Subgroups — #15770927

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123400	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718125	C₂	1	2	yes	maximal
3	#13114575	C₂	1	1	yes	maximal, characteristic
4	#13114530	C₂	1	2	yes	maximal
5	C₁₃×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₁₃×#389	—	2	2	no	—
12	C₁₃×#385	C₄	1	4	yes	—
13	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
14	#11718660	—	2	4	no	—
15	#11718585	—	2	2	no	—
16	#11718125	—	2	4	no	—
17	#11718122	—	2	2	no	—
18	#11718116	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×#117	—	2	2	no	—
22	C₁₃×#117	—	2	1	no	—
23	C₁₃×#117	—	2	1	no	—
24	C₁₃×#117	—	2	2	no	—
25	C₁₃×#117	D₄	1	2	yes	—
26	C₁₃×#117	Q₈	1	2	yes	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
29	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	—	2	1	no	—
35	C₂×C₄²×C₁₃	—	2	2	no	—
36	C₂×C₄²×C₁₃	—	2	2	no	—
37	C₂×C₄²×C₁₃	D₄	1	2	yes	—
38	C₂×#6099	—	2	2	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	Q₈	1	4	yes	—
50	C₂²×#1055	—	2	2	no	—
51	C₂²×#1055	D₄	1	4	yes	—
52	C₁₃×#95	—	2	4	no	—
53	C₄⋊#1055	—	2	8	no	—
54	#11223	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	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×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	4	8	no	—
75	C₁₃×C₂²⋊C₄	—	2	2	no	—
76	C₁₃×C₂²⋊C₄	—	2	4	no	—
77	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	4	1	no	—
83	C₄²×C₁₃	—	4	2	no	—
84	C₄²×C₁₃	—	4	1	no	—
85	#6099	—	2	4	no	—
86	#6099	—	2	4	no	—
87	#6099	—	4	2	no	—
88	#6099	—	4	2	no	—
89	#6099	—	4	2	no	—
90	#6099	—	4	2	no	—
91	#6099	—	4	2	no	—
92	#6099	—	4	2	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	4	2	no	—
98	C₂×#1055	—	4	2	no	—
99	#2905	—	13	1	no	—
100	#2296	D₁₃	1	1	yes	characteristic
101	C₂×#344	—	13	2	no	—
102	#1742	—	13	1	no	—
103	#1728	—	13	2	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₃	#117	1	1	yes	characteristic
108	C₂³×C₁₃	C₄×D₄	1	2	yes	—
109	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	8	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#121	1	2	yes	—
121	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#121	1	2	yes	—
124	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#123	1	2	yes	—
130	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#125	1	2	yes	—
133	C₂×C₄×C₁₃	#125	1	2	yes	—
134	C₂×C₄×C₁₃	#126	1	2	yes	—
135	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
137	#1055	—	4	4	no	—
138	#1055	—	4	4	no	—
139	C₂²×M₁₆	—	13	2	no	—
140	C₂²×M₁₆	—	13	2	no	—
141	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
142	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
143	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
144	#424	—	26	4	no	—
145	#423	—	26	2	no	—
146	#389	—	2	2	no	—
147	#385	Dic₁₃	1	4	yes	—
148	C₄×C₄⋊C₄	—	2	2	no	—
149	#344	—	26	4	no	—
150	#343	—	26	2	no	—
151	#336	—	26	4	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	4	no	—
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	—	4	1	no	—
156	C₂²×C₁₃	—	4	2	no	—
157	C₂²×C₁₃	#380	1	1	yes	characteristic
158	C₂²×C₁₃	#386	1	1	yes	characteristic
159	C₂²×C₁₃	#387	1	2	yes	—
160	C₂²×C₁₃	#389	1	2	yes	—
161	C₂²×C₁₃	#391	1	1	yes	characteristic
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	—	4	4	no	—
164	C₄×C₁₃	—	4	4	no	—
165	C₄×C₁₃	#394	1	1	yes	characteristic
166	C₄×C₁₃	#395	1	1	yes	characteristic
167	C₄×C₁₃	#396	1	2	yes	—
168	C₄×C₁₃	#398	1	1	yes	characteristic
169	C₄×C₁₃	#398	1	2	yes	—
170	C₄×C₁₃	#400	1	1	yes	characteristic
171	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
172	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	—	26	2	no	—
182	C₂×M₁₆	—	26	2	no	—
183	C₂×M₁₆	—	26	2	no	—
184	C₂×M₁₆	—	26	2	no	—
185	C₂²×C₈	—	13	4	no	—
186	C₂²×C₈	—	26	2	no	—
187	#117	—	2	1	no	—
188	#117	—	2	1	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	Dic₂₆	1	2	yes	—
192	#117	D₅₂	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
197	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
204	C₄⋊C₈	—	26	8	no	—
205	#95	—	2	4	no	—
206	C₂×C₁₃	—	2	1	no	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	—	2	2	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	#1441	1	1	yes	characteristic
211	C₂×C₁₃	#1891	1	2	yes	—
212	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₂²⋊Dic₁₃	1	4	yes	—
218	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
220	C₂²×C₄	#6127	1	1	yes	characteristic
221	C₂²×C₄	#6127	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₁₃	1	2	yes	—
223	C₂²×C₄	#6129	1	2	yes	—
224	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
225	C₂²×C₄	#6131	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	M₁₆	—	26	4	no	—
229	M₁₆	—	26	4	no	—
230	M₁₆	—	52	2	no	—
231	M₁₆	—	52	2	no	—
232	M₁₆	—	52	2	no	—
233	M₁₆	—	52	2	no	—
234	M₁₆	—	52	2	no	—
235	M₁₆	—	52	2	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	—	26	4	no	—
240	C₂×C₈	—	52	2	no	—
241	C₂×C₈	—	52	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	8	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	2	no	—
256	C₁₃	#11223	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
259	C₂³	#88180	1	1	yes	characteristic
260	C₂³	D₄×Dic₁₃	1	2	yes	—
261	C₂³	Q₈×Dic₁₃	1	2	yes	—
262	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	#88142	1	2	yes	—
273	C₂×C₄	#88143	1	2	yes	—
274	C₂×C₄	#88144	1	2	yes	—
275	C₂×C₄	#88148	1	2	yes	—
276	C₂×C₄	#88149	1	2	yes	—
277	C₂×C₄	#88150	1	2	yes	—
278	C₂×C₄	#88151	1	2	yes	—
279	C₂×C₄	#88152	1	2	yes	—
280	C₂×C₄	#88178	1	1	yes	characteristic
281	C₂×C₄	#88178	1	1	yes	characteristic
282	C₂×C₄	#88184	1	1	yes	characteristic
283	C₂×C₄	#88184	1	1	yes	characteristic
284	C₂×C₄	#88189	1	1	yes	characteristic
285	C₂×C₄	#88191	1	1	yes	characteristic
286	C₂×C₄	#88193	1	2	yes	—
287	C₂×C₄	#88198	1	1	yes	characteristic
288	C₂×C₄	#88200	1	2	yes	—
289	C₂×C₄	#88201	1	1	yes	characteristic
290	C₈	—	52	4	no	—
291	C₈	—	52	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	2	no	—
297	C₂²	#11718642	1	2	yes	—
298	C₂²	#11718646	1	2	yes	—
299	C₂²	#11718873	1	1	yes	characteristic
300	C₂²	#11718881	1	1	yes	characteristic
301	C₂²	#11718891	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	#11718640	1	1	yes	characteristic
306	C₄	#11718644	1	1	yes	characteristic
307	C₄	#11718645	1	2	yes	—
308	C₄	#11718647	1	1	yes	characteristic
309	C₄	#11718652	1	1	yes	characteristic
310	C₄	#11718655	1	2	yes	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	2	no	—
315	C₂	#13113089	1	1	yes	characteristic
316	C₂	#13115255	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88191
commutator, prime radical	C₂⁴×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#2296	C₂
socle	C₂²×C₁₃	#386
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#2296	D₁₃
hypocenter	C₁₃	#11223
LCS	G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2296

Complements

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

N	H ≅ G/N	type
256. C₁₃	#11223	semidirect

Maximal subgroups

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

subgroup	index
#13123400	2
C₂×#11718125	2
#13114575	2
#13114530	2
C₁₃×#2296	2
#11223	13