Subgroups — #13124624

Order 1664.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
2	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718894	C₂	1	2	yes	maximal
4	#11718683	C₂	1	2	yes	maximal, complemented
5	#11718682	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718681	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718670	C₂	1	2	yes	maximal, complemented
8	#11718669	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718668	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718659	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₃	—	2	1	no	—
16	C₄×D₄×C₁₃	—	2	1	no	—
17	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#117	—	2	1	no	—
20	C₁₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	—	2	1	no	—
22	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	#88180	C₂²	1	2	yes	—
24	#88178	—	2	1	no	—
25	#88178	—	2	1	no	—
26	C₂²×#1055	C₂²	1	2	yes	—
27	#88124	C₂²	1	2	yes	—
28	#88123	C₂²	1	1	yes	characteristic
29	#88123	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₆	—	2	1	no	—
31	C₄×Dic₂₆	—	2	1	no	—
32	#88075	C₂²	1	2	yes	—
33	#88075	C₂²	1	2	yes	—
34	#88072	C₂²	1	2	yes	—
35	#88072	C₂²	1	2	yes	—
36	#88048	C₂²	1	2	yes	—
37	#88048	C₂²	1	2	yes	—
38	#88047	C₂²	1	2	yes	—
39	#88047	C₂²	1	2	yes	—
40	C₄⋊#1055	C₂²	1	1	yes	characteristic
41	C₄⋊#1055	C₂²	1	1	yes	characteristic
42	C₄⋊#1055	C₂²	1	2	yes	—
43	C₁₃×#41	—	2	2	no	—
44	C₁₃×#41	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₆	—	2	1	no	—
55	C₂×Dic₂₆	—	2	1	no	—
56	C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₁₃×C₂²⋊C₄	—	2	1	no	—
61	C₁₃×C₂²⋊C₄	—	2	1	no	—
62	C₁₃×C₂²⋊C₄	—	2	1	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	D₄	1	1	yes	characteristic
67	C₄²×C₁₃	D₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂³	1	1	yes	characteristic
69	C₄²×C₁₃	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₁₃	—	2	2	no	—
71	C₄⋊Dic₁₃	—	2	1	no	—
72	C₄⋊Dic₁₃	—	2	1	no	—
73	C₄⋊Dic₁₃	C₂³	1	2	yes	—
74	C₄⋊Dic₁₃	C₂³	1	2	yes	—
75	#6101	—	2	2	no	—
76	#6101	—	2	2	no	—
77	C₄×Dic₁₃	—	2	2	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	C₂³	1	2	yes	—
80	C₂×#1055	C₂³	1	2	yes	—
81	#3076	—	13	1	no	maximal, Sylow
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₃	—	2	1	no	—
85	Q₈×C₁₃	C₂×Q₈	1	2	yes	—
86	D₄×C₁₃	—	2	4	no	—
87	D₄×C₁₃	—	2	1	no	—
88	D₄×C₁₃	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	4	1	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
103	C₂×C₄×C₁₃	#41	1	2	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	Dic₂₆	—	4	2	no	—
108	Dic₂₆	—	4	2	no	—
109	#1055	—	2	2	no	—
110	#1055	—	2	2	no	—
111	#533	—	13	1	no	—
112	C₄×#41	D₁₃	1	1	yes	characteristic
113	#479	—	13	2	no	—
114	#478	—	13	2	no	—
115	#477	—	13	1	no	—
116	#476	—	13	1	no	—
117	#475	—	13	1	no	—
118	#474	—	13	1	no	—
119	#427	—	13	2	no	—
120	C₂×C₄⋊C₈	—	13	1	no	—
121	#416	—	13	2	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	4	1	no	—
126	C₂²×C₁₃	—	4	1	no	—
127	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
128	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	2	1	no	—
133	C₄×C₁₃	—	4	1	no	—
134	C₄×C₁₃	—	4	1	no	—
135	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
136	C₄×C₁₃	#135	1	2	yes	—
137	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
138	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	Dic₁₃	—	4	2	no	—
140	Dic₁₃	—	4	2	no	—
141	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
142	C₂²×C₈	—	13	2	no	—
143	C₄⋊Q₈	—	13	1	no	—
144	C₄⋊Q₈	—	13	1	no	—
145	#125	—	13	2	no	—
146	C₂²⋊Q₈	—	26	1	no	—
147	C₂²⋊Q₈	—	26	1	no	—
148	C₄×Q₈	—	26	1	no	—
149	C₄×Q₈	—	26	1	no	—
150	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
151	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
155	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
156	#117	—	2	1	no	—
157	#117	—	13	2	no	—
158	#117	C₂×D₁₃	1	1	yes	characteristic
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
161	#107	—	13	2	no	—
162	#107	—	13	2	no	—
163	#106	—	13	2	no	—
164	#106	—	13	2	no	—
165	C₄⋊C₈	—	13	1	no	—
166	C₄⋊C₈	—	13	1	no	—
167	C₄⋊C₈	—	13	2	no	—
168	#103	—	13	2	no	—
169	#103	—	13	2	no	—
170	#102	—	13	2	no	—
171	#102	—	13	2	no	—
172	C₂×C₁₃	—	2	2	no	—
173	C₂×C₁₃	—	4	2	no	—
174	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
176	C₂×C₁₃	#575	1	1	yes	characteristic
177	#41	—	2	2	no	—
178	#41	Dic₂₆	1	4	yes	complemented
179	C₂×Q₈	—	26	1	no	—
180	C₂×Q₈	—	26	1	no	—
181	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	13	2	no	—
192	C₂×C₈	—	26	4	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	13	2	no	—
196	C₄⋊C₄	—	13	2	no	—
197	C₄⋊C₄	—	26	2	no	—
198	C₄⋊C₄	—	26	2	no	—
199	C₄⋊C₄	—	26	1	no	—
200	C₄⋊C₄	—	26	1	no	—
201	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	26	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	26	2	no	—
211	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
212	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₄²	C₂²×D₁₃	1	1	yes	characteristic
214	C₄²	C₂²×D₁₃	1	1	yes	characteristic
215	C₁₃	#3076	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	2	1	no	—
217	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
218	Q₈	—	2	1	no	—
219	Q₈	—	52	2	no	—
220	Q₈	—	52	2	no	—
221	Q₈	C₂×Dic₂₆	1	2	yes	—
222	D₄	—	2	4	no	—
223	D₄	—	2	1	no	—
224	D₄	C₂×Dic₂₆	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	—	26	2	no	—
235	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
236	C₂×C₄	#6127	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
243	C₈	—	26	2	no	—
244	C₈	—	26	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	#88178	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	52	2	no	—
259	C₄	—	52	2	no	—
260	C₄	#88178	1	2	yes	—
261	C₄	#88204	1	2	yes	—
262	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
263	C₄	C₂×#6127	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#88178	1	1	yes	characteristic
267	C₂	C₂×#88204	1	1	yes	characteristic
268	C₂	#11719527	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#41	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×#41	C₂²
hypercenter	C₄×#41	D₁₃
hypocenter	C₁₃	#3076
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #11719175	C₂	semidirect
4. #11718683	C₂	semidirect
5. #11718682	C₂	semidirect
6. #11718681	C₂	semidirect
7. #11718670	C₂	semidirect
8. #11718669	C₂	semidirect
9. #11718668	C₂	semidirect
10. #11718659	C₂	semidirect
11. C₂×C₄⋊#1055	C₂	semidirect
44. C₁₃×#41	Q₈	semidirect
178. #41	Dic₂₆	semidirect
215. C₁₃	#3076	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#41	2
#11719175	2
#11718894	2
#11718683	2
#11718682	2
#11718681	2
#11718670	2
#11718669	2
#11718668	2
#11718659	2
C₂×C₄⋊#1055	2
#3076	13