Subgroups — #13123842

Order 1664.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 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	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718501	C₂	1	2	yes	maximal, complemented
5	#11718499	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718499	C₂	1	2	yes	maximal, complemented
7	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718369	C₂	1	2	yes	maximal, complemented
9	#11718367	C₂	1	2	yes	maximal, complemented
10	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₅₂	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	1	yes	characteristic
16	C₂×#6094	C₂²	1	2	yes	—
17	C₂×#6094	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₃	C₂²	1	2	yes	—
19	#88152	C₄	1	2	yes	—
20	#88149	C₄	1	1	yes	characteristic
21	#88148	C₄	1	2	yes	—
22	#88144	C₄	1	1	yes	characteristic
23	C₄×D₅₂	C₂²	1	1	yes	characteristic
24	C₄×D₅₂	C₂²	1	2	yes	—
25	#88126	C₂²	1	2	yes	—
26	C₄²×D₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₃×#97	C₂²	1	2	yes	—
30	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
31	#88060	C₂²	1	2	yes	—
32	#88060	C₂²	1	2	yes	—
33	#88060	C₂²	1	2	yes	—
34	#88055	C₂²	1	1	yes	characteristic
35	#88055	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃	C₂²	1	2	yes	—
37	C₄⋊#1055	C₂²	1	1	yes	characteristic
38	C₄⋊#1055	C₂²	1	2	yes	—
39	C₄×#1055	C₂²	1	1	yes	characteristic
40	C₄×#1055	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
43	C₂×D₅₂	C₂×C₄	1	2	yes	—
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	2	yes	—
68	C₂×#1055	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	#6094	—	2	2	no	—
71	#6094	—	2	2	no	—
72	#6094	—	2	4	no	—
73	C₈×D₁₃	—	2	4	no	—
74	#2969	—	13	1	no	maximal, Sylow
75	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
77	Q₈×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂⁴	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	1	yes	characteristic
86	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
87	D₅₂	—	2	2	no	—
88	D₅₂	—	4	2	no	—
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	—	2	2	no	—
91	C₄×D₁₃	—	2	1	no	—
92	C₄×D₁₃	—	2	1	no	—
93	C₄×D₁₃	—	2	2	no	—
94	C₄×D₁₃	—	4	1	no	—
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	#41	1	2	yes	—
98	#1055	—	2	2	no	—
99	#1055	—	2	1	no	—
100	#1055	#41	1	2	yes	—
101	#529	—	13	1	no	—
102	#446	D₁₃	1	1	yes	characteristic, complemented
103	C₈×Q₈	—	13	1	no	—
104	#436	—	13	1	no	—
105	#436	—	13	2	no	—
106	#435	—	13	2	no	—
107	C₈×D₄	—	13	1	no	—
108	#433	—	13	2	no	—
109	#431	—	13	1	no	—
110	#405	—	13	2	no	—
111	C₄×M₁₆	—	13	1	no	—
112	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×D₁₃	—	2	2	no	—
114	C₂×D₁₃	—	2	1	no	—
115	C₂×D₁₃	—	2	1	no	—
116	C₂×D₁₃	—	2	2	no	—
117	C₂×D₁₃	—	2	1	no	—
118	C₂×D₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	#131	1	2	yes	—
123	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	—	2	1	no	—
128	Dic₁₃	—	2	1	no	—
129	C₂×M₁₆	—	13	1	no	—
130	C₂×M₁₆	—	13	1	no	—
131	C₂×M₁₆	—	13	2	no	—
132	C₂²×C₈	—	13	2	no	—
133	#126	—	13	2	no	—
134	#125	—	13	1	no	—
135	#124	—	13	1	no	—
136	#123	—	13	2	no	—
137	C₂²⋊Q₈	—	13	1	no	—
138	#121	—	13	1	no	—
139	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
140	C₄×D₄	—	13	1	no	—
141	C₄×D₄	—	13	2	no	—
142	#117	—	13	2	no	—
143	C₂×C₄²	—	13	1	no	—
144	C₄⋊C₈	—	13	1	no	—
145	C₄⋊C₈	—	13	2	no	—
146	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
148	C₂²⋊C₈	—	13	2	no	—
149	C₂²⋊C₈	—	13	2	no	—
150	C₂²⋊C₈	—	13	2	no	—
151	#97	—	13	1	no	—
152	#97	—	13	1	no	—
153	#97	C₂×D₁₃	1	2	yes	—
154	C₄×C₈	—	13	2	no	—
155	C₄×C₈	—	13	1	no	—
156	C₄×C₈	—	13	2	no	—
157	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
158	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
159	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
160	C₂×C₁₃	#568	1	1	yes	characteristic
161	D₁₃	—	4	1	no	—
162	D₁₃	—	4	2	no	—
163	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
164	C₂×D₄	—	13	1	no	—
165	C₂×D₄	—	13	2	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	13	2	no	—
169	C₂²×C₄	—	13	1	no	—
170	M₁₆	—	26	2	no	—
171	M₁₆	—	26	4	no	—
172	M₁₆	—	26	2	no	—
173	C₂×C₈	—	13	1	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	1	no	—
176	C₂×C₈	—	26	4	no	—
177	C₂×C₈	C₂²×D₁₃	1	2	yes	—
178	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₄⋊C₄	—	13	2	no	—
181	C₄⋊C₄	—	13	2	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	—	13	2	no	—
184	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
185	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
186	C₂²⋊C₄	—	13	2	no	—
187	C₂²⋊C₄	—	13	2	no	—
188	C₂²⋊C₄	—	13	2	no	—
189	C₂²⋊C₄	—	13	2	no	—
190	C₂²⋊C₄	—	13	2	no	—
191	C₄²	—	13	1	no	—
192	C₄²	—	13	1	no	—
193	C₄²	—	13	1	no	—
194	C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄²	C₂²×D₁₃	1	2	yes	—
196	C₁₃	#2969	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	13	2	no	—
198	C₂³	—	13	1	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	26	2	no	—
201	D₄	—	52	2	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	1	no	—
204	C₂×C₄	—	13	1	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	1	no	—
209	C₂×C₄	—	26	1	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	—	52	1	no	—
212	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	13	2	no	—
220	C₈	—	26	2	no	—
221	C₈	—	26	1	no	—
222	C₈	#6131	1	2	yes	—
223	C₂²	—	26	1	no	—
224	C₂²	—	26	2	no	—
225	C₂²	—	26	1	no	—
226	C₂²	—	26	2	no	—
227	C₂²	—	26	2	no	—
228	C₂²	—	26	1	no	—
229	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	26	1	no	—
234	C₄	—	26	1	no	—
235	C₄	—	26	1	no	—
236	C₄	—	26	2	no	—
237	C₄	#88161	1	2	yes	—
238	C₄	C₂×#6131	1	1	yes	characteristic
239	C₄	#41×D₁₃	1	1	yes	characteristic
240	C₂	—	52	1	no	—
241	C₂	—	52	2	no	—
242	C₂	C₄×#6131	1	1	yes	characteristic
243	C₂	C₂×#88161	1	1	yes	characteristic
244	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#446	C₂
prime radical	#88149	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#446	D₁₃
hypocenter	C₁₃	#2969
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₁₃×#446	C₂	semidirect
3. Q₈×#1055	C₂	semidirect
4. #11718501	C₂	semidirect
5. #11718499	C₂	semidirect
6. #11718499	C₂	semidirect
7. #11718494	C₂	semidirect
8. #11718369	C₂	semidirect
9. #11718367	C₂	semidirect
10. C₄×#6094	C₂	semidirect
11. C₈×D₅₂	C₂	semidirect
102. #446	D₁₃	semidirect
196. C₁₃	#2969	semidirect

Maximal subgroups

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

subgroup	index
#11719285	2
C₁₃×#446	2
Q₈×#1055	2
#11718501	2
#11718499	2
#11718494	2
#11718369	2
#11718367	2
C₄×#6094	2
C₈×D₅₂	2
#2969	13