Subgroups — #13118523

Order 1664.

508 conjugacy classes of subgroups.

The subgroup lattice has 282 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×#88146	C₂	1	1	yes	maximal, characteristic
3	#11718892	C₂	1	2	yes	maximal
4	#11718891	C₂	1	2	yes	maximal, complemented
5	#11718655	C₂	1	2	yes	maximal
6	#11718648	C₂	1	1	yes	maximal, characteristic
7	#11718646	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718639	C₂	1	2	yes	maximal, complemented
9	#11718638	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#6103	C₂	1	1	yes	maximal, characteristic
11	#11718589	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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×#6103	C₂²	1	2	yes	—
20	C₂×#6103	C₂²	1	1	yes	characteristic
21	C₂×#6103	C₂²	1	1	yes	characteristic
22	C₂×#6103	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
26	C₂×#6101	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
29	#88146	—	2	4	no	—
30	#88071	C₂²	1	2	yes	—
31	#88071	C₂²	1	1	yes	characteristic
32	#88071	C₂²	1	1	yes	characteristic
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	2	yes	—
36	#88071	C₂²	1	2	yes	—
37	C₂³×D₁₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₁₃	—	2	4	no	—
39	C₂×Q₈×C₁₃	—	2	4	no	—
40	C₂×Q₈×C₁₃	—	4	2	no	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂³	1	2	yes	—
48	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₃	—	2	2	no	—
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	Q₈	1	4	yes	complemented
54	C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
57	C₄²×C₁₃	—	2	2	no	—
58	#6103	—	2	4	no	—
59	#6103	—	2	2	no	—
60	#6103	—	2	2	no	—
61	#6103	—	4	2	no	—
62	C₄⋊Dic₁₃	—	2	2	no	—
63	C₄⋊Dic₁₃	—	2	2	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	#6101	—	2	4	no	—
66	C₄×Dic₁₃	—	2	4	no	—
67	C₄×Dic₁₃	—	2	2	no	—
68	#2439	—	13	1	no	maximal, Sylow
69	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×D₁₃	—	2	1	no	—
71	C₂²×D₁₃	—	2	1	no	—
72	C₂²×D₁₃	—	2	1	no	—
73	C₂²×D₁₃	—	2	1	no	—
74	C₂²×D₁₃	—	2	1	no	—
75	C₂²×D₁₃	—	2	1	no	—
76	C₂²×D₁₃	C₂×Q₈	1	2	yes	—
77	Q₈×C₁₃	—	4	8	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	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	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₁₃	#41	1	2	yes	—
91	C₂×C₄×C₁₃	#41	1	2	yes	—
92	C₂×C₄×C₁₃	#41	1	2	yes	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	—	2	1	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	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	2	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
108	C₄×D₁₃	—	2	8	no	—
109	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
110	C₂×#117	—	13	1	no	—
111	#396	—	13	2	no	—
112	#393	—	13	2	no	—
113	#391	—	13	2	no	—
114	#389	—	13	1	no	—
115	#389	—	13	1	no	—
116	#387	—	13	2	no	—
117	#382	—	13	1	no	—
118	#380	—	13	1	no	—
119	C₄×C₂²⋊C₄	—	13	1	no	—
120	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×D₁₃	—	2	2	no	—
128	C₂×D₁₃	—	2	2	no	—
129	C₂×D₁₃	—	2	2	no	—
130	C₂×D₁₃	—	4	1	no	—
131	C₂×D₁₃	—	4	1	no	—
132	C₂×D₁₃	—	4	1	no	—
133	C₂×D₁₃	—	4	1	no	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	4	2	no	—
138	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	—	4	2	no	—
141	Dic₁₃	—	4	1	no	—
142	Dic₁₃	—	4	2	no	—
143	Dic₁₃	—	4	1	no	—
144	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
145	C₂³×C₄	—	13	1	no	—
146	C₄⋊Q₈	—	2	4	no	—
147	#117	—	26	4	no	—
148	C₂×C₄⋊C₄	—	13	1	no	—
149	C₂×C₄⋊C₄	—	13	2	no	—
150	C₂×C₄⋊C₄	—	13	1	no	—
151	C₂×C₄⋊C₄	—	13	2	no	—
152	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	—	13	2	no	—
156	C₂×C₂²⋊C₄	—	13	1	no	—
157	C₂×C₂²⋊C₄	—	13	1	no	—
158	C₂×C₂²⋊C₄	—	13	2	no	—
159	C₂×C₄²	—	13	2	no	—
160	C₂×C₄²	—	13	1	no	—
161	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
162	#95	—	13	1	no	—
163	#95	—	13	1	no	—
164	#95	—	13	2	no	—
165	#95	—	13	2	no	—
166	#95	—	13	2	no	—
167	#95	—	13	2	no	—
168	#95	—	13	2	no	—
169	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
170	C₂×C₁₃	#533	1	1	yes	characteristic
171	C₂×C₁₃	#544	1	1	yes	characteristic
172	C₂×C₁₃	#546	1	1	yes	characteristic
173	C₂×C₁₃	#548	1	1	yes	characteristic
174	C₂×C₁₃	#555	1	1	yes	characteristic
175	C₂×C₁₃	#556	1	1	yes	characteristic
176	D₁₃	—	4	2	no	—
177	C₂⁴	—	13	1	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂²×C₄	—	13	1	no	—
182	C₂²×C₄	—	13	4	no	—
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	1	no	—
185	C₂²×C₄	—	13	2	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	—	26	2	no	—
188	C₂²×C₄	—	26	2	no	—
189	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
190	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
193	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	26	4	no	—
197	C₄⋊C₄	—	26	2	no	—
198	C₄⋊C₄	—	26	2	no	—
199	C₄⋊C₄	—	52	2	no	—
200	C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
201	C₂²⋊C₄	—	26	2	no	—
202	C₂²⋊C₄	—	26	2	no	—
203	C₂²⋊C₄	—	26	4	no	—
204	C₂²⋊C₄	—	52	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	26	2	no	—
207	C₄²	—	26	4	no	—
208	C₁₃	#2439	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	13	2	no	—
210	C₂³	—	26	1	no	—
211	C₂³	—	26	1	no	—
212	C₂³	—	26	1	no	—
213	C₂³	—	26	1	no	—
214	C₂³	—	26	1	no	—
215	C₂³	—	26	1	no	—
216	C₂³	C₂³×D₁₃	1	1	yes	characteristic
217	Q₈	—	4	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	13	2	no	—
228	C₂×C₄	—	26	1	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	—	26	8	no	—
231	C₂×C₄	—	26	1	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	26	1	no	—
234	C₂×C₄	—	26	1	no	—
235	C₂×C₄	—	26	1	no	—
236	C₂×C₄	—	26	1	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	26	1	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	#6129	1	2	yes	—
244	C₂×C₄	Q₈×D₁₃	1	2	yes	—
245	C₂×C₄	#6131	1	2	yes	—
246	C₂×C₄	#6131	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
248	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
249	C₂²	—	26	2	no	—
250	C₂²	—	26	2	no	—
251	C₂²	—	26	2	no	—
252	C₂²	—	52	1	no	—
253	C₂²	—	52	1	no	—
254	C₂²	—	52	1	no	—
255	C₂²	—	52	1	no	—
256	C₂²	C₂×#6129	1	1	yes	characteristic
257	C₂²	#88251	1	1	yes	characteristic
258	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
259	C₂²	C₂×#6131	1	1	yes	characteristic
260	C₂²	C₂×#6131	1	1	yes	characteristic
261	C₂²	#88254	1	1	yes	characteristic
262	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	26	2	no	—
268	C₄	—	52	1	no	—
269	C₄	—	52	2	no	—
270	C₄	—	52	1	no	—
271	C₄	—	52	2	no	—
272	C₄	#88200	1	4	yes	—
273	C₂	—	52	2	no	—
274	C₂	#11719433	1	1	yes	characteristic
275	C₂	#11719434	1	1	yes	characteristic
276	C₂	#11719437	1	1	yes	characteristic
277	C₂	#11719439	1	1	yes	characteristic
278	C₂	#11719511	1	1	yes	characteristic
279	C₂	C₂×#88200	1	1	yes	characteristic
280	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	C₂³×D₁₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2439
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
4. #11718891	C₂	semidirect
7. #11718646	C₂	semidirect
8. #11718639	C₂	semidirect
9. #11718638	C₂	semidirect
11. #11718589	C₂	semidirect
53. C₂×C₄×D₁₃	Q₈	semidirect
109. C₂×C₄⋊Q₈	D₁₃	semidirect
208. C₁₃	#2439	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×#88146	2
#11718892	2
#11718891	2
#11718655	2
#11718648	2
#11718646	2
#11718639	2
#11718638	2
C₄×#6103	2
#11718589	2
#2439	13