Subgroups — #13124964

Order 1664.

380 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719283	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718847	C₂	1	1	yes	maximal, characteristic
5	#11718843	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718691	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718691	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718523	C₂	1	1	yes	maximal, characteristic
9	#11718515	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718469	C₂	1	1	yes	maximal, characteristic
11	#11718466	C₂	1	1	yes	maximal, characteristic
12	#11718454	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718452	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718377	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718366	C₂	1	1	yes	maximal, characteristic
16	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
20	#88200	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	#88198	C₂²	1	1	yes	characteristic
23	C₂×#6107	C₂²	1	1	yes	characteristic
24	C₂×#6107	C₂²	1	1	yes	characteristic
25	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
26	#88151	C₂²	1	1	yes	characteristic
27	#88150	—	2	1	no	—
28	C₄⋊C₄×D₁₃	—	2	1	no	—
29	C₄⋊C₄×D₁₃	C₂²	1	1	yes	characteristic
30	#88143	—	2	1	no	—
31	#88142	C₂²	1	1	yes	characteristic
32	#88141	—	2	1	no	—
33	#88141	C₂²	1	1	yes	characteristic
34	#88126	—	2	1	no	—
35	C₄²×D₁₃	—	2	1	no	—
36	C₄²×D₁₃	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₆	—	2	1	no	—
38	C₄×Dic₂₆	—	2	1	no	—
39	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
40	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
41	C₁₃×#106	C₂²	1	1	yes	characteristic
42	C₁₃×#103	C₂²	1	1	yes	characteristic
43	C₁₃×#103	C₂²	1	1	yes	characteristic
44	C₁₃×#97	C₂²	1	1	yes	characteristic
45	#88075	C₂²	1	1	yes	characteristic
46	#88075	C₂²	1	1	yes	characteristic
47	#88060	C₂²	1	1	yes	characteristic
48	#88060	C₂²	1	1	yes	characteristic
49	#88057	C₂²	1	1	yes	characteristic
50	#88056	C₂²	1	1	yes	characteristic
51	#88056	C₂²	1	1	yes	characteristic
52	#88054	C₂²	1	1	yes	characteristic
53	#88054	C₂²	1	1	yes	characteristic
54	#88050	C₂²	1	1	yes	characteristic
55	#88050	C₂²	1	1	yes	characteristic
56	#88048	C₂²	1	1	yes	characteristic
57	#88048	C₂²	1	1	yes	characteristic
58	#88043	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
61	Q₈×D₁₃	—	2	2	no	—
62	Q₈×D₁₃	—	2	2	no	—
63	C₂×C₄×D₁₃	—	2	1	no	—
64	C₂×C₄×D₁₃	—	2	1	no	—
65	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₂₆	—	2	1	no	—
69	C₂×Dic₂₆	—	2	1	no	—
70	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
71	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
72	C₁₃×Q₁₆	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	1	no	—
76	C₁₃×C₄⋊C₄	—	2	1	no	—
77	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	C₂³	1	1	yes	characteristic
82	#6107	—	2	2	no	—
83	#6107	—	2	2	no	—
84	#6103	—	2	1	no	—
85	#6103	—	2	1	no	—
86	C₄⋊Dic₁₃	—	2	1	no	—
87	C₄⋊Dic₁₃	—	2	1	no	—
88	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
90	#6101	—	2	1	no	—
91	#6101	—	2	1	no	—
92	#6101	—	2	1	no	—
93	#6101	—	2	1	no	—
94	#6101	—	2	1	no	—
95	#6101	—	2	1	no	—
96	#6101	—	2	1	no	—
97	#6101	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	—	2	1	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	C₄×Dic₁₃	D₄	1	1	yes	characteristic
103	C₄×Dic₁₃	D₄	1	1	yes	characteristic
104	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
105	C₂×#1055	C₂³	1	1	yes	characteristic
106	C₂×#1055	C₂³	1	1	yes	characteristic
107	Dic₅₂	—	2	2	no	—
108	#3113	—	13	1	no	maximal, Sylow
109	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₃	—	2	1	no	—
111	Q₈×C₁₃	—	2	1	no	—
112	Q₈×C₁₃	—	2	1	no	—
113	Q₈×C₁₃	#41	1	2	yes	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	2	1	no	—
124	C₂×Dic₁₃	—	2	1	no	—
125	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₃	—	2	4	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	2	1	no	—
131	C₄×D₁₃	—	2	1	no	—
132	C₄×D₁₃	—	2	2	no	—
133	C₄×D₁₃	—	4	2	no	—
134	Dic₂₆	—	2	4	no	—
135	Dic₂₆	—	2	1	no	—
136	Dic₂₆	—	2	1	no	—
137	Dic₂₆	—	2	1	no	—
138	Dic₂₆	—	4	2	no	—
139	Dic₂₆	#41	1	2	yes	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	2	1	no	—
142	#1055	—	2	1	no	—
143	#1055	—	2	1	no	—
144	#533	—	13	1	no	—
145	C₂×C₄×Q₈	—	13	1	no	—
146	#483	—	13	1	no	—
147	#483	—	13	1	no	—
148	#475	—	13	1	no	—
149	#475	—	13	1	no	—
150	#462	—	13	1	no	—
151	#462	—	13	1	no	—
152	#451	—	13	1	no	—
153	#451	—	13	1	no	—
154	#441	—	13	1	no	—
155	#441	—	13	1	no	—
156	#441	—	13	1	no	—
157	#441	D₁₃	1	1	yes	characteristic, complemented
158	#432	—	13	1	no	—
159	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₃	—	2	1	no	—
161	C₂×D₁₃	—	2	1	no	—
162	C₂×D₁₃	#137	1	2	yes	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	4	1	no	—
167	C₄×C₁₃	—	4	1	no	—
168	C₄×C₁₃	—	4	1	no	—
169	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
171	Dic₁₃	—	2	1	no	—
172	Dic₁₃	—	2	2	no	—
173	Dic₁₃	—	2	1	no	—
174	Dic₁₃	—	4	1	no	—
175	Dic₁₃	—	4	1	no	—
176	Dic₁₃	—	4	1	no	—
177	Dic₁₃	#137	1	2	yes	—
178	C₂²×Q₈	—	13	1	no	—
179	C₂×Q₁₆	—	13	1	no	—
180	C₂×Q₁₆	—	13	1	no	—
181	C₂×Q₁₆	—	13	1	no	—
182	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	13	1	no	—
184	C₄⋊Q₈	—	13	1	no	—
185	#125	—	26	1	no	—
186	C₂²⋊Q₈	—	13	1	no	—
187	C₂²⋊Q₈	—	13	1	no	—
188	C₂²⋊Q₈	—	26	1	no	—
189	C₄×Q₈	—	13	1	no	—
190	C₄×Q₈	—	13	1	no	—
191	C₄×Q₈	—	13	1	no	—
192	C₄×Q₈	—	13	1	no	—
193	C₄×Q₈	—	26	1	no	—
194	C₄×Q₈	—	26	1	no	—
195	C₄×Q₈	—	26	1	no	—
196	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
197	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
198	#117	—	26	1	no	—
199	C₂×C₄⋊C₄	—	13	1	no	—
200	C₂×C₄⋊C₄	—	26	1	no	—
201	C₂×C₄²	—	13	1	no	—
202	C₂×C₄²	—	26	1	no	—
203	#106	—	13	1	no	—
204	#106	—	13	1	no	—
205	#106	—	13	1	no	—
206	#106	C₂×D₁₃	1	1	yes	characteristic
207	C₄⋊C₈	—	13	1	no	—
208	C₄⋊C₈	—	13	1	no	—
209	#103	—	13	1	no	—
210	#103	—	13	1	no	—
211	#103	—	13	1	no	—
212	#103	—	13	1	no	—
213	#103	—	13	1	no	—
214	#103	—	13	1	no	—
215	#103	C₂×D₁₃	1	1	yes	characteristic
216	#103	C₂×D₁₃	1	1	yes	characteristic
217	C₂²⋊C₈	—	13	1	no	—
218	C₂²⋊C₈	—	13	1	no	—
219	#97	—	13	1	no	—
220	#97	C₂×D₁₃	1	1	yes	characteristic
221	C₂×C₁₃	#525	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
224	D₁₃	—	2	2	no	—
225	C₂×Q₈	—	13	1	no	—
226	C₂×Q₈	—	13	1	no	—
227	C₂×Q₈	—	26	1	no	—
228	C₂×Q₈	—	26	1	no	—
229	C₂×Q₈	—	26	2	no	—
230	C₂×Q₈	—	26	2	no	—
231	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	26	1	no	—
237	C₂²×C₄	—	26	1	no	—
238	Q₁₆	—	2	2	no	—
239	Q₁₆	—	26	2	no	—
240	Q₁₆	—	26	2	no	—
241	Q₁₆	—	26	2	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	13	1	no	—
249	C₄⋊C₄	—	13	1	no	—
250	C₄⋊C₄	—	26	1	no	—
251	C₄⋊C₄	—	26	1	no	—
252	C₄⋊C₄	—	26	1	no	—
253	C₄⋊C₄	—	26	1	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	—	26	1	no	—
256	C₄⋊C₄	—	26	1	no	—
257	C₄⋊C₄	—	26	1	no	—
258	C₄⋊C₄	—	26	1	no	—
259	C₄⋊C₄	—	26	1	no	—
260	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
262	C₂²⋊C₄	—	26	1	no	—
263	C₂²⋊C₄	—	26	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	13	1	no	—
267	C₄²	—	13	1	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	26	1	no	—
270	C₄²	—	26	1	no	—
271	C₄²	—	26	1	no	—
272	C₄²	—	26	1	no	—
273	C₄²	C₂²×D₁₃	1	1	yes	characteristic
274	C₁₃	#3113	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	13	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	13	2	no	—
280	Q₈	—	26	4	no	—
281	Q₈	—	26	1	no	—
282	Q₈	—	26	1	no	—
283	Q₈	—	26	1	no	—
284	Q₈	—	52	2	no	—
285	Q₈	#6127	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	1	no	—
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	26	4	no	—
296	C₂×C₄	—	26	1	no	—
297	C₂×C₄	—	26	1	no	—
298	C₂×C₄	—	26	1	no	—
299	C₂×C₄	—	26	1	no	—
300	C₂×C₄	—	26	1	no	—
301	C₂×C₄	—	26	1	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	26	1	no	—
309	C₈	—	26	1	no	—
310	C₂²	—	13	2	no	—
311	C₂²	—	26	1	no	—
312	C₂²	—	26	1	no	—
313	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	13	2	no	—
321	C₄	—	26	1	no	—
322	C₄	—	26	1	no	—
323	C₄	—	26	2	no	—
324	C₄	—	52	1	no	—
325	C₄	—	52	1	no	—
326	C₄	—	52	1	no	—
327	C₄	C₂×#6127	1	1	yes	characteristic
328	C₄	#41×D₁₃	1	1	yes	characteristic
329	C₂	—	26	2	no	—
330	C₂	#11719252	1	1	yes	characteristic
331	C₂	D₁₃×#137	1	1	yes	characteristic
332	C₂	D₁₃×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#441	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂²×D₁₃	C₂×D₄
hypercenter	#441	D₁₃
hypocenter	C₁₃	#3113
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
3. C₁₃×#441	C₂	semidirect
5. #11718843	C₂	semidirect
6. #11718691	C₂	semidirect
7. #11718691	C₂	semidirect
9. #11718515	C₂	semidirect
12. #11718454	C₂	semidirect
13. #11718452	C₂	semidirect
14. #11718377	C₂	semidirect
157. #441	D₁₃	semidirect
274. C₁₃	#3113	semidirect

Maximal subgroups

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

subgroup	index
#11719283	2
C₄×Q₈×D₁₃	2
C₁₃×#441	2
#11718847	2
#11718843	2
#11718691	2
#11718523	2
#11718515	2
#11718469	2
#11718466	2
#11718454	2
#11718452	2
#11718377	2
#11718366	2
#3113	13