Subgroups — #15768112

Order 3328.

926 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718658	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718124	C₂	1	2	yes	maximal
5	C₁₃×#1735	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	—	2	4	no	—
12	C₂×C₈×Dic₁₃	C₄	1	4	yes	complemented
13	#11718658	—	2	4	no	—
14	#11718658	C₄	1	8	yes	complemented
15	C₂×#88043	—	2	2	no	—
16	C₂×C₄×#1055	—	2	2	no	—
17	C₁₃×#340	C₄	1	4	yes	complemented
18	#11718144	C₄	1	4	yes	complemented
19	#11718124	C₄	1	8	yes	complemented
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₁₃×#117	—	2	2	no	—
28	C₁₃×#117	—	2	2	no	—
29	C₁₃×#117	C₂×C₄	1	4	yes	—
30	C₁₃×#117	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₄×Dic₁₃	—	2	2	no	—
35	C₂×C₄×Dic₁₃	—	2	1	no	—
36	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
38	C₂×#6099	—	2	2	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	C₂×C₄	1	4	yes	—
42	C₂×#6099	C₂×C₄	1	4	yes	—
43	C₂×#6099	C₂×C₄	1	4	yes	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	C₂×C₄	1	4	yes	—
46	C₂²×#1055	C₂×C₄	1	2	yes	—
47	C₂²×#1055	D₄	1	4	yes	—
48	C₂²×#1055	D₄	1	1	yes	characteristic
49	C₂²×#1055	D₄	1	1	yes	characteristic
50	C₂²×#1055	Q₈	1	2	yes	—
51	C₈×Dic₁₃	—	2	8	no	—
52	#88043	—	2	8	no	—
53	C₄×#1055	—	2	8	no	—
54	#10706	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	#41	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	2	8	no	—
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₂²×Dic₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	C₄²	1	4	yes	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
77	C₁₃×C₂²⋊C₄	—	2	4	no	—
78	C₁₃×C₂²⋊C₄	—	2	4	no	—
79	C₄²×C₁₃	—	2	4	no	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄×Dic₁₃	—	2	2	no	—
82	C₄×Dic₁₃	—	2	4	no	—
83	C₄×Dic₁₃	C₄²	1	4	yes	—
84	#6099	—	2	4	no	—
85	#6099	—	2	4	no	—
86	#6099	—	2	4	no	—
87	#6099	C₄²	1	8	yes	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	C₄²	1	4	yes	—
92	C₂×#1055	C₂²⋊C₄	1	4	yes	—
93	C₂×#1055	C₂²⋊C₄	1	4	yes	—
94	C₂×#1055	C₂²⋊C₄	1	4	yes	—
95	C₂×#1055	C₄⋊C₄	1	8	yes	—
96	C₂×#1055	C₄⋊C₄	1	2	yes	—
97	C₂×#1055	C₄⋊C₄	1	2	yes	—
98	C₂×#405	—	13	2	no	—
99	C₂²×C₄×C₈	—	13	1	no	—
100	C₂×#341	—	13	1	no	—
101	#1735	D₁₃	1	1	yes	characteristic
102	C₂×#339	—	13	2	no	—
103	C₂³×C₁₃	—	2	4	no	—
104	C₂³×C₁₃	#117	1	2	yes	—
105	C₂³×C₁₃	#117	1	1	yes	characteristic
106	C₂³×C₁₃	C₄×D₄	1	2	yes	—
107	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
108	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	4	8	no	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
129	C₂×Dic₁₃	—	2	16	no	—
130	C₂×Dic₁₃	—	2	8	no	—
131	C₂×Dic₁₃	—	2	4	no	—
132	C₈×C₁₃	—	2	4	no	—
133	#1055	—	2	8	no	—
134	#1055	#95	1	4	yes	—
135	#1055	#95	1	4	yes	—
136	C₂²×M₁₆	—	13	2	no	—
137	C₂³×C₈	—	13	1	no	—
138	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₂×#117	C₂×D₁₃	1	2	yes	—
140	C₂²×C₄²	—	13	1	no	—
141	#405	—	13	8	no	—
142	#405	—	26	4	no	—
143	C₂×#97	—	26	2	no	—
144	C₂×C₄×C₈	—	13	4	no	—
145	C₂×C₄×C₈	—	26	2	no	—
146	C₂×C₄×C₈	—	26	4	no	—
147	#341	—	13	4	no	—
148	#340	Dic₁₃	1	4	yes	complemented
149	#339	—	13	8	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	#376	1	1	yes	characteristic
155	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
157	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
158	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₃	#443	1	4	yes	—
160	C₄×C₁₃	—	4	8	no	—
161	C₄×C₁₃	C₄³	1	1	yes	characteristic
162	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
163	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
164	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
167	Dic₁₃	—	2	8	no	—
168	C₂³×C₄	—	26	1	no	—
169	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×M₁₆	—	13	4	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	—	26	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	—	13	4	no	—
180	C₂²×C₈	—	13	4	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	26	2	no	—
185	C₂²×C₈	C₄×D₁₃	1	2	yes	—
186	C₂²×C₈	C₄×D₁₃	1	2	yes	—
187	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	C₄×D₁₃	1	4	yes	—
191	#117	C₄×D₁₃	1	4	yes	—
192	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	13	4	no	—
196	C₂×C₄²	—	13	2	no	—
197	C₂×C₄²	—	26	2	no	—
198	C₂×C₄²	—	26	1	no	—
199	#97	—	26	8	no	—
200	C₄×C₈	—	26	8	no	—
201	C₄×C₈	—	26	8	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	C₄×#95	1	1	yes	characteristic
206	C₂×C₁₃	#1764	1	4	yes	—
207	C₂×C₁₃	#1772	1	2	yes	—
208	C₂⁴	#6129	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	26	8	no	—
211	C₂²×C₄	—	26	4	no	—
212	C₂²×C₄	—	26	2	no	—
213	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
217	C₂²×C₄	D₄×D₁₃	1	2	yes	—
218	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	#6129	1	2	yes	—
220	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
221	C₂²×C₄	#6131	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
223	M₁₆	—	13	8	no	—
224	M₁₆	—	26	4	no	—
225	M₁₆	—	26	4	no	—
226	M₁₆	—	26	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	13	4	no	—
232	C₂×C₈	—	13	4	no	—
233	C₂×C₈	—	13	4	no	—
234	C₂×C₈	—	13	4	no	—
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	8	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	—	26	4	no	—
240	C₂×C₈	—	26	4	no	—
241	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄×Dic₁₃	1	8	yes	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	13	4	no	—
249	C₄²	—	26	2	no	—
250	C₄²	—	26	4	no	—
251	C₁₃	#10706	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	2	4	no	—
253	C₂³	#88131	1	2	yes	—
254	C₂³	#88135	1	2	yes	—
255	C₂³	#88141	1	1	yes	characteristic
256	C₂³	#88146	1	1	yes	characteristic
257	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	26	16	no	—
261	C₂×C₄	—	26	8	no	—
262	C₂×C₄	—	26	4	no	—
263	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₄²×D₁₃	1	2	yes	—
266	C₂×C₄	#88131	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₁₃	1	2	yes	—
268	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
269	C₂×C₄	#88135	1	2	yes	—
270	C₂×C₄	#88135	1	2	yes	—
271	C₂×C₄	#88141	1	2	yes	—
272	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
274	C₂×C₄	#88146	1	2	yes	—
275	C₂×C₄	#88147	1	1	yes	characteristic
276	C₂×C₄	#88147	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
279	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
280	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	13	4	no	—
283	C₈	—	13	4	no	—
284	C₈	—	26	8	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#11718309	1	1	yes	characteristic
290	C₂²	#11718310	1	2	yes	—
291	C₂²	#11718327	1	1	yes	characteristic
292	C₂²	C₂²⋊C₄×Dic₁₃	1	2	yes	—
293	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
294	C₂²	#11718857	1	4	yes	—
295	C₄	—	4	8	no	—
296	C₄	—	26	8	no	—
297	C₄	#11718310	1	1	yes	characteristic
298	C₄	D₁₃×#95	1	1	yes	characteristic
299	C₄	#11718327	1	2	yes	—
300	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
301	C₄	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
302	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	#95×Dic₁₃	1	1	yes	characteristic
307	C₂	#13114676	1	4	yes	—
308	C₂	#13114717	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₄³
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#1735	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#1735	D₁₃
hypocenter	C₁₃	#10706
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
12. C₂×C₈×Dic₁₃	C₄	semidirect
14. #11718658	C₄	semidirect
17. C₁₃×#340	C₄	semidirect
18. #11718144	C₄	semidirect
19. #11718124	C₄	semidirect
148. #340	Dic₁₃	semidirect
251. C₁₃	#10706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718658	2
C₂²×C₈×Dic₁₃	2
C₂×#11718144	2
C₂×#11718124	2
C₁₃×#1735	2
#10706	13