Subgroups — #13124626

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×#6127	C₂	1	1	yes	maximal, characteristic
2	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718774	C₂	1	2	yes	maximal
5	#11718767	C₂	1	2	yes	maximal, complemented
6	#11718512	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718511	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718508	C₂	1	2	yes	maximal, complemented
9	#11718507	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718503	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718490	C₂	1	2	yes	maximal, complemented
12	C₂×#6127	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₃	—	2	1	no	—
16	C₄×C₂²⋊D₁₃	—	2	1	no	—
17	#88180	C₂²	1	2	yes	—
18	#88180	C₂²	1	1	yes	characteristic
19	#88178	—	2	1	no	—
20	#88178	—	2	1	no	—
21	C₄×D₅₂	C₂²	1	1	yes	characteristic
22	C₄×D₅₂	C₂²	1	1	yes	characteristic
23	#88126	—	2	1	no	—
24	C₄²×D₁₃	—	2	1	no	—
25	#88124	C₂²	1	2	yes	—
26	#88123	C₂²	1	1	yes	characteristic
27	#88123	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆	—	2	1	no	—
29	C₄×Dic₂₆	—	2	1	no	—
30	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	#88061	C₂²	1	2	yes	—
36	#88061	C₂²	1	2	yes	—
37	#88058	C₂²	1	2	yes	—
38	#88058	C₂²	1	2	yes	—
39	#88057	C₂²	1	2	yes	—
40	#88057	C₂²	1	2	yes	—
41	#88056	C₂²	1	2	yes	—
42	#88056	C₂²	1	2	yes	—
43	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃	—	2	1	no	—
47	#6127	—	2	2	no	—
48	#6127	Q₈	1	4	yes	complemented
49	C₂×D₅₂	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₃	—	2	1	no	—
51	C₂×C₄×D₁₃	—	2	1	no	—
52	C₂×C₄×D₁₃	—	2	1	no	—
53	C₂×Dic₂₆	—	2	1	no	—
54	C₂×Dic₂₆	—	2	1	no	—
55	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃	—	2	4	no	—
57	C₂×C₈×C₁₃	C₂³	1	2	yes	—
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₄²×C₁₃	D₄	1	1	yes	characteristic
60	C₄²×C₁₃	D₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₃	—	2	2	no	—
64	C₂²⋊Dic₁₃	—	2	1	no	—
65	#6103	—	2	1	no	—
66	#6103	—	2	1	no	—
67	C₄⋊Dic₁₃	—	2	1	no	—
68	C₄⋊Dic₁₃	—	2	1	no	—
69	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₃	C₂³	1	2	yes	—
72	C₄⋊Dic₁₃	C₂³	1	2	yes	—
73	#6101	—	2	2	no	—
74	#6101	—	2	2	no	—
75	#6101	—	2	1	no	—
76	#6101	—	2	1	no	—
77	C₄×Dic₁₃	—	2	2	no	—
78	C₄×Dic₁₃	—	2	1	no	—
79	C₄×Dic₁₃	—	2	1	no	—
80	C₄×Dic₁₃	—	2	1	no	—
81	#3076	—	13	1	no	maximal, Sylow
82	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	4	1	no	—
86	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₁₃	#41	1	2	yes	—
93	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₁₃	—	2	4	no	—
95	C₂×Dic₁₃	—	2	1	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	D₅₂	—	2	1	no	—
101	D₅₂	C₂×Q₈	1	2	yes	—
102	C₄×D₁₃	—	2	4	no	—
103	C₄×D₁₃	—	4	2	no	—
104	C₄×D₁₃	—	4	2	no	—
105	Dic₂₆	—	2	1	no	—
106	Dic₂₆	—	4	2	no	—
107	Dic₂₆	—	4	2	no	—
108	Dic₂₆	C₂×Q₈	1	2	yes	—
109	C₈×C₁₃	—	2	2	no	—
110	C₈×C₁₃	—	2	2	no	—
111	#533	—	13	1	no	—
112	C₄×#41	—	13	1	no	—
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	D₁₃	1	1	yes	characteristic, complemented
121	#416	—	13	2	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
125	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×D₁₃	—	2	2	no	—
127	C₂×D₁₃	—	4	1	no	—
128	C₂×D₁₃	—	4	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
133	C₄×C₁₃	#135	1	2	yes	—
134	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
135	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
136	Dic₁₃	—	2	2	no	—
137	Dic₁₃	—	4	2	no	—
138	Dic₁₃	—	4	1	no	—
139	Dic₁₃	—	4	2	no	—
140	Dic₁₃	—	4	1	no	—
141	C₂×#41	—	13	1	no	—
142	C₂²×C₈	C₂×D₁₃	1	2	yes	—
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	—	13	1	no	—
149	C₄×Q₈	—	13	1	no	—
150	C₄×Q₈	—	26	1	no	—
151	C₄×Q₈	—	26	1	no	—
152	C₄×D₄	—	13	1	no	—
153	C₄×D₄	—	13	1	no	—
154	C₄×D₄	—	26	1	no	—
155	C₄×D₄	—	26	1	no	—
156	#117	—	13	1	no	—
157	#117	—	13	2	no	—
158	#117	—	26	1	no	—
159	C₂×C₄²	—	26	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	C₂×D₁₃	1	1	yes	characteristic
166	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
167	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
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	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
175	C₂×C₁₃	#575	1	1	yes	characteristic
176	D₁₃	—	4	2	no	—
177	#41	—	13	4	no	—
178	#41	—	26	2	no	—
179	C₂×Q₈	—	13	1	no	—
180	C₂×Q₈	—	26	1	no	—
181	C₂×Q₈	—	26	1	no	—
182	C₂×D₄	—	13	1	no	—
183	C₂×D₄	—	26	1	no	—
184	C₂²×C₄	—	26	1	no	—
185	C₂²×C₄	—	26	1	no	—
186	C₂²×C₄	—	26	1	no	—
187	C₂²×C₄	D₅₂	1	1	yes	characteristic
188	C₂²×C₄	D₅₂	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	C₂²×D₁₃	1	2	yes	—
192	C₂×C₈	C₂²×D₁₃	1	2	yes	—
193	C₄⋊C₄	—	13	2	no	—
194	C₄⋊C₄	—	13	2	no	—
195	C₄⋊C₄	—	13	1	no	—
196	C₄⋊C₄	—	13	1	no	—
197	C₄⋊C₄	—	26	1	no	—
198	C₄⋊C₄	—	26	1	no	—
199	C₄⋊C₄	—	26	2	no	—
200	C₄⋊C₄	—	26	2	no	—
201	C₄⋊C₄	—	26	1	no	—
202	C₄⋊C₄	—	26	1	no	—
203	C₂²⋊C₄	—	26	1	no	—
204	C₂²⋊C₄	—	26	2	no	—
205	C₂²⋊C₄	—	26	1	no	—
206	C₂²⋊C₄	—	26	1	no	—
207	C₄²	—	26	1	no	—
208	C₄²	—	26	1	no	—
209	C₄²	—	26	1	no	—
210	C₄²	—	26	2	no	—
211	C₄²	D₅₂	1	1	yes	characteristic
212	C₄²	D₅₂	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	—	26	1	no	—
217	C₂³	C₂×D₅₂	1	1	yes	characteristic
218	Q₈	—	13	2	no	—
219	Q₈	—	26	1	no	—
220	Q₈	—	52	2	no	—
221	Q₈	—	52	2	no	—
222	D₄	—	13	2	no	—
223	D₄	—	26	4	no	—
224	D₄	—	26	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	26	1	no	—
228	C₂×C₄	—	26	4	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	—	26	1	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	52	2	no	—
234	C₂×C₄	—	52	2	no	—
235	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
236	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
237	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
238	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
239	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
240	C₂×C₄	#6129	1	2	yes	—
241	C₂×C₄	Q₈×D₁₃	1	2	yes	—
242	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	26	2	no	—
248	C₂²	—	52	1	no	—
249	C₂²	—	52	1	no	—
250	C₂²	#88151	1	2	yes	—
251	C₂²	C₂²×D₅₂	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	26	2	no	—
256	C₄	—	52	1	no	—
257	C₄	—	52	2	no	—
258	C₄	—	52	1	no	—
259	C₄	—	52	2	no	—
260	C₄	#88151	1	2	yes	—
261	C₄	#88158	1	2	yes	—
262	C₄	C₂×#6129	1	1	yes	characteristic
263	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	52	2	no	—
266	C₂	C₂×#88151	1	1	yes	characteristic
267	C₂	C₂×#88158	1	1	yes	characteristic
268	C₂	#11719453	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₈	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#6127	C₂²
hypercenter	C₂×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#3076
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #11719175	C₂	semidirect
3. C₂×C₁₃×C₄⋊C₈	C₂	semidirect
5. #11718767	C₂	semidirect
6. #11718512	C₂	semidirect
7. #11718511	C₂	semidirect
8. #11718508	C₂	semidirect
9. #11718507	C₂	semidirect
10. #11718503	C₂	semidirect
11. #11718490	C₂	semidirect
48. #6127	Q₈	semidirect
120. C₂×C₄⋊C₈	D₁₃	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×#6127	2
#11719175	2
C₂×C₁₃×C₄⋊C₈	2
#11718774	2
#11718767	2
#11718512	2
#11718511	2
#11718508	2
#11718507	2
#11718503	2
#11718490	2
#3076	13