Subgroups — #15761811

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 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×C₁₃×#322	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	C₁₃×#322	—	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	4	no	—
31	C₂×Dic₅₂	—	2	2	no	—
32	C₄×Dic₂₆	—	2	2	no	—
33	C₄×Dic₂₆	—	2	2	no	—
34	C₄×Dic₂₆	—	2	2	no	—
35	C₄×Dic₂₆	—	2	2	no	—
36	C₄×C₈×C₁₃	—	2	1	no	—
37	C₄×C₈×C₁₃	—	2	1	no	—
38	C₄×C₈×C₁₃	—	2	1	no	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	C₄×C₈×C₁₃	D₄	1	2	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	#88058	—	2	2	no	—
43	#88056	—	2	4	no	—
44	C₄⋊#1055	—	2	2	no	—
45	C₄⋊#1055	—	2	2	no	—
46	#9662	—	13	1	no	maximal, Sylow
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	2	2	no	—
51	C₂²×Dic₁₃	—	2	2	no	—
52	C₂×Dic₂₆	—	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	M₁₆	1	4	yes	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	4	2	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
67	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₃	#41	1	1	yes	characteristic
71	C₄⋊Dic₁₃	—	2	2	no	—
72	C₄⋊Dic₁₃	—	2	2	no	—
73	#6101	—	4	2	no	—
74	#6101	—	4	2	no	—
75	C₄×Dic₁₃	—	4	2	no	—
76	C₄×Dic₁₃	—	4	2	no	—
77	C₂×#1055	—	4	2	no	—
78	C₂×#1055	—	4	4	no	—
79	Dic₅₂	—	2	8	no	—
80	C₂×C₄×Q₁₆	—	13	1	no	—
81	#1984	—	13	2	no	—
82	C₂×#335	—	13	1	no	—
83	C₂×Q₈⋊C₈	—	13	2	no	—
84	C₂×#322	D₁₃	1	1	yes	characteristic
85	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₁₃	#131	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₁₃	#135	1	2	yes	—
97	C₂×Dic₁₃	—	2	4	no	—
98	C₂×Dic₁₃	—	4	2	no	—
99	C₂×Dic₁₃	—	4	2	no	—
100	C₂×Dic₁₃	—	4	2	no	—
101	C₂×Dic₁₃	—	4	4	no	—
102	Dic₂₆	—	2	4	no	—
103	Dic₂₆	—	2	4	no	—
104	Dic₂₆	—	4	2	no	—
105	Dic₂₆	—	4	2	no	—
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	4	no	—
109	C₈×C₁₃	—	4	2	no	—
110	#1055	—	8	2	no	—
111	C₂²×Q₁₆	—	13	1	no	—
112	C₂×C₄×Q₈	—	13	2	no	—
113	C₄×Q₁₆	—	26	2	no	—
114	C₄×Q₁₆	—	26	2	no	—
115	C₂×#107	—	13	1	no	—
116	C₂×C₄⋊C₈	—	13	2	no	—
117	C₂×#103	—	13	2	no	—
118	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
121	#336	—	26	4	no	—
122	#335	—	26	2	no	—
123	Q₈⋊C₈	—	26	4	no	—
124	#322	—	2	2	no	—
125	C₂²×C₁₃	#390	1	1	yes	characteristic
126	C₂²×C₁₃	#435	1	2	yes	—
127	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
128	C₂²×C₁₃	#444	1	2	yes	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	#407	1	1	yes	characteristic
132	C₄×C₁₃	#408	1	1	yes	characteristic
133	C₄×C₁₃	#435	1	2	yes	—
134	C₄×C₁₃	#467	1	2	yes	—
135	C₄×C₁₃	#494	1	1	yes	characteristic
136	C₄×C₁₃	#495	1	1	yes	characteristic
137	Dic₁₃	—	4	4	no	—
138	Dic₁₃	—	8	2	no	—
139	C₂²×Q₈	—	13	2	no	—
140	C₂×Q₁₆	—	26	4	no	—
141	C₂×Q₁₆	—	26	2	no	—
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	—	26	2	no	—
144	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
145	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
146	C₂²×C₈	D₅₂	1	2	yes	—
147	C₄×Q₈	—	26	2	no	—
148	C₄×Q₈	—	26	2	no	—
149	C₄×Q₈	—	26	2	no	—
150	C₄×Q₈	—	26	2	no	—
151	C₂×C₄⋊C₄	—	13	2	no	—
152	C₂×C₄⋊C₄	—	26	2	no	—
153	C₂×C₄²	—	26	2	no	—
154	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
155	#107	—	26	2	no	—
156	C₄⋊C₈	—	26	2	no	—
157	C₄⋊C₈	—	26	2	no	—
158	#103	—	26	4	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	D₅₂	1	2	yes	—
163	C₄×C₈	D₅₂	1	2	yes	—
164	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
165	C₂×C₁₃	#1584	1	4	yes	—
166	C₂×C₁₃	#1955	1	1	yes	characteristic
167	C₂×C₁₃	#1969	1	1	yes	characteristic
168	C₂×C₁₃	#1975	1	1	yes	characteristic
169	C₂×Q₈	—	13	4	no	—
170	C₂×Q₈	—	26	2	no	—
171	C₂×Q₈	—	26	2	no	—
172	C₂×Q₈	—	26	2	no	—
173	C₂×Q₈	—	26	2	no	—
174	C₂×Q₈	—	26	2	no	—
175	C₂²×C₄	—	26	2	no	—
176	C₂²×C₄	—	26	2	no	—
177	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
179	C₂²×C₄	#6127	1	1	yes	characteristic
180	Q₁₆	—	26	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	52	2	no	—
188	C₂×C₈	—	52	4	no	—
189	C₂×C₈	Dic₅₂	1	4	yes	complemented
190	C₂×C₈	#6103	1	2	yes	—
191	C₂×C₈	#6103	1	2	yes	—
192	C₄⋊C₄	—	26	2	no	—
193	C₄⋊C₄	—	26	2	no	—
194	C₄⋊C₄	—	52	2	no	—
195	C₄⋊C₄	—	52	2	no	—
196	C₄²	—	52	2	no	—
197	C₄²	—	52	2	no	—
198	C₄²	C₂×D₅₂	1	1	yes	characteristic
199	C₄²	C₂×D₅₂	1	1	yes	characteristic
200	C₄²	#6127	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₁₃	#9662	1	1	yes	characteristic, complemented, Sylow
203	C₂³	C₄×D₅₂	1	1	yes	characteristic
204	Q₈	—	26	4	no	—
205	Q₈	—	26	4	no	—
206	Q₈	—	52	2	no	—
207	Q₈	—	52	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	26	4	no	—
210	C₂×C₄	—	52	2	no	—
211	C₂×C₄	—	52	2	no	—
212	C₂×C₄	—	52	2	no	—
213	C₂×C₄	—	52	4	no	—
214	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
215	C₂×C₄	#88128	1	1	yes	characteristic
216	C₂×C₄	#88129	1	1	yes	characteristic
217	C₂×C₄	#88158	1	2	yes	—
218	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
219	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
220	C₂×C₄	#88161	1	2	yes	—
221	C₂×C₄	C₂×#6103	1	1	yes	characteristic
222	C₂×C₄	#88184	1	1	yes	characteristic
223	C₂×C₄	#88184	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	104	2	no	—
229	C₂²	C₄×Dic₅₂	1	2	yes	—
230	C₂²	#11718367	1	2	yes	—
231	C₂²	#11718378	1	2	yes	—
232	C₂²	#11718600	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	52	4	no	—
236	C₄	—	104	2	no	—
237	C₄	#11718357	1	1	yes	characteristic
238	C₄	#11718360	1	1	yes	characteristic
239	C₄	#11718367	1	2	yes	—
240	C₄	#11718777	1	2	yes	—
241	C₄	#11718787	1	1	yes	characteristic
242	C₄	#11718790	1	1	yes	characteristic
243	C₂	#13113688	1	4	yes	—
244	C₂	#13115634	1	1	yes	characteristic
245	C₂	#13115697	1	1	yes	characteristic
246	C₂	#13115729	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#9662
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
66. C₂×C₈×C₁₃	Q₁₆	semidirect
189. C₂×C₈	Dic₅₂	semidirect
202. C₁₃	#9662	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×C₁₃×#322	2
#9662	13