Subgroups — #15779413

Order 3328.

376 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127938	C₂	1	1	yes	maximal, characteristic
2	#13116601	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116463	C₂	1	1	yes	maximal, characteristic
5	#13114673	C₂	1	1	yes	maximal, characteristic
6	#13112923	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88155	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718774	C₄	1	1	yes	characteristic
13	#11718762	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	1	yes	characteristic
16	C₂×#88045	C₄	1	1	yes	characteristic
17	#11718568	—	2	2	no	—
18	D₁₃×M₃₂	—	2	2	no	—
19	#11718562	—	2	2	no	—
20	#11718144	C₄	1	1	yes	characteristic
21	#11718115	C₄	1	1	yes	characteristic
22	#11718103	—	2	2	no	—
23	#11718100	—	2	2	no	—
24	C₄⋊#6090	—	2	2	no	—
25	C₂×#6127	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	#88180	C₂×C₄	1	1	yes	characteristic
30	C₂×#6099	C₂×C₄	1	1	yes	characteristic
31	C₂×#6099	C₂×C₄	1	1	yes	characteristic
32	#88155	—	2	1	no	—
33	#88155	—	2	1	no	—
34	#88155	D₄	1	2	yes	—
35	#88155	Q₈	1	2	yes	—
36	C₂×#6094	—	2	1	no	—
37	C₂×C₈×D₁₃	—	2	1	no	—
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	—	2	1	no	—
40	C₁₃×M₃₂	—	2	1	no	—
41	C₁₃×M₃₂	D₄	1	2	yes	—
42	C₂×C₁₃×C₁₆	—	2	1	no	—
43	C₂×C₁₃×C₁₆	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	—	2	1	no	—
46	#88061	—	2	1	no	—
47	#88056	—	2	1	no	—
48	#88055	—	2	1	no	—
49	C₈×Dic₁₃	—	2	1	no	—
50	#88052	—	2	1	no	—
51	#88052	—	2	1	no	—
52	#88052	—	2	1	no	—
53	#88052	—	2	1	no	—
54	C₂×#6090	—	2	1	no	—
55	C₂×#6090	D₄	1	2	yes	—
56	#88045	C₈	1	2	yes	—
57	#88045	C₈	1	2	yes	—
58	#88038	—	2	4	no	—
59	C₁₆×D₁₃	—	2	4	no	—
60	#12918	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	2	1	no	—
64	#6127	—	2	1	no	—
65	#6127	—	2	1	no	—
66	#6127	C₄⋊C₄	1	2	yes	—
67	#6127	C₂×C₈	1	2	yes	—
68	C₂×D₅₂	M₁₆	1	1	yes	characteristic
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×Dic₂₆	M₁₆	1	1	yes	characteristic
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂²⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	—	2	1	no	—
85	C₄×Dic₁₃	—	2	1	no	—
86	#6099	—	2	1	no	—
87	#6099	—	2	1	no	—
88	#6099	C₄⋊C₄	1	2	yes	—
89	#6099	C₂×C₈	1	2	yes	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	—	2	1	no	—
92	#6094	—	2	2	no	—
93	#6094	—	2	2	no	—
94	C₈×D₁₃	—	2	2	no	—
95	C₈×D₁₃	—	2	2	no	—
96	C₁₃×C₁₆	—	2	2	no	—
97	C₁₃×C₁₆	—	4	1	no	—
98	#6090	—	2	2	no	—
99	#6090	—	4	1	no	—
100	#3407	—	13	1	no	—
101	#2153	—	13	1	no	—
102	#2150	—	13	1	no	—
103	#2117	—	13	1	no	—
104	#1764	—	13	1	no	—
105	#1378	—	13	1	no	—
106	#1376	D₁₃	1	1	yes	characteristic, complemented
107	C₂³×C₁₃	#131	1	1	yes	characteristic
108	C₂²×D₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
118	C₂²⋊D₁₃	—	2	2	no	—
119	C₂²⋊D₁₃	—	2	2	no	—
120	C₂×Dic₁₃	—	2	1	no	—
121	C₂×Dic₁₃	—	4	1	no	—
122	D₅₂	—	2	1	no	—
123	D₅₂	C₄⋊C₈	1	2	yes	—
124	C₄×D₁₃	—	2	2	no	—
125	C₄×D₁₃	—	2	2	no	—
126	Dic₂₆	—	2	1	no	—
127	Dic₂₆	C₄⋊C₈	1	2	yes	—
128	C₈×C₁₃	—	4	1	no	—
129	C₈×C₁₃	#120	1	1	yes	characteristic
130	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₁₃	#123	1	1	yes	characteristic
133	#1055	—	2	2	no	—
134	#1055	—	2	2	no	—
135	C₂×#131	—	13	1	no	—
136	#504	—	26	2	no	—
137	#504	—	26	2	no	—
138	C₂×M₃₂	—	13	1	no	—
139	C₂×M₃₂	—	13	1	no	—
140	C₂×M₃₂	—	26	2	no	—
141	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
142	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
143	C₂×#104	—	13	1	no	—
144	#416	—	13	1	no	—
145	#405	—	13	1	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#364	—	26	2	no	—
148	C₄⋊C₁₆	—	26	2	no	—
149	#350	—	26	2	no	—
150	#341	—	13	1	no	—
151	#339	—	13	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	#424	1	1	yes	characteristic
154	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
155	C₂²×C₁₃	#435	1	1	yes	characteristic
156	C₂×D₁₃	—	2	2	no	—
157	C₂×D₁₃	—	8	1	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	4	1	no	—
160	C₄×C₁₃	#385	1	1	yes	characteristic
161	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
162	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
163	C₄×C₁₃	#435	1	1	yes	characteristic
164	Dic₁₃	—	2	2	no	—
165	Dic₁₃	—	8	1	no	—
166	C₂×#41	—	13	1	no	—
167	#131	—	13	2	no	—
168	#131	—	13	2	no	—
169	#131	—	26	1	no	—
170	#131	—	26	1	no	—
171	C₂×M₁₆	—	13	1	no	—
172	C₂×M₁₆	—	13	1	no	—
173	C₂×M₁₆	—	26	1	no	—
174	C₂²×C₈	—	26	1	no	—
175	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	#117	—	13	1	no	—
178	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	26	4	no	—
183	M₃₂	—	26	1	no	—
184	M₃₂	—	26	1	no	—
185	M₃₂	—	26	1	no	—
186	M₃₂	—	26	1	no	—
187	M₃₂	D₅₂	1	2	yes	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	13	2	no	—
191	C₂×C₁₆	—	26	4	no	—
192	C₂×C₁₆	—	26	1	no	—
193	#104	—	13	2	no	—
194	#104	—	13	2	no	—
195	#103	—	26	1	no	—
196	#102	—	26	1	no	—
197	#97	—	26	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	26	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	#1850	1	1	yes	characteristic
205	D₁₃	—	4	2	no	—
206	#41	—	13	2	no	—
207	#41	—	13	2	no	—
208	#41	—	26	1	no	—
209	#41	—	26	1	no	—
210	C₂×Q₈	—	13	1	no	—
211	C₂×D₄	—	13	1	no	—
212	C₂×D₄	—	26	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	26	1	no	—
215	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
216	M₁₆	—	13	2	no	—
217	M₁₆	—	13	2	no	—
218	M₁₆	—	26	1	no	—
219	M₁₆	—	26	1	no	—
220	M₁₆	—	26	2	no	—
221	M₁₆	—	26	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	26	1	no	—
226	C₂×C₈	—	26	1	no	—
227	C₂×C₈	—	26	2	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
230	C₂×C₈	#6127	1	1	yes	characteristic
231	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
232	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
233	C₂×C₈	#6129	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
235	C₄⋊C₄	—	26	1	no	—
236	C₂²⋊C₄	—	26	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	26	1	no	—
239	C₄²	C₈×D₁₃	1	1	yes	characteristic
240	C₄²	C₈×D₁₃	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	4	1	no	—
243	C₁₆	—	26	2	no	—
244	C₁₆	—	52	1	no	—
245	C₁₃	#12918	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	26	1	no	—
247	C₂³	#88161	1	1	yes	characteristic
248	Q₈	—	13	2	no	—
249	Q₈	—	26	1	no	—
250	D₄	—	13	2	no	—
251	D₄	—	26	2	no	—
252	D₄	—	26	2	no	—
253	D₄	—	26	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	26	1	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	52	1	no	—
261	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
262	C₂×C₄	#88135	1	1	yes	characteristic
263	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
265	C₂×C₄	#88155	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
267	C₈	—	4	1	no	—
268	C₈	—	26	2	no	—
269	C₈	—	26	2	no	—
270	C₈	#88136	1	1	yes	characteristic
271	C₈	#88137	1	1	yes	characteristic
272	C₈	#88150	1	1	yes	characteristic
273	C₈	#88151	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	26	2	no	—
276	C₂²	—	104	1	no	—
277	C₂²	#11718367	1	1	yes	characteristic
278	C₂²	#11718386	1	1	yes	characteristic
279	C₂²	#11718500	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	26	2	no	—
283	C₄	—	104	1	no	—
284	C₄	#11718329	1	1	yes	characteristic
285	C₄	C₈×D₅₂	1	1	yes	characteristic
286	C₄	#11718389	1	1	yes	characteristic
287	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	52	2	no	—
292	C₂	#13115015	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#88136
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1376	C₂
socle	C₂×C₁₃	#1850
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1376	D₁₃
hypocenter	C₁₃	#12918
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
2. #13116601	C₂	semidirect
3. #13116592	C₂	semidirect
6. #13112923	C₂	semidirect
7. C₁₃×#1376	C₂	semidirect
106. #1376	D₁₃	semidirect
245. C₁₃	#12918	semidirect

Maximal subgroups

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

subgroup	index
#13127938	2
#13116601	2
#13116592	2
#13116463	2
#13114673	2
#13112923	2
C₁₃×#1376	2
#12918	13