₃	#1954	1	1	yes	characteristic
176	C₂×C₁₃	#1969	1	1	yes	characteristic
177	C₂×C₁₃	#1974	1	1	yes	characteristic
178	C₂×Q₈	—	13	4	no	—
179	C₂×Q₈	—	26	2	no	—
180	C₂×Q₈	—	26	2	no	—
181	C₂×Q₈	—	26	2	no	—
182	C₂×Q₈	—	26	2	no	—
183	C₂×Q₈	—	26	2	no	—
184	C₂²×C₄	—	26	2	no	—
185	C₂²×C₄	—	26	2	no	—
186	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
188	C₂²×C₄	#6127	1	1	yes	characteristic
189	Q₁₆	—	26	8	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	52	2	no	—
197	C₂×C₈	—	52	4	no	—
198	C₂×C₈	C₈×D₁₃	1	2	yes	—
199	C₂×C₈	#6094	1	2	yes	—
200	C₂×C₈	Dic₅₂	1	4	yes	complemented
201	C₂×C₈	#6103	1	2	yes	—
202	C₂×C₈	#6103	1	2	yes	—
203	C₄⋊C₄	—	26	2	no	—
204	C₄⋊C₄	—	26	2	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₄²	—	52	2	no	—
208	C₄²	—	52	2	no	—
209	C₄²	C₂×D₅₂	1	1	yes	characteristic
210	C₄²	C₂×D₅₂	1	1	yes	characteristic
211	C₄²	#6127	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	C₁₃	#9656	1	1	yes	characteristic, complemented, Sylow
214	C₂³	C₄×D₅₂	1	1	yes	characteristic
215	Q₈	—	26	4	no	—
216	Q₈	—	26	4	no	—
217	Q₈	—	52	2	no	—
218	Q₈	—	52	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	26	4	no	—
221	C₂×C₄	—	52	2	no	—
222	C₂×C₄	—	52	2	no	—
223	C₂×C₄	—	52	2	no	—
224	C₂×C₄	—	52	4	no	—
225	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
226	C₂×C₄	#88128	1	1	yes	characteristic
227	C₂×C₄	#88129	1	1	yes	characteristic
228	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂×#6094	1	1	yes	characteristic
230	C₂×C₄	#88155	1	1	yes	characteristic
231	C₂×C₄	#88155	1	1	yes	characteristic
232	C₂×C₄	#88158	1	2	yes	—
233	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
234	C₂×C₄	C₂×#6103	1	1	yes	characteristic
235	C₂×C₄	#88184	1	1	yes	characteristic
236	C₂×C₄	#88184	1	1	yes	characteristic
237	C₈	—	2	4	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	4	no	—
240	C₈	—	104	2	no	—
241	C₈	#88060	1	4	yes	—
242	C₂²	C₈×D₅₂	1	1	yes	characteristic
243	C₂²	#11718349	1	1	yes	characteristic
244	C₂²	C₄×Dic₅₂	1	2	yes	—
245	C₂²	#11718361	1	2	yes	—
246	C₂²	#11718600	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	52	4	no	—
250	C₄	—	104	2	no	—
251	C₄	C₈×D₅₂	1	1	yes	characteristic
252	C₄	#11718349	1	1	yes	characteristic
253	C₄	#11718357	1	1	yes	characteristic
254	C₄	#11718360	1	1	yes	characteristic
255	C₄	C₂×#88060	1	1	yes	characteristic
256	C₄	#11718771	1	1	yes	characteristic
257	C₄	#11718777	1	2	yes	—
258	C₂	C₈×Dic₅₂	1	2	yes	—
259	C₂	#13113734	1	2	yes	—
260	C₂	#13115630	1	1	yes	characteristic
261	C₂	#13115697	1	1	yes	characteristic
262	C₂	#13115724	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#9656
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
32. C₂×Dic₅₂	C₈	semidirect
69. C₂×C₈×C₁₃	Q₁₆	semidirect
200. C₂×C₈	Dic₅₂	semidirect
213. C₁₃	#9656	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
#13115772	2
C₂×#11718050	2
C₂×#11718048	2
C₂×C₈²×C₁₃	2
#9656	13