Subgroups — #13123928

Order 1664.

322 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719222	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#433	C₂	1	1	yes	maximal, characteristic
3	#11718661	C₂	1	1	yes	maximal, characteristic
4	#11718492	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718491	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718489	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718489	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718381	C₂	1	1	yes	maximal, characteristic
9	#11718381	C₂	1	1	yes	maximal, characteristic
10	#11718380	C₂	1	1	yes	maximal, characteristic
11	#11718379	C₂	1	1	yes	maximal, characteristic
12	#11718362	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718362	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718340	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₃×#117	C₂²	1	1	yes	characteristic
17	#88180	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
19	#88144	C₄	1	1	yes	characteristic
20	#88144	C₄	1	1	yes	characteristic
21	#88142	C₄	1	1	yes	characteristic
22	#88142	C₄	1	1	yes	characteristic
23	#88132	C₄	1	1	yes	characteristic
24	#88132	C₄	1	1	yes	characteristic
25	#88132	C₄	1	1	yes	characteristic
26	#88132	C₄	1	1	yes	characteristic
27	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×#97	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
37	C₂²⋊#1055	C₂²	1	1	yes	characteristic
38	C₂²⋊#1055	C₂²	1	1	yes	characteristic
39	#88055	C₂²	1	1	yes	characteristic
40	#88055	C₂²	1	1	yes	characteristic
41	#88055	C₂²	1	1	yes	characteristic
42	#88055	C₂²	1	1	yes	characteristic
43	#88054	C₂²	1	1	yes	characteristic
44	#88054	C₂²	1	1	yes	characteristic
45	#88054	C₂²	1	1	yes	characteristic
46	#88054	C₂²	1	1	yes	characteristic
47	#88054	C₂²	1	1	yes	characteristic
48	#88054	C₂²	1	1	yes	characteristic
49	#88054	C₂²	1	1	yes	characteristic
50	#88054	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
55	C₄⋊#1055	C₂²	1	1	yes	characteristic
56	C₄⋊#1055	C₂²	1	1	yes	characteristic
57	#88043	C₂²	1	1	yes	characteristic
58	C₄×#1055	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
63	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
72	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂³	1	1	yes	characteristic
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
84	#6101	C₂×C₄	1	1	yes	characteristic
85	#6101	C₂×C₄	1	1	yes	characteristic
86	#6101	C₂×C₄	1	1	yes	characteristic
87	#6101	C₂×C₄	1	1	yes	characteristic
88	#6101	C₂×C₄	1	1	yes	characteristic
89	#6101	C₂×C₄	1	1	yes	characteristic
90	#6101	C₂×C₄	1	1	yes	characteristic
91	#6101	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
98	C₂×#1055	C₂³	1	1	yes	characteristic
99	C₂×#1055	C₂³	1	1	yes	characteristic
100	C₂×#1055	C₂³	1	1	yes	characteristic
101	C₂×#1055	C₂³	1	1	yes	characteristic
102	#2985	—	13	1	no	maximal, Sylow
103	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	2	no	—
113	C₂×Dic₁₃	—	4	1	no	—
114	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
122	Dic₂₆	—	2	2	no	—
123	Dic₂₆	—	2	2	no	—
124	Dic₂₆	—	4	1	no	—
125	Dic₂₆	—	4	1	no	—
126	C₈×C₁₃	—	2	1	no	—
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	#1055	—	2	1	no	—
131	#1055	—	2	1	no	—
132	#1055	—	2	1	no	—
133	#1055	—	2	1	no	—
134	#533	—	13	1	no	—
135	#446	—	13	1	no	—
136	#446	—	13	1	no	—
137	#446	—	13	1	no	—
138	#446	—	13	1	no	—
139	#446	—	13	1	no	—
140	#446	—	13	1	no	—
141	C₈×Q₈	—	13	1	no	—
142	C₈×Q₈	—	13	1	no	—
143	#433	—	13	1	no	—
144	#433	—	13	1	no	—
145	#433	—	13	1	no	—
146	#433	D₁₃	1	1	yes	characteristic
147	#432	—	13	1	no	—
148	#431	—	13	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
159	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₁₃	—	2	1	no	—
161	Dic₁₃	—	2	1	no	—
162	Dic₁₃	—	2	1	no	—
163	Dic₁₃	—	2	1	no	—
164	Dic₁₃	—	2	1	no	—
165	Dic₁₃	—	2	1	no	—
166	Dic₁₃	—	2	1	no	—
167	Dic₁₃	#131	1	2	yes	—
168	C₄⋊Q₈	—	13	1	no	—
169	C₄⋊Q₈	—	13	1	no	—
170	#125	—	13	1	no	—
171	#125	—	13	1	no	—
172	C₂²⋊Q₈	—	13	1	no	—
173	C₂²⋊Q₈	—	13	1	no	—
174	C₂²⋊Q₈	—	13	1	no	—
175	C₂²⋊Q₈	—	13	1	no	—
176	C₄×Q₈	—	13	1	no	—
177	C₄×Q₈	—	13	1	no	—
178	C₄×Q₈	—	13	1	no	—
179	C₄×Q₈	—	13	1	no	—
180	#117	—	13	1	no	—
181	#117	C₂×D₁₃	1	1	yes	characteristic
182	C₂×C₄²	—	13	1	no	—
183	C₄⋊C₈	—	13	1	no	—
184	C₄⋊C₈	—	13	1	no	—
185	C₄⋊C₈	—	13	1	no	—
186	C₄⋊C₈	—	13	1	no	—
187	C₄⋊C₈	—	13	1	no	—
188	C₄⋊C₈	—	13	1	no	—
189	C₄⋊C₈	—	13	1	no	—
190	C₄⋊C₈	—	13	1	no	—
191	C₄⋊C₈	—	13	1	no	—
192	C₄⋊C₈	—	13	1	no	—
193	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
194	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
195	C₂²⋊C₈	—	13	1	no	—
196	C₂²⋊C₈	—	13	1	no	—
197	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
198	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
199	#97	—	13	1	no	—
200	#97	—	13	1	no	—
201	#97	—	13	1	no	—
202	#97	—	13	1	no	—
203	#97	—	13	1	no	—
204	#97	C₂×D₁₃	1	1	yes	characteristic
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	13	1	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	1	no	—
209	C₄×C₈	—	13	1	no	—
210	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
211	C₂×C₁₃	—	4	1	no	—
212	C₂×C₁₃	#519	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
214	C₂×C₁₃	#568	1	1	yes	characteristic
215	C₂×Q₈	—	13	1	no	—
216	C₂×Q₈	—	13	1	no	—
217	C₂×Q₈	—	13	1	no	—
218	C₂×Q₈	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₈	—	13	1	no	—
223	C₂×C₈	—	13	1	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	13	1	no	—
226	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	13	1	no	—
231	C₄⋊C₄	—	13	1	no	—
232	C₄⋊C₄	—	13	1	no	—
233	C₄⋊C₄	—	13	1	no	—
234	C₄⋊C₄	—	13	1	no	—
235	C₄⋊C₄	—	13	1	no	—
236	C₄⋊C₄	—	13	1	no	—
237	C₄⋊C₄	—	13	1	no	—
238	C₄⋊C₄	—	13	1	no	—
239	C₄⋊C₄	—	13	1	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₄⋊C₄	—	13	1	no	—
242	C₄⋊C₄	—	13	1	no	—
243	C₄⋊C₄	—	13	1	no	—
244	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
245	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	13	1	no	—
247	C₂²⋊C₄	—	13	1	no	—
248	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
249	C₂²⋊C₄	C₄×D₁₃	1	1	yes	characteristic
250	C₄²	—	13	1	no	—
251	C₄²	—	13	1	no	—
252	C₄²	—	13	1	no	—
253	C₄²	—	13	1	no	—
254	C₄²	—	13	1	no	—
255	C₄²	—	13	1	no	—
256	C₄²	C₂²×D₁₃	1	1	yes	characteristic
257	C₄²	C₂²×D₁₃	1	1	yes	characteristic
258	C₁₃	#2985	1	1	yes	characteristic, complemented, Sylow
259	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
260	Q₈	—	26	2	no	—
261	Q₈	—	26	2	no	—
262	Q₈	—	52	1	no	—
263	Q₈	—	52	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	13	1	no	—
267	C₂×C₄	—	13	1	no	—
268	C₂×C₄	—	13	1	no	—
269	C₂×C₄	—	13	1	no	—
270	C₂×C₄	—	13	1	no	—
271	C₂×C₄	—	13	1	no	—
272	C₂×C₄	—	13	1	no	—
273	C₂×C₄	—	13	1	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	—	52	1	no	—
276	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	26	1	no	—
287	C₈	—	26	1	no	—
288	C₈	—	26	1	no	—
289	C₈	—	26	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	13	2	no	—
300	C₄	—	26	1	no	—
301	C₄	—	26	1	no	—
302	C₄	—	26	1	no	—
303	C₄	—	26	1	no	—
304	C₄	—	26	1	no	—
305	C₄	—	26	1	no	—
306	C₄	—	26	1	no	—
307	C₄	#88257	1	1	yes	characteristic
308	C₄	#88257	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	#11719221	1	1	yes	characteristic
311	C₂	D₁₃×#131	1	1	yes	characteristic
312	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#433	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#433	D₁₃
hypocenter	C₁₃	#2985
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
4. #11718492	C₂	semidirect
5. #11718491	C₂	semidirect
6. #11718489	C₂	semidirect
7. #11718489	C₂	semidirect
12. #11718362	C₂	semidirect
13. #11718362	C₂	semidirect
14. #11718340	C₂	semidirect
15. C₈×Dic₂₆	C₂	semidirect
258. C₁₃	#2985	semidirect

Maximal subgroups

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

subgroup	index
#11719222	2
C₁₃×#433	2
#11718661	2
#11718492	2
#11718491	2
#11718489	2
#11718381	2
#11718380	2
#11718379	2
#11718362	2
#11718340	2
C₈×Dic₂₆	2
#2985	13