Subgroups — #15863541

Order 3328.

718 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	#13123473	C₂	1	2	yes	maximal
2	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
3	#13115008	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114976	C₂	1	2	yes	maximal, complemented
5	#13114875	C₂	1	2	yes	maximal
6	#13114871	C₂	1	2	yes	maximal
7	#13114744	C₂	1	2	yes	maximal, complemented
8	#108×Dic₁₃	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#6094	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88126	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	2	yes	—
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₁₃×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718787	C₂²	1	4	yes	—
18	M₁₆×Dic₁₃	C₂²	1	4	yes	—
19	C₂×#88059	C₂²	1	2	yes	—
20	C₂×#88059	C₂²	1	2	yes	—
21	C₂×#88055	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
23	#11718585	C₂²	1	2	yes	—
24	C₂×#88043	C₂²	1	1	yes	characteristic
25	#11718501	—	2	8	no	—
26	#11718350	C₄	1	8	yes	—
27	#11718150	C₂²	1	4	yes	—
28	#11718150	C₂²	1	4	yes	—
29	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6103	Q₈	1	2	yes	—
35	C₂×#6101	Q₈	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
37	C₂×#6099	C₂³	1	4	yes	—
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	C₂²×#1055	C₂³	1	1	yes	characteristic
40	C₂×#6094	—	2	2	no	—
41	C₂×#6094	—	2	2	no	—
42	C₂×#6094	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₁₃	—	2	2	no	—
44	C₂×C₈×D₁₃	—	2	2	no	—
45	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
46	#88126	C₂×C₄	1	4	yes	—
47	#88126	D₄	1	4	yes	—
48	C₁₃×#108	—	2	4	no	—
49	C₁₃×#108	—	2	4	no	—
50	C₁₃×C₄⋊C₈	—	2	4	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
52	#88060	—	2	8	no	—
53	#88059	—	2	4	no	—
54	#88059	—	2	4	no	—
55	#88055	—	2	4	no	—
56	#88055	C₂×C₄	1	4	yes	—
57	C₈×Dic₁₃	—	2	4	no	—
58	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
59	C₄⋊#1055	—	2	4	no	—
60	#88043	C₂×C₄	1	4	yes	—
61	#20828	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
67	C₂×C₄×D₁₃	—	2	1	no	—
68	C₂×C₄×D₁₃	—	2	1	no	—
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₂×C₈×C₁₃	#41	1	2	yes	—
80	C₂×C₈×C₁₃	#41	1	2	yes	—
81	C₂×C₈×C₁₃	#41	1	2	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃	C₂×D₄	1	2	yes	—
84	#6103	C₄⋊C₄	1	8	yes	—
85	#6101	C₄⋊C₄	1	8	yes	—
86	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
88	#6099	—	2	4	no	—
89	#6099	—	2	4	no	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	—	2	1	no	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	—	2	1	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	C₂²×C₄	1	2	yes	—
96	C₂×#1055	C₂²×C₄	1	2	yes	—
97	#6094	—	2	8	no	—
98	C₈×D₁₃	—	2	8	no	—
99	#2921	—	13	2	no	—
100	C₂×#405	—	13	1	no	—
101	#1848	—	13	1	no	—
102	#1848	D₁₃	1	1	yes	characteristic, complemented
103	#1838	—	13	2	no	—
104	#1811	—	13	2	no	—
105	#1810	—	13	2	no	—
106	#1778	—	13	2	no	—
107	C₄×#108	—	13	2	no	—
108	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×D₁₃	—	2	1	no	—
110	C₂²×D₁₃	—	2	1	no	—
111	C₂²×D₁₃	—	4	2	no	—
112	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₄×D₁₃	—	2	2	no	—
127	C₄×D₁₃	—	2	2	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₈×C₁₃	—	4	4	no	—
131	C₈×C₁₃	#117	1	4	yes	—
132	C₈×C₁₃	#117	1	4	yes	—
133	#1055	—	2	2	no	—
134	#1055	—	2	2	no	—
135	#1055	—	4	4	no	—
136	C₂²×M₁₆	—	13	1	no	—
137	C₂³×C₈	—	13	1	no	—
138	C₂×#117	—	13	1	no	—
139	#433	—	26	8	no	—
140	C₂×#108	—	13	2	no	—
141	C₂×#108	—	13	2	no	—
142	C₂×#108	C₂×D₁₃	1	2	yes	—
143	C₂×#108	C₂×D₁₃	1	2	yes	—
144	#423	—	13	2	no	—
145	#423	C₂×D₁₃	1	2	yes	—
146	#407	—	13	4	no	—
147	#405	—	13	8	no	—
148	C₄×M₁₆	—	13	4	no	—
149	C₂×#97	—	13	1	no	—
150	C₂×#97	—	13	1	no	—
151	C₂×C₄×C₈	—	13	1	no	—
152	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	#341	—	13	4	no	—
154	#341	—	13	4	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₁₃	#533	1	1	yes	characteristic
158	C₂²×C₁₃	#533	1	1	yes	characteristic
159	C₂×D₁₃	—	2	4	no	—
160	C₂×D₁₃	—	2	2	no	—
161	C₂×D₁₃	—	4	2	no	—
162	C₂×D₁₃	—	4	2	no	—
163	C₄×C₁₃	—	2	4	no	—
164	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
165	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
166	C₄×C₁₃	#532	1	1	yes	characteristic
167	C₄×C₁₃	#532	1	1	yes	characteristic
168	Dic₁₃	—	2	4	no	—
169	Dic₁₃	—	2	2	no	—
170	C₂³×C₄	—	13	1	no	—
171	C₂×M₁₆	—	13	4	no	—
172	C₂×M₁₆	—	13	4	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	4	no	—
179	C₂²×C₈	—	26	2	no	—
180	C₂²×C₈	—	26	2	no	—
181	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	#117	—	13	4	no	—
184	#117	—	13	4	no	—
185	C₂×C₄⋊C₄	—	13	2	no	—
186	C₂×C₂²⋊C₄	—	13	2	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
189	#108	—	2	4	no	—
190	#108	—	2	4	no	—
191	#108	—	26	4	no	—
192	#108	—	26	4	no	—
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	26	4	no	—
195	C₂²⋊C₈	—	26	8	no	—
196	#97	—	13	4	no	—
197	#97	—	13	4	no	—
198	#97	—	26	4	no	—
199	C₄×C₈	—	13	4	no	—
200	C₄×C₈	—	26	4	no	—
201	C₄×C₈	C₄×D₁₃	1	4	yes	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	#2285	1	1	yes	characteristic
205	C₂×C₁₃	#2916	1	2	yes	—
206	D₁₃	—	4	2	no	—
207	C₂⁴	—	13	1	no	—
208	C₂²×C₄	—	13	1	no	—
209	C₂²×C₄	—	13	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	26	1	no	—
213	C₂²×C₄	—	26	1	no	—
214	C₂²×C₄	—	26	1	no	—
215	C₂²×C₄	—	26	1	no	—
216	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
217	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	26	8	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	—	26	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	26	1	no	—
227	C₂×C₈	—	26	1	no	—
228	C₂×C₈	—	26	1	no	—
229	C₂×C₈	—	26	1	no	—
230	C₂×C₈	—	26	8	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
234	C₂×C₈	#6129	1	2	yes	—
235	C₂×C₈	#6129	1	2	yes	—
236	C₂×C₈	#6131	1	2	yes	—
237	C₂×C₈	#6131	1	2	yes	—
238	C₄⋊C₄	—	13	8	no	—
239	C₂²⋊C₄	—	13	8	no	—
240	C₄²	—	13	2	no	—
241	C₄²	—	13	2	no	—
242	C₄²	C₂×C₄×D₁₃	1	2	yes	—
243	C₄²	D₄×D₁₃	1	2	yes	—
244	C₁₃	#20828	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	13	2	no	—
246	C₂³	—	26	1	no	—
247	C₂³	—	26	1	no	—
248	C₂³	—	52	2	no	—
249	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	13	4	no	—
252	C₂×C₄	—	13	2	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	2	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	52	2	no	—
260	C₂×C₄	C₄⋊C₄×D₁₃	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#6129	1	1	yes	characteristic
264	C₂×C₄	C₂×#6129	1	1	yes	characteristic
265	C₂×C₄	C₂×#6131	1	1	yes	characteristic
266	C₂×C₄	C₂×#6131	1	1	yes	characteristic
267	C₈	—	4	4	no	—
268	C₈	—	26	2	no	—
269	C₈	—	26	2	no	—
270	C₈	—	52	4	no	—
271	C₈	#88146	1	4	yes	—
272	C₈	#88146	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	26	2	no	—
275	C₂²	—	26	4	no	—
276	C₂²	—	52	2	no	—
277	C₂²	—	52	2	no	—
278	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
279	C₂²	#11719433	1	1	yes	characteristic
280	C₂²	#11719433	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	26	2	no	—
283	C₄	—	26	4	no	—
284	C₄	C₂×#88146	1	1	yes	characteristic
285	C₄	C₂×#88146	1	1	yes	characteristic
286	C₄	#11719421	1	1	yes	characteristic
287	C₄	#11719430	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	52	2	no	—
291	C₂	#13117241	1	1	yes	characteristic
292	C₂	#13123444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#20828
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
3. #13115008	C₂	semidirect
4. #13114976	C₂	semidirect
7. #13114744	C₂	semidirect
8. #108×Dic₁₃	C₂	semidirect
9. C₁₃×#1848	C₂	semidirect
102. #1848	D₁₃	semidirect
244. C₁₃	#20828	semidirect

Maximal subgroups

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

subgroup	index
#13123473	2
C₂×#11718350	2
#13115008	2
#13114976	2
#13114875	2
#13114871	2
#13114744	2
#108×Dic₁₃	2
C₁₃×#1848	2
#20828	13