Subgroups — #15772056

Order 3328.

392 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116447	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
3	#13112776	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
8	C₁₃×#424	—	2	1	no	—
9	C₁₃×#424	—	2	1	no	—
10	C₁₃×#424	—	2	1	no	—
11	C₁₃×#424	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×#88052	—	2	1	no	—
15	C₂²×#6090	—	2	1	no	—
16	C₂²⋊#6090	—	2	1	no	—
17	C₂²⋊#6090	—	2	1	no	—
18	C₂²⋊#6090	C₂²	1	2	yes	—
19	C₂²⋊#6090	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	1	no	—
22	C₂×C₁₃×M₁₆	—	2	1	no	—
23	C₂×C₁₃×M₁₆	—	4	2	no	—
24	C₂²×C₈×C₁₃	—	4	2	no	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#117	—	2	1	no	—
33	C₁₃×#117	—	2	1	no	—
34	C₁₃×#117	—	2	1	no	—
35	C₁₃×#117	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
38	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	#88052	—	4	2	no	—
45	C₂×#6090	—	2	2	no	—
46	C₂×#6090	—	2	2	no	—
47	C₂×#6090	—	4	2	no	—
48	#11441	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₁₃×M₁₆	—	4	2	no	—
61	C₁₃×M₁₆	—	4	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	4	1	no	—
66	C₂×C₈×C₁₃	—	4	1	no	—
67	C₂×C₈×C₁₃	—	4	1	no	—
68	C₂×C₈×C₁₃	—	4	1	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	1	no	—
75	C₁₃×C₄⋊C₄	—	2	1	no	—
76	C₁₃×C₄⋊C₄	—	2	1	no	—
77	C₁₃×C₄⋊C₄	—	2	1	no	—
78	C₁₃×C₂²⋊C₄	—	2	2	no	—
79	C₁₃×C₂²⋊C₄	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	#6090	—	4	2	no	—
83	#6090	—	4	2	no	—
84	C₂×#424	D₁₃	1	1	yes	characteristic
85	#2112	—	13	1	no	—
86	C₂×C₂²⋊C₁₆	—	13	1	no	—
87	#1322	—	13	4	no	—
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	4	1	no	—
91	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
92	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
93	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#99	1	2	yes	—
108	C₂×C₄×C₁₃	#101	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	4	2	no	—
116	C₂²×M₁₆	Dic₁₃	1	1	yes	characteristic
117	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₂×#117	Dic₁₃	1	1	yes	characteristic
119	C₂×M₃₂	—	26	1	no	—
120	C₂²×C₁₆	—	26	1	no	—
121	#424	—	2	1	no	—
122	#424	—	2	1	no	—
123	#424	—	2	1	no	—
124	#424	—	2	1	no	—
125	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
126	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
127	C₂²⋊C₁₆	—	13	2	no	—
128	C₂²⋊C₁₆	—	13	2	no	—
129	C₂²⋊C₁₆	—	26	1	no	—
130	C₂²⋊C₁₆	—	26	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	4	4	no	—
139	C₂²×C₁₃	#324	1	2	yes	—
140	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₁₃	#504	1	2	yes	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	4	2	no	—
145	C₄×C₁₃	—	4	2	no	—
146	C₄×C₁₃	#324	1	2	yes	—
147	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
148	C₄×C₁₃	C₂×#101	1	1	yes	characteristic
149	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
150	C₂×M₁₆	—	2	1	no	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	4	2	no	—
153	C₂²×C₈	—	4	2	no	—
154	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
155	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	—	2	1	no	—
163	#117	—	2	1	no	—
164	#117	—	2	1	no	—
165	C₂×C₄⋊C₄	#1055	1	1	yes	characteristic
166	C₂×C₄⋊C₄	#1055	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	#1055	1	2	yes	—
168	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
169	M₃₂	—	52	2	no	—
170	C₂×C₁₆	—	26	2	no	—
171	C₂×C₁₆	—	26	2	no	—
172	C₂×C₁₆	—	52	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	—	2	2	no	—
179	C₂×C₁₃	—	4	1	no	—
180	C₂×C₁₃	C₂×#324	1	1	yes	characteristic
181	C₂×C₁₃	#2113	1	1	yes	characteristic
182	C₂×C₁₃	#2115	1	1	yes	characteristic
183	C₂⁴	C₂×#1055	1	1	yes	characteristic
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂×#1055	1	2	yes	—
187	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
188	C₂²×C₄	C₂×#1055	1	2	yes	—
189	C₂²×C₄	#6099	1	1	yes	characteristic
190	C₂²×C₄	#6099	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₁₆	—	52	2	no	—
217	C₁₆	—	52	2	no	—
218	C₁₃	#11441	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	C₂²⋊#1055	1	1	yes	characteristic
223	C₂³	C₂²⋊#1055	1	1	yes	characteristic
224	C₂³	C₂²×#1055	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
237	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
238	C₂×C₄	#88074	1	2	yes	—
239	C₂×C₄	#88076	1	2	yes	—
240	C₂×C₄	C₂×#6099	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	#11718120	1	2	yes	—
256	C₂²	#11718801	1	2	yes	—
257	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	#11718120	1	2	yes	—
263	C₄	C₂×#88074	1	1	yes	characteristic
264	C₄	C₂×#88076	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	2	2	no	—
267	C₂	—	4	1	no	—
268	C₂	C₂×#11718120	1	1	yes	characteristic
269	C₂	#13116450	1	1	yes	characteristic
270	C₂	#13116457	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₃
commutator	C₂³×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#424	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#424	D₁₃
hypocenter	C₁₃	#11441
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
3. #13112776	C₂	semidirect
218. C₁₃	#11441	semidirect

Maximal subgroups

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

subgroup	index
#13116447	2
C₂×C₂²⋊#6090	2
#13112776	2
C₂×C₁₃×#424	2
#11441	13