Subgroups — #13123896

Order 1664.

402 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719496	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718868	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718789	C₂	1	2	yes	maximal, complemented
5	#11718788	C₂	1	2	yes	maximal, complemented
6	#11718390	C₂	1	2	yes	maximal, complemented
7	#11718389	C₂	1	2	yes	maximal, complemented
8	#11718388	C₂	1	2	yes	maximal, complemented
9	#11718381	C₂	1	2	yes	maximal, complemented
10	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6127	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
13	#88207	C₄	1	2	yes	—
14	#88184	C₄	1	1	yes	characteristic
15	#88184	C₄	1	1	yes	characteristic
16	#88183	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
18	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
19	#88180	C₂²	1	1	yes	characteristic
20	#88178	C₄	1	1	yes	characteristic
21	#88178	C₄	1	1	yes	characteristic
22	C₂×#6099	C₂²	1	2	yes	—
23	C₂×#6094	C₂²	1	2	yes	complemented
24	C₂×C₈×D₁₃	C₂²	1	2	yes	complemented
25	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²⋊#1055	C₂²	1	1	yes	characteristic
29	C₂²⋊#1055	C₂²	1	1	yes	characteristic
30	C₂²⋊#1055	C₂²	1	2	yes	—
31	#88060	C₂²	1	2	yes	complemented
32	#88060	C₂²	1	2	yes	complemented
33	#88055	C₂²	1	2	yes	complemented
34	#88054	C₂²	1	2	yes	complemented
35	#88054	C₂²	1	2	yes	complemented
36	C₈×Dic₁₃	C₂²	1	2	yes	complemented
37	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	4	2	no	—
39	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
48	#6127	—	4	2	no	—
49	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃	C₂³	1	2	yes	—
51	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
52	C₁₃×M₁₆	—	4	2	no	—
53	C₂×C₈×C₁₃	C₂³	1	2	yes	—
54	C₂×C₈×C₁₃	C₂³	1	2	yes	—
55	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₃	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	1	yes	characteristic
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	C₂³	1	2	yes	—
65	#6099	—	4	2	no	—
66	C₂×#1055	C₂³	1	2	yes	—
67	C₂×#1055	C₂³	1	2	yes	—
68	#6094	—	4	2	no	—
69	C₈×D₁₃	—	4	2	no	—
70	#2978	—	13	1	no	maximal, Sylow
71	C₂³×C₁₃	—	4	1	no	—
72	C₂³×C₁₃	—	4	1	no	—
73	C₂³×C₁₃	—	4	1	no	—
74	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	4	1	no	—
82	C₂×C₄×C₁₃	—	4	1	no	—
83	C₂×C₄×C₁₃	—	4	1	no	—
84	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₁₃	—	4	2	no	—
89	C₂²⋊D₁₃	—	4	2	no	—
90	C₂²⋊D₁₃	—	4	2	no	—
91	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
93	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
94	D₅₂	—	4	1	no	—
95	C₄×D₁₃	—	2	2	no	—
96	C₄×D₁₃	—	2	2	no	—
97	Dic₂₆	—	4	1	no	—
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	—	2	2	no	—
100	#1055	—	2	2	no	—
101	#1055	—	2	2	no	—
102	#525	—	13	1	no	—
103	#436	—	13	2	no	—
104	#436	—	13	2	no	—
105	#435	—	13	2	no	—
106	#435	—	13	2	no	—
107	#433	—	13	2	no	—
108	#408	—	13	2	no	—
109	#407	—	13	1	no	—
110	#407	D₁₃	1	1	yes	characteristic, complemented
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	—	2	1	no	—
116	C₂²×C₁₃	—	2	1	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	4	4	no	—
121	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₁₃	—	2	2	no	—
123	C₂×D₁₃	—	2	2	no	—
124	C₂×D₁₃	—	2	2	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
129	Dic₁₃	—	2	2	no	—
130	Dic₁₃	—	2	2	no	—
131	Dic₁₃	—	2	2	no	—
132	C₂×#41	—	13	1	no	—
133	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
134	C₂×M₁₆	—	13	2	no	—
135	C₂×M₁₆	—	13	2	no	—
136	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
137	C₂²×C₈	—	13	2	no	—
138	#123	—	13	2	no	—
139	C₂²⋊Q₈	—	13	1	no	—
140	C₂²⋊Q₈	—	13	1	no	—
141	#121	—	13	1	no	—
142	#121	—	13	1	no	—
143	#120	—	13	2	no	—
144	C₄×D₄	—	13	2	no	—
145	C₄×D₄	—	13	2	no	—
146	#117	—	13	1	no	—
147	C₄⋊C₈	—	13	2	no	—
148	C₄⋊C₈	—	13	2	no	—
149	C₂²⋊C₈	—	13	2	no	—
150	C₂²⋊C₈	—	13	1	no	—
151	C₂²⋊C₈	—	13	1	no	—
152	C₂²⋊C₈	—	13	2	no	—
153	C₂²⋊C₈	—	13	2	no	—
154	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
155	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
157	#97	—	13	2	no	—
158	C₄×C₈	—	13	2	no	—
159	C₂×C₁₃	—	4	1	no	—
160	C₂×C₁₃	—	4	1	no	—
161	C₂×C₁₃	—	4	1	no	—
162	C₂×C₁₃	#518	1	1	yes	characteristic
163	C₂×C₁₃	#568	1	1	yes	characteristic
164	C₂×C₁₃	#568	1	1	yes	characteristic
165	D₁₃	—	4	2	no	—
166	C₂⁴	C₄×D₁₃	1	1	yes	characteristic
167	#41	—	52	2	no	—
168	C₂×Q₈	—	13	1	no	—
169	C₂×D₄	—	13	2	no	—
170	C₂×D₄	—	13	2	no	—
171	C₂×D₄	—	13	2	no	—
172	C₂×D₄	—	13	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	13	2	no	—
175	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	52	2	no	—
183	M₁₆	—	52	2	no	—
184	C₂×C₈	—	13	2	no	—
185	C₂×C₈	—	13	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₂×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂×C₈	C₂²×D₁₃	1	2	yes	—
189	C₄⋊C₄	—	13	1	no	—
190	C₄⋊C₄	—	13	1	no	—
191	C₄⋊C₄	—	13	2	no	—
192	C₄⋊C₄	—	13	2	no	—
193	C₂²⋊C₄	—	13	2	no	—
194	C₂²⋊C₄	—	13	2	no	—
195	C₂²⋊C₄	—	13	2	no	—
196	C₂²⋊C₄	—	13	2	no	—
197	C₂²⋊C₄	—	13	2	no	—
198	C₄²	—	13	2	no	—
199	C₁₃	#2978	1	1	yes	characteristic, complemented, Sylow
200	C₂³	—	4	1	no	—
201	C₂³	—	4	1	no	—
202	C₂³	—	4	1	no	—
203	C₂³	—	13	2	no	—
204	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
206	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
207	Q₈	—	52	1	no	—
208	D₄	—	52	1	no	—
209	D₄	—	52	2	no	—
210	D₄	—	52	2	no	—
211	D₄	—	52	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	13	2	no	—
219	C₂×C₄	—	13	2	no	—
220	C₂×C₄	—	13	2	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	26	2	no	—
230	C₈	—	26	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	26	2	no	—
242	C₂²	—	26	2	no	—
243	C₂²	—	26	2	no	—
244	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	26	2	no	—
249	C₄	—	26	2	no	—
250	C₄	—	26	2	no	—
251	C₄	#88251	1	2	yes	—
252	C₂	—	4	1	no	—
253	C₂	—	4	1	no	—
254	C₂	—	4	1	no	—
255	C₂	—	52	2	no	—
256	C₂	#11719191	1	1	yes	characteristic
257	C₂	#11719450	1	1	yes	characteristic
258	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#407	C₂
prime radical	#11719496	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#407	D₁₃
hypocenter	C₁₃	#2978
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. C₁₃×#407	C₂	semidirect
3. #11718868	C₂	semidirect
4. #11718789	C₂	semidirect
5. #11718788	C₂	semidirect
6. #11718390	C₂	semidirect
7. #11718389	C₂	semidirect
8. #11718388	C₂	semidirect
9. #11718381	C₂	semidirect
23. C₂×#6094	C₂²	semidirect
24. C₂×C₈×D₁₃	C₂²	semidirect
31. #88060	C₂²	semidirect
32. #88060	C₂²	semidirect
33. #88055	C₂²	semidirect
34. #88054	C₂²	semidirect
35. #88054	C₂²	semidirect
36. C₈×Dic₁₃	C₂²	semidirect
110. #407	D₁₃	semidirect
199. C₁₃	#2978	semidirect

Maximal subgroups

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

subgroup	index
#11719496	2
C₁₃×#407	2
#11718868	2
#11718789	2
#11718788	2
#11718390	2
#11718389	2
#11718388	2
#11718381	2
#2978	13