Subgroups — #13118682

Order 1664.

620 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88200	C₂	1	2	yes	maximal, complemented
3	C₂×#88147	C₂	1	1	yes	maximal, characteristic
4	#11718892	C₂	1	2	yes	maximal, complemented
5	#11718655	C₂	1	2	yes	maximal, complemented
6	#11718650	C₂	1	2	yes	maximal, complemented
7	#11718649	C₂	1	1	yes	maximal, characteristic
8	#11718646	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718634	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718600	C₂	1	1	yes	maximal, characteristic
11	C₄×#6101	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂²×D₅₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
15	C₁₃×C₄⋊Q₈	—	2	4	no	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	#88200	—	2	8	no	—
21	C₂×#6103	C₂²	1	2	yes	—
22	C₂×#6103	C₂²	1	2	yes	—
23	C₂×#6103	C₂²	1	2	yes	—
24	C₂×#6103	C₂²	1	2	yes	—
25	C₂×#6103	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
28	C₂×#6101	C₂²	1	2	yes	—
29	C₂×#6101	C₂²	1	1	yes	characteristic
30	C₂×#6101	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
33	#88147	—	2	4	no	—
34	#88071	C₂²	1	1	yes	characteristic
35	#88071	C₂²	1	1	yes	characteristic
36	#88071	C₂²	1	2	yes	—
37	#88071	C₂²	1	2	yes	—
38	C₂³×D₁₃	C₂³	1	2	yes	—
39	C₂×Q₈×C₁₃	—	2	4	no	—
40	C₂×Q₈×C₁₃	—	2	4	no	—
41	C₂×Q₈×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
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×D₅₂	—	2	2	no	—
53	C₂×D₅₂	—	2	2	no	—
54	C₂×D₅₂	Q₈	1	4	yes	complemented
55	C₂×C₄×D₁₃	—	2	4	no	—
56	C₂×C₄×D₁₃	—	2	4	no	—
57	C₂×C₄×D₁₃	—	4	2	no	—
58	C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
61	C₄²×C₁₃	—	2	2	no	—
62	#6103	—	2	4	no	—
63	#6103	—	2	4	no	—
64	#6103	—	2	4	no	—
65	#6103	—	2	4	no	—
66	#6103	—	4	2	no	—
67	C₄⋊Dic₁₃	—	2	2	no	—
68	C₄⋊Dic₁₃	—	2	4	no	—
69	#6101	—	2	2	no	—
70	#6101	—	2	4	no	—
71	#6101	D₄	1	4	yes	complemented
72	C₄×Dic₁₃	—	2	2	no	—
73	C₄×Dic₁₃	—	2	2	no	—
74	#2452	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₂²×D₁₃	—	2	2	no	—
77	C₂²×D₁₃	—	2	2	no	—
78	C₂²×D₁₃	—	2	2	no	—
79	C₂²×D₁₃	—	2	2	no	—
80	C₂²×D₁₃	—	2	2	no	—
81	C₂²×D₁₃	—	2	2	no	—
82	C₂²×D₁₃	C₂×Q₈	1	4	yes	—
83	Q₈×C₁₃	—	4	8	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₁₃	#41	1	2	yes	—
98	C₂×C₄×C₁₃	#41	1	2	yes	—
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
113	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
114	D₅₂	—	2	8	no	—
115	C₄×D₁₃	—	4	8	no	—
116	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
117	C₂×C₂²⋊Q₈	—	13	2	no	—
118	C₂×C₄×D₄	—	13	1	no	—
119	#396	—	13	2	no	—
120	#393	—	13	2	no	—
121	#390	—	13	1	no	—
122	#390	—	13	1	no	—
123	#389	—	13	1	no	—
124	#389	—	13	1	no	—
125	#385	—	13	2	no	—
126	C₄×C₄⋊C₄	—	13	1	no	—
127	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×D₁₃	—	2	4	no	—
135	C₂×D₁₃	—	2	4	no	—
136	C₂×D₁₃	—	2	4	no	—
137	C₂×D₁₃	—	4	2	no	—
138	C₂×D₁₃	—	4	2	no	—
139	C₂×D₁₃	—	4	2	no	—
140	C₂×D₁₃	—	4	2	no	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	4	2	no	—
145	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	complemented
146	Dic₁₃	—	2	2	no	—
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	—	4	2	no	—
149	Dic₁₃	—	4	1	no	—
150	Dic₁₃	—	4	1	no	—
151	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
152	C₂²×D₄	—	13	1	no	—
153	C₂³×C₄	—	13	2	no	—
154	C₄⋊Q₈	—	2	4	no	—
155	C₂²⋊Q₈	—	26	8	no	—
156	C₄×D₄	—	26	4	no	—
157	C₂×C₄⋊C₄	—	13	1	no	—
158	C₂×C₄⋊C₄	—	13	2	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄⋊C₄	—	13	1	no	—
161	C₂×C₄⋊C₄	—	13	1	no	—
162	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	13	2	no	—
166	C₂×C₂²⋊C₄	—	13	2	no	—
167	C₂×C₂²⋊C₄	—	13	2	no	—
168	C₂×C₂²⋊C₄	—	13	2	no	—
169	C₂×C₂²⋊C₄	—	13	2	no	—
170	C₂×C₄²	—	13	1	no	—
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
173	#95	—	13	1	no	—
174	#95	—	13	1	no	—
175	#95	—	13	2	no	—
176	#95	—	13	2	no	—
177	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₁₃	#532	1	1	yes	characteristic
179	C₂×C₁₃	#542	1	1	yes	characteristic
180	C₂×C₁₃	D₄²	1	1	yes	characteristic
181	C₂×C₁₃	#548	1	1	yes	characteristic
182	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
183	C₂×C₁₃	#554	1	1	yes	characteristic
184	D₁₃	—	4	4	no	—
185	C₂⁴	—	13	2	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×D₄	—	13	4	no	—
190	C₂×D₄	—	26	2	no	—
191	C₂×D₄	—	26	2	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	1	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	2	no	—
197	C₂²×C₄	—	26	4	no	—
198	C₂²×C₄	—	26	4	no	—
199	C₂²×C₄	—	52	2	no	—
200	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
204	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	13	4	no	—
208	C₄⋊C₄	—	26	4	no	—
209	C₄⋊C₄	—	26	4	no	—
210	C₄⋊C₄	—	26	2	no	—
211	C₄⋊C₄	—	26	2	no	—
212	C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
213	C₂²⋊C₄	—	26	4	no	—
214	C₂²⋊C₄	—	26	4	no	—
215	C₂²⋊C₄	—	26	4	no	—
216	C₂²⋊C₄	—	26	4	no	—
217	C₂²⋊C₄	—	52	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	26	2	no	—
220	C₄²	—	26	2	no	—
221	C₁₃	#2452	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	13	4	no	—
223	C₂³	—	26	2	no	—
224	C₂³	—	26	2	no	—
225	C₂³	—	26	2	no	—
226	C₂³	—	26	2	no	—
227	C₂³	—	26	2	no	—
228	C₂³	—	26	2	no	—
229	C₂³	C₂³×D₁₃	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	D₄	—	26	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	13	2	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	26	1	no	—
244	C₂×C₄	—	26	1	no	—
245	C₂×C₄	—	26	1	no	—
246	C₂×C₄	—	26	1	no	—
247	C₂×C₄	—	26	2	no	—
248	C₂×C₄	—	26	1	no	—
249	C₂×C₄	—	26	1	no	—
250	C₂×C₄	—	26	2	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	52	8	no	—
257	C₂×C₄	D₄×D₁₃	1	2	yes	—
258	C₂×C₄	Q₈×D₁₃	1	2	yes	—
259	C₂×C₄	#6131	1	2	yes	—
260	C₂×C₄	#6131	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
262	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
263	C₂²	—	26	4	no	—
264	C₂²	—	26	4	no	—
265	C₂²	—	26	4	no	—
266	C₂²	—	52	2	no	—
267	C₂²	—	52	2	no	—
268	C₂²	—	52	2	no	—
269	C₂²	—	52	2	no	—
270	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
271	C₂²	#88251	1	1	yes	characteristic
272	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
273	C₂²	C₂×#6131	1	1	yes	characteristic
274	C₂²	C₂×#6131	1	1	yes	characteristic
275	C₂²	#88254	1	1	yes	characteristic
276	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	26	2	no	—
282	C₄	—	26	2	no	—
283	C₄	—	52	2	no	—
284	C₄	—	52	1	no	—
285	C₄	—	52	1	no	—
286	C₄	#88200	1	4	yes	complemented
287	C₂	—	52	4	no	—
288	C₂	#11719430	1	1	yes	characteristic
289	C₂	#11719432	1	1	yes	characteristic
290	C₂	#11719435	1	1	yes	characteristic
291	C₂	#11719438	1	1	yes	characteristic
292	C₂	D₄×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₂	C₂×#88200	1	1	yes	characteristic
294	C₂	#11719522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2452
LCS	G ▷ C₂³×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×C₁₃×C₄⋊Q₈	C₂	semidirect
2. C₂×#88200	C₂	semidirect
4. #11718892	C₂	semidirect
5. #11718655	C₂	semidirect
6. #11718650	C₂	semidirect
8. #11718646	C₂	semidirect
9. #11718634	C₂	semidirect
11. C₄×#6101	C₂	semidirect
54. C₂×D₅₂	Q₈	semidirect
71. #6101	D₄	semidirect
116. C₂×C₄⋊Q₈	D₁₃	semidirect
145. C₄×C₁₃	C₂²⋊Q₈	semidirect
221. C₁₃	#2452	semidirect
286. C₄	#88200	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88200	2
C₂×#88147	2
#11718892	2
#11718655	2
#11718650	2
#11718649	2
#11718646	2
#11718634	2
#11718600	2
C₄×#6101	2
#2452	13