Subgroups — #13117837

Order 1664.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88200	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
4	#11718891	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718650	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718633	C₂	1	2	yes	maximal, complemented
8	C₄×#6103	C₂	1	2	yes	maximal
9	C₄×#6103	C₂	1	1	yes	maximal, characteristic
10	#11718594	C₂	1	2	yes	maximal, complemented
11	#11718589	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₃	—	2	4	no	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
20	#88200	C₄	1	8	yes	complemented
21	C₂×#6103	C₂²	1	1	yes	characteristic
22	C₂×#6103	C₂²	1	2	yes	—
23	C₂×#6103	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×#6101	C₂²	1	2	yes	—
27	C₂×#6101	C₂²	1	2	yes	—
28	C₂×#6101	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
31	C₄⋊C₄×D₁₃	—	2	4	no	—
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	1	yes	characteristic
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	2	yes	—
36	#88071	C₂²	1	1	yes	characteristic
37	#88071	C₂²	1	2	yes	—
38	C₂³×D₁₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₃	—	2	2	no	—
40	C₂×Q₈×C₁₃	—	2	2	no	—
41	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂³	1	2	yes	—
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	—	2	2	no	—
54	C₂×C₄×D₁₃	—	2	4	no	—
55	C₂×C₄×D₁₃	—	2	4	no	—
56	C₂×C₄×D₁₃	—	4	2	no	—
57	C₂×C₄×D₁₃	C₂×C₄	1	4	yes	—
58	C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
60	C₄²×C₁₃	—	2	4	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	#6103	—	2	2	no	—
63	#6103	C₂×C₄	1	8	yes	—
64	#6103	Q₈	1	4	yes	complemented
65	C₄⋊Dic₁₃	—	2	2	no	—
66	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	complemented
67	#6101	—	2	4	no	—
68	#6101	—	4	2	no	—
69	#6101	C₂×C₄	1	8	yes	complemented
70	C₄×Dic₁₃	—	2	2	no	—
71	C₄×Dic₁₃	—	4	2	no	—
72	#2360	—	13	1	no	maximal, Sylow
73	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂²×D₁₃	—	2	1	no	—
75	C₂²×D₁₃	—	2	1	no	—
76	C₂²×D₁₃	—	2	1	no	—
77	C₂²×D₁₃	—	2	1	no	—
78	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×D₁₃	C₂×Q₈	1	2	yes	—
80	C₂²×D₁₃	#41	1	2	yes	—
81	Q₈×C₁₃	—	2	8	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₁₃	#41	1	2	yes	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
110	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
111	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
112	C₄×D₁₃	—	2	8	no	—
113	C₄×D₁₃	—	4	8	no	—
114	C₂×C₂²⋊Q₈	—	13	1	no	—
115	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
116	C₂²×C₄⋊C₄	—	13	1	no	—
117	#391	—	13	1	no	—
118	#389	—	13	2	no	—
119	#389	—	13	1	no	—
120	#387	—	13	2	no	—
121	#385	—	13	2	no	—
122	C₄×C₄⋊C₄	—	13	1	no	—
123	C₄×C₂²⋊C₄	—	13	1	no	—
124	C₄×C₂²⋊C₄	—	13	2	no	—
125	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂×D₁₃	—	2	2	no	—
133	C₂×D₁₃	—	2	2	no	—
134	C₂×D₁₃	—	2	2	no	—
135	C₂×D₁₃	—	2	2	no	—
136	C₂×D₁₃	—	2	2	no	—
137	C₂×D₁₃	—	2	2	no	—
138	C₂×D₁₃	C₄×Q₈	1	4	yes	complemented
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	C₄×D₄	1	4	yes	—
144	Dic₁₃	—	2	2	no	—
145	Dic₁₃	—	2	4	no	—
146	Dic₁₃	—	2	2	no	—
147	Dic₁₃	—	4	2	no	—
148	Dic₁₃	—	4	1	no	—
149	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
150	C₂³×C₄	—	13	1	no	—
151	C₂³×C₄	—	13	2	no	—
152	C₂²⋊Q₈	—	13	8	no	—
153	C₄×Q₈	—	2	4	no	—
154	C₂×C₄⋊C₄	—	13	2	no	—
155	C₂×C₄⋊C₄	—	13	1	no	—
156	C₂×C₄⋊C₄	—	13	2	no	—
157	C₂×C₄⋊C₄	—	13	1	no	—
158	C₂×C₄⋊C₄	—	13	2	no	—
159	C₂×C₄⋊C₄	—	26	4	no	—
160	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	—	13	2	no	—
163	C₂×C₂²⋊C₄	—	13	1	no	—
164	C₂×C₂²⋊C₄	—	13	1	no	—
165	C₂×C₄²	—	13	1	no	—
166	C₂×C₄²	—	13	2	no	—
167	C₂×C₄²	C₂×D₁₃	1	2	yes	—
168	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
169	#95	—	13	1	no	—
170	#95	—	13	2	no	—
171	#95	—	13	1	no	—
172	#95	—	13	2	no	—
173	#95	—	13	2	no	—
174	#95	—	13	2	no	—
175	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₂×C₁₃	#520	1	1	yes	characteristic
178	C₂×C₁₃	#546	1	1	yes	characteristic
179	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
180	C₂×C₁₃	#552	1	1	yes	characteristic
181	C₂×C₁₃	#554	1	1	yes	characteristic
182	D₁₃	—	2	4	no	—
183	C₂⁴	—	13	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	C₄×D₁₃	1	4	yes	complemented
187	C₂²×C₄	—	13	1	no	—
188	C₂²×C₄	—	13	2	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	4	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	26	4	no	—
194	C₂²×C₄	—	26	2	no	—
195	C₂²×C₄	—	26	2	no	—
196	C₂²×C₄	—	26	4	no	—
197	C₂²×C₄	—	52	2	no	—
198	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
200	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	13	8	no	—
205	C₄⋊C₄	—	13	4	no	—
206	C₄⋊C₄	—	26	4	no	—
207	C₄⋊C₄	—	26	2	no	—
208	C₄⋊C₄	—	52	2	no	—
209	C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
210	C₂²⋊C₄	—	13	8	no	—
211	C₂²⋊C₄	—	13	4	no	—
212	C₂²⋊C₄	—	26	2	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	26	2	no	—
216	C₄²	—	52	2	no	—
217	C₁₃	#2360	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	13	2	no	—
219	C₂³	—	13	2	no	—
220	C₂³	—	13	2	no	—
221	C₂³	—	26	1	no	—
222	C₂³	—	26	1	no	—
223	C₂³	—	26	1	no	—
224	C₂³	—	26	1	no	—
225	C₂³	C₂³×D₁₃	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	2	no	—
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	26	1	no	—
240	C₂×C₄	—	26	1	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	—	26	1	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	26	1	no	—
245	C₂×C₄	—	26	8	no	—
246	C₂×C₄	—	26	1	no	—
247	C₂×C₄	—	26	1	no	—
248	C₂×C₄	—	26	1	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	2	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	52	8	no	—
253	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
254	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
255	C₂×C₄	#6127	1	2	yes	—
256	C₂×C₄	Q₈×D₁₃	1	2	yes	—
257	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
258	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
259	C₂²	—	13	4	no	—
260	C₂²	—	26	2	no	—
261	C₂²	—	26	2	no	—
262	C₂²	—	26	2	no	—
263	C₂²	—	26	2	no	—
264	C₂²	—	26	2	no	—
265	C₂²	—	26	2	no	—
266	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
267	C₂²	C₂×#6127	1	1	yes	characteristic
268	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
269	C₂²	#88254	1	1	yes	characteristic
270	C₂²	#41×D₁₃	1	1	yes	characteristic
271	C₂²	#88256	1	1	yes	characteristic
272	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	1	no	—
277	C₄	—	26	2	no	—
278	C₄	—	26	4	no	—
279	C₄	—	26	2	no	—
280	C₄	—	52	2	no	—
281	C₄	—	52	1	no	—
282	C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
283	C₂	—	26	4	no	—
284	C₂	C₄×Q₈×D₁₃	1	1	yes	characteristic
285	C₂	#11719279	1	1	yes	characteristic
286	C₂	#11719284	1	1	yes	characteristic
287	C₂	#11719286	1	1	yes	characteristic
288	C₂	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₂	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₂	#11719534	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₃	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
prime radical	C₂³×D₁₃	C₂³
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	#2360
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₁₃	C₂	semidirect
4. #11718891	C₂	semidirect
6. C₄⋊C₄×Dic₁₃	C₂	semidirect
7. #11718633	C₂	semidirect
10. #11718594	C₂	semidirect
11. #11718589	C₂	semidirect
20. #88200	C₄	semidirect
41. C₂×Q₈×C₁₃	C₂×C₄	semidirect
64. #6103	Q₈	semidirect
66. C₄⋊Dic₁₃	C₂×C₄	semidirect
69. #6101	C₂×C₄	semidirect
115. C₂×C₄×Q₈	D₁₃	semidirect
138. C₂×D₁₃	C₄×Q₈	semidirect
186. C₂×Q₈	C₄×D₁₃	semidirect
217. C₁₃	#2360	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
C₂×#88200	2
C₂×C₄⋊C₄×D₁₃	2
#11718891	2
#11718650	2
C₄⋊C₄×Dic₁₃	2
#11718633	2
C₄×#6103	2
#11718594	2
#11718589	2
#2360	13