Subgroups — #13123995

Order 1664.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719285	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718685	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718500	C₂	1	2	yes	maximal, complemented
5	#11718497	C₂	1	2	yes	maximal, complemented
6	#11718495	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718351	C₂	1	2	yes	maximal, complemented
9	#11718349	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₅₂	C₂	1	2	yes	maximal, complemented
11	#11718346	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	#88201	C₄	1	1	yes	characteristic
14	#88200	C₄	1	1	yes	characteristic
15	C₂×#6094	C₂²	1	2	yes	—
16	C₂×C₈×D₁₃	C₂²	1	2	yes	—
17	C₂×C₈×D₁₃	C₂²	1	2	yes	—
18	#88152	C₄	1	2	yes	—
19	#88149	C₄	1	1	yes	characteristic
20	#88148	C₄	1	2	yes	—
21	#88144	C₄	1	1	yes	characteristic
22	C₄×D₅₂	C₂²	1	1	yes	characteristic
23	C₄×D₅₂	C₂²	1	2	yes	—
24	#88126	C₂²	1	2	yes	—
25	C₄²×D₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂²	1	2	yes	—
30	#88060	C₂²	1	2	yes	—
31	#88060	C₂²	1	1	yes	characteristic
32	#88060	C₂²	1	1	yes	characteristic
33	#88060	C₂²	1	2	yes	—
34	#88054	C₂²	1	2	yes	—
35	#88054	C₂²	1	1	yes	characteristic
36	#88054	C₂²	1	1	yes	characteristic
37	C₄⋊#1055	C₂²	1	1	yes	characteristic
38	C₄⋊#1055	C₂²	1	2	yes	—
39	#88043	C₂²	1	2	yes	—
40	C₄×#1055	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₅₂	C₂×C₄	1	2	yes	—
43	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₃	C₂³	1	2	yes	—
48	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂³	1	2	yes	—
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₄²×C₁₃	C₂³	1	2	yes	—
55	#6103	C₂×C₄	1	2	yes	—
56	#6103	C₂×C₄	1	2	yes	—
57	#6103	C₂×C₄	1	2	yes	—
58	#6103	C₂×C₄	1	2	yes	—
59	#6103	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	#6101	C₂×C₄	1	2	yes	—
63	#6101	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
67	C₂×#1055	C₂³	1	1	yes	characteristic
68	C₂×#1055	C₂³	1	2	yes	—
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	#6094	—	4	2	no	—
71	C₈×D₁₃	—	4	2	no	—
72	C₈×D₁₃	—	4	2	no	—
73	#2994	—	13	1	no	maximal, Sylow
74	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
86	D₅₂	—	2	4	no	—
87	D₅₂	—	2	2	no	—
88	C₄×D₁₃	—	2	2	no	—
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	—	2	1	no	—
91	C₄×D₁₃	—	2	2	no	—
92	C₄×D₁₃	—	2	1	no	—
93	C₄×D₁₃	—	2	2	no	—
94	C₈×C₁₃	—	2	2	no	—
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	1	no	—
97	#1055	—	2	1	no	—
98	#1055	—	2	2	no	—
99	#1055	—	2	1	no	—
100	#529	—	13	1	no	—
101	#446	—	13	1	no	—
102	C₈×Q₈	D₁₃	1	1	yes	characteristic, complemented
103	#436	—	13	1	no	—
104	#435	—	13	2	no	—
105	C₈×D₄	—	13	1	no	—
106	C₈×D₄	—	13	2	no	—
107	#433	—	13	2	no	—
108	#431	—	13	1	no	—
109	#424	—	13	2	no	—
110	C₂×C₄⋊C₈	—	13	1	no	—
111	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₁₃	—	2	1	no	—
113	C₂×D₁₃	—	2	1	no	—
114	C₂×D₁₃	—	2	2	no	—
115	C₂×D₁₃	—	2	2	no	—
116	C₂×D₁₃	—	2	2	no	—
117	C₂×D₁₃	#131	1	2	yes	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	#131	1	2	yes	—
122	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
124	Dic₁₃	—	2	1	no	—
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	—	2	1	no	—
128	C₂×M₁₆	—	13	2	no	—
129	C₂²×C₈	—	13	2	no	—
130	C₂²×C₈	—	13	2	no	—
131	#126	—	13	2	no	—
132	#125	—	13	1	no	—
133	#124	—	13	1	no	—
134	#123	—	13	2	no	—
135	C₂²⋊Q₈	—	13	1	no	—
136	#121	—	13	1	no	—
137	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
138	C₄×D₄	—	13	2	no	—
139	C₄×D₄	—	13	1	no	—
140	#117	—	13	2	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₄⋊C₈	—	13	2	no	—
143	C₄⋊C₈	—	13	1	no	—
144	C₄⋊C₈	—	13	1	no	—
145	C₄⋊C₈	—	13	1	no	—
146	C₄⋊C₈	—	13	2	no	—
147	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
149	C₂²⋊C₈	—	13	1	no	—
150	C₂²⋊C₈	—	13	1	no	—
151	C₂²⋊C₈	—	13	2	no	—
152	C₂²⋊C₈	—	13	2	no	—
153	#97	—	13	2	no	—
154	C₄×C₈	—	13	1	no	—
155	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₄×C₈	C₂×D₁₃	1	2	yes	—
157	C₂×C₁₃	#520	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
159	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
160	D₁₃	—	2	2	no	—
161	D₁₃	—	4	2	no	—
162	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
163	C₂×D₄	—	13	2	no	—
164	C₂×D₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	13	2	no	—
169	M₁₆	—	52	2	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	52	2	no	—
174	C₂×C₈	—	52	2	no	—
175	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₁₃	1	2	yes	—
177	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊C₄	—	13	2	no	—
179	C₄⋊C₄	—	13	2	no	—
180	C₄⋊C₄	—	13	1	no	—
181	C₄⋊C₄	—	13	2	no	—
182	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
183	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
184	C₂²⋊C₄	—	13	2	no	—
185	C₂²⋊C₄	—	13	2	no	—
186	C₂²⋊C₄	—	13	2	no	—
187	C₂²⋊C₄	—	13	2	no	—
188	C₂²⋊C₄	—	13	2	no	—
189	C₄²	—	13	1	no	—
190	C₄²	—	13	1	no	—
191	C₄²	—	13	1	no	—
192	C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	C₄²	C₂²×D₁₃	1	2	yes	—
194	C₁₃	#2994	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	13	1	no	—
196	C₂³	—	13	2	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	26	4	no	—
199	D₄	—	26	2	no	—
200	C₂×C₄	—	13	1	no	—
201	C₂×C₄	—	13	1	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	2	no	—
204	C₂×C₄	—	26	2	no	—
205	C₂×C₄	—	26	2	no	—
206	C₂×C₄	—	26	1	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	1	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	26	1	no	—
219	C₈	—	26	1	no	—
220	C₈	—	26	2	no	—
221	C₂²	—	13	2	no	—
222	C₂²	—	26	1	no	—
223	C₂²	—	26	1	no	—
224	C₂²	—	26	2	no	—
225	C₂²	—	26	2	no	—
226	C₂²	—	26	2	no	—
227	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	26	1	no	—
232	C₄	—	26	1	no	—
233	C₄	—	26	1	no	—
234	C₄	—	26	2	no	—
235	C₄	#88155	1	2	yes	—
236	C₄	#88254	1	1	yes	characteristic
237	C₄	#88256	1	1	yes	characteristic
238	C₂	—	26	2	no	—
239	C₂	—	52	2	no	—
240	C₂	#11719279	1	1	yes	characteristic
241	C₂	C₂×#88155	1	1	yes	characteristic
242	C₂	D₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₈×Q₈×C₁₃	C₂
prime radical	#88149	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₈×Q₈	D₁₃
hypocenter	C₁₃	#2994
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₁₃	C₂	semidirect
3. #11718685	C₂	semidirect
4. #11718500	C₂	semidirect
5. #11718497	C₂	semidirect
6. #11718495	C₂	semidirect
7. D₁₃×C₄⋊C₈	C₂	semidirect
8. #11718351	C₂	semidirect
9. #11718349	C₂	semidirect
10. C₈×D₅₂	C₂	semidirect
11. #11718346	C₂	semidirect
102. C₈×Q₈	D₁₃	semidirect
194. C₁₃	#2994	semidirect

Maximal subgroups

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

subgroup	index
#11719285	2
C₈×Q₈×C₁₃	2
#11718685	2
#11718500	2
#11718497	2
#11718495	2
D₁₃×C₄⋊C₈	2
#11718351	2
#11718349	2
C₈×D₅₂	2
#11718346	2
#2994	13