Subgroups — #13123575

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88155	C₂	1	2	yes	maximal, complemented
2	#11719253	C₂	1	1	yes	maximal, characteristic
3	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718772	C₂	1	2	yes	maximal, complemented
5	#11718673	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718495	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718389	C₂	1	2	yes	maximal, complemented
9	#11718388	C₂	1	2	yes	maximal, complemented
10	#11718349	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6127	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	#88194	C₄	1	1	yes	characteristic
16	#88192	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
18	C₂×#6099	C₂²	1	2	yes	—
19	#88155	—	2	8	no	—
20	C₂×#6094	C₂²	1	2	yes	—
21	C₂×#6094	C₂²	1	1	yes	characteristic
22	C₂×#6094	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₃	C₂²	1	2	yes	—
24	C₂×C₈×D₁₃	C₂²	1	2	yes	—
25	#88149	C₄	1	1	yes	characteristic
26	#88142	C₄	1	1	yes	characteristic
27	#88139	C₄	1	2	yes	—
28	#88138	C₄	1	2	yes	—
29	C₄×D₅₂	C₂²	1	1	yes	characteristic
30	C₄²×D₁₃	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
34	C₂²⋊#1055	C₂²	1	2	yes	—
35	#88060	C₂²	1	2	yes	—
36	#88060	C₂²	1	2	yes	—
37	#88055	C₂²	1	1	yes	characteristic
38	#88055	C₂²	1	1	yes	characteristic
39	#88054	C₂²	1	2	yes	—
40	C₄⋊#1055	C₂²	1	1	yes	characteristic
41	C₄×#1055	C₂²	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
47	#6127	—	2	4	no	—
48	#6127	—	2	4	no	—
49	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
50	C₂×D₅₂	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₃	C₂³	1	2	yes	—
54	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
56	C₂×C₈×C₁₃	—	2	4	no	—
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
64	#6103	C₂×C₄	1	2	yes	—
65	#6103	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
67	#6101	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
71	#6099	—	2	4	no	—
72	C₂×#1055	C₂³	1	1	yes	characteristic
73	C₂×#1055	C₂³	1	2	yes	—
74	C₂×#1055	C₂³	1	1	yes	characteristic
75	#6094	—	2	2	no	—
76	#6094	—	2	4	no	—
77	#6094	D₄	1	4	yes	complemented
78	C₈×D₁₃	—	2	4	no	—
79	C₈×D₁₃	—	2	4	no	—
80	#2935	—	13	1	no	maximal, Sylow
81	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
84	D₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂²⋊D₁₃	—	2	4	no	—
92	C₂²⋊D₁₃	—	2	4	no	—
93	C₂²⋊D₁₃	—	4	2	no	—
94	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
98	D₅₂	—	2	4	no	—
99	D₅₂	—	2	2	no	—
100	C₄×D₁₃	—	2	2	no	—
101	C₄×D₁₃	—	2	2	no	—
102	C₄×D₁₃	—	2	2	no	—
103	C₄×D₁₃	—	4	1	no	—
104	C₄×D₁₃	C₂×D₄	1	2	yes	—
105	C₄×D₁₃	#41	1	2	yes	—
106	Dic₂₆	—	2	4	no	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	1	no	—
109	C₈×C₁₃	C₂×D₄	1	2	yes	—
110	#1055	—	2	2	no	—
111	#1055	—	2	1	no	—
112	#1055	C₂×D₄	1	2	yes	—
113	C₂×#131	—	13	2	no	—
114	#532	—	13	1	no	—
115	#436	—	13	1	no	—
116	#436	—	13	1	no	—
117	#435	—	13	2	no	—
118	#435	—	13	2	no	—
119	C₈×D₄	—	13	1	no	—
120	C₈×D₄	D₁₃	1	1	yes	characteristic, complemented
121	C₂×C₄⋊C₈	—	13	1	no	—
122	#408	—	13	2	no	—
123	C₄×M₁₆	—	13	1	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×D₁₃	—	2	1	no	—
129	C₂×D₁₃	—	2	1	no	—
130	C₂×D₁₃	—	2	2	no	—
131	C₂×D₁₃	—	2	2	no	—
132	C₂×D₁₃	—	2	2	no	—
133	C₂×D₁₃	C₄×D₄	1	2	yes	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	#131	1	2	yes	complemented
137	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	—	2	1	no	—
141	Dic₁₃	—	2	1	no	—
142	Dic₁₃	C₄×D₄	1	2	yes	—
143	C₂×#41	—	13	2	no	—
144	#131	—	26	8	no	—
145	C₂×M₁₆	—	13	2	no	—
146	C₂×M₁₆	—	13	1	no	—
147	C₂×M₁₆	—	13	2	no	—
148	C₂×M₁₆	—	13	1	no	—
149	C₂²×C₈	—	13	2	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	C₂×D₁₃	1	2	yes	—
152	C₄⋊Q₈	—	13	1	no	—
153	#127	—	13	1	no	—
154	#124	—	13	2	no	—
155	#121	—	13	1	no	—
156	#121	—	13	1	no	—
157	#121	—	13	2	no	—
158	C₄×D₄	—	13	2	no	—
159	C₄×D₄	—	13	1	no	—
160	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
161	C₂×C₄²	—	13	1	no	—
162	C₄⋊C₈	—	13	2	no	—
163	C₄⋊C₈	—	13	1	no	—
164	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
165	C₂²⋊C₈	—	13	2	no	—
166	C₂²⋊C₈	—	13	2	no	—
167	C₂²⋊C₈	—	13	2	no	—
168	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
169	#97	—	13	1	no	—
170	#97	—	13	1	no	—
171	C₄×C₈	—	13	1	no	—
172	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
173	C₂×C₁₃	—	4	2	no	—
174	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
176	C₂×C₁₃	#568	1	1	yes	characteristic
177	D₁₃	—	2	2	no	—
178	D₁₃	—	4	2	no	—
179	#41	—	26	4	no	—
180	#41	—	26	4	no	—
181	C₂×Q₈	—	13	2	no	—
182	C₂×D₄	—	13	2	no	—
183	C₂×D₄	—	13	2	no	—
184	C₂×D₄	—	13	2	no	—
185	C₂×D₄	—	13	1	no	—
186	C₂×D₄	—	13	2	no	—
187	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
193	M₁₆	—	13	4	no	—
194	M₁₆	—	26	2	no	—
195	M₁₆	—	26	4	no	—
196	M₁₆	—	26	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	13	1	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	26	4	no	—
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₁₃	1	2	yes	—
205	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
206	C₄⋊C₄	—	13	2	no	—
207	C₄⋊C₄	—	13	1	no	—
208	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
209	C₂²⋊C₄	—	13	2	no	—
210	C₂²⋊C₄	—	13	2	no	—
211	C₂²⋊C₄	—	13	2	no	—
212	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
213	C₄²	—	13	1	no	—
214	C₄²	—	13	1	no	—
215	C₄²	—	13	1	no	—
216	C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	C₁₃	#2935	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	13	1	no	—
219	C₂³	—	13	2	no	—
220	C₂³	C₂×C₄×D₁₃	1	2	yes	—
221	Q₈	—	26	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	26	4	no	—
224	D₄	—	26	4	no	—
225	D₄	—	26	4	no	—
226	D₄	—	26	2	no	—
227	D₄	—	52	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	13	1	no	—
231	C₂×C₄	—	13	2	no	—
232	C₂×C₄	—	13	2	no	—
233	C₂×C₄	—	13	1	no	—
234	C₂×C₄	—	13	1	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	26	2	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	52	1	no	—
240	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	13	2	no	—
247	C₈	—	26	2	no	—
248	C₈	—	26	1	no	—
249	C₈	D₄×D₁₃	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	13	2	no	—
254	C₂²	—	26	1	no	—
255	C₂²	—	26	1	no	—
256	C₂²	—	26	2	no	—
257	C₂²	—	26	2	no	—
258	C₂²	—	26	2	no	—
259	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	13	2	no	—
263	C₄	—	26	1	no	—
264	C₄	—	26	1	no	—
265	C₄	—	26	2	no	—
266	C₄	#88155	1	2	yes	complemented
267	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
268	C₄	#41×D₁₃	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	26	2	no	—
271	C₂	—	52	2	no	—
272	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
273	C₂	C₂×#88155	1	1	yes	characteristic
274	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₈×D₄×C₁₃	C₂
prime radical	#11719253	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₈×D₄	D₁₃
hypocenter	C₁₃	#2935
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#88155	C₂	semidirect
3. C₈×D₄×C₁₃	C₂	semidirect
4. #11718772	C₂	semidirect
5. #11718673	C₂	semidirect
6. #11718495	C₂	semidirect
7. D₁₃×C₄⋊C₈	C₂	semidirect
8. #11718389	C₂	semidirect
9. #11718388	C₂	semidirect
10. #11718349	C₂	semidirect
11. C₄×#6094	C₂	semidirect
77. #6094	D₄	semidirect
120. C₈×D₄	D₁₃	semidirect
136. C₄×C₁₃	#131	semidirect
217. C₁₃	#2935	semidirect
266. C₄	#88155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88155	2
#11719253	2
C₈×D₄×C₁₃	2
#11718772	2
#11718673	2
#11718495	2
D₁₃×C₄⋊C₈	2
#11718389	2
#11718388	2
#11718349	2
C₄×#6094	2
#2935	13