Subgroups — #13123542

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719533	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6094	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718787	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718783	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718763	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718391	C₂	1	4	yes	maximal, complemented
9	#11718390	C₂	1	4	yes	maximal, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#88202	—	2	4	no	—
15	#88200	C₄	1	2	yes	—
16	#88192	C₄	1	2	yes	—
17	#88191	C₄	1	2	yes	—
18	#88189	C₄	1	2	yes	—
19	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
20	#88180	C₂²	1	1	yes	characteristic
21	C₂×#6099	C₂²	1	1	yes	characteristic
22	C₂×#6099	C₂²	1	2	yes	complemented
23	C₂²×#1055	C₂²	1	1	yes	characteristic
24	C₂²×#1055	C₂²	1	2	yes	complemented
25	C₂×#6094	—	2	4	no	—
26	C₂×#6094	C₂²	1	2	yes	—
27	C₂×#6094	C₂²	1	2	yes	—
28	C₁₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
29	#88060	C₂²	1	4	yes	—
30	#88055	C₂²	1	2	yes	complemented
31	#88054	C₂²	1	4	yes	complemented
32	C₈×Dic₁₃	C₂²	1	2	yes	complemented
33	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
34	C₁₃×#41	—	2	2	no	—
35	C₁₃×#41	—	2	2	no	—
36	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
42	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃	—	2	2	no	—
44	C₂×C₄×D₁₃	—	2	2	no	—
45	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃	C₂³	1	2	yes	—
47	C₁₃×M₁₆	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₂³	1	2	yes	—
50	C₂×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
52	C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
53	#6103	C₂×C₄	1	4	yes	—
54	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
55	#6101	C₂×C₄	1	4	yes	—
56	C₄×Dic₁₃	C₂³	1	2	yes	—
57	#6099	—	2	2	no	—
58	#6099	—	2	4	no	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	D₄	1	4	yes	complemented
61	C₂×#1055	C₂³	1	2	yes	—
62	C₂×#1055	C₂³	1	1	yes	characteristic
63	C₂×#1055	C₂³	1	1	yes	characteristic
64	#6094	—	2	4	no	—
65	#6094	—	2	4	no	—
66	#2930	—	13	1	no	maximal, Sylow
67	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₁₃	—	2	2	no	—
70	C₂²×D₁₃	—	4	2	no	—
71	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
72	Q₈×C₁₃	—	2	2	no	—
73	D₄×C₁₃	—	2	4	no	—
74	D₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	#41	1	2	yes	—
81	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₁₃	—	4	4	no	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
85	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
86	C₄×D₁₃	—	2	4	no	—
87	C₄×D₁₃	—	2	2	no	—
88	C₄×D₁₃	—	2	2	no	—
89	C₈×C₁₃	—	2	2	no	—
90	C₈×C₁₃	—	2	2	no	—
91	#1055	—	2	2	no	—
92	#1055	C₂×D₄	1	2	yes	—
93	#1055	C₂×D₄	1	2	yes	—
94	C₂×#131	—	13	1	no	—
95	C₂²×M₁₆	—	13	1	no	—
96	#525	—	13	1	no	—
97	#436	—	13	4	no	—
98	#435	—	13	4	no	—
99	#424	—	13	1	no	—
100	#408	D₁₃	1	1	yes	characteristic, complemented
101	#407	—	13	1	no	—
102	#405	—	13	1	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	C₄×D₄	1	2	yes	—
107	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₁₃	—	2	2	no	—
109	C₂×D₁₃	—	2	2	no	—
110	C₂×D₁₃	—	2	2	no	—
111	C₂×D₁₃	—	4	4	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
116	Dic₁₃	—	2	2	no	—
117	Dic₁₃	—	2	4	no	—
118	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
119	C₂³×C₄	—	13	1	no	—
120	#131	—	26	4	no	—
121	C₂×M₁₆	—	13	1	no	—
122	C₂×M₁₆	—	13	2	no	—
123	C₂×M₁₆	—	13	2	no	—
124	C₂×M₁₆	—	13	2	no	—
125	C₂×M₁₆	—	26	4	no	—
126	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	2	no	—
129	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
130	#123	—	13	2	no	—
131	C₂²⋊Q₈	—	13	2	no	—
132	#121	—	13	2	no	—
133	#120	—	13	2	no	—
134	C₄×D₄	—	13	4	no	—
135	#117	—	13	1	no	—
136	C₄⋊C₈	—	13	4	no	—
137	C₂²⋊C₈	—	13	4	no	—
138	C₂²⋊C₈	C₂×D₁₃	1	4	yes	complemented
139	#97	—	13	2	no	—
140	C₄×C₈	—	13	2	no	—
141	C₂×C₁₃	—	2	2	no	—
142	C₂×C₁₃	—	4	2	no	—
143	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂×C₁₃	#568	1	2	yes	—
145	D₁₃	—	4	2	no	—
146	C₂⁴	—	13	1	no	—
147	#41	—	2	2	no	—
148	#41	—	2	2	no	—
149	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
150	C₂×D₄	—	13	4	no	—
151	C₂×D₄	C₄×D₁₃	1	2	yes	—
152	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
153	C₂²×C₄	—	13	1	no	—
154	C₂²×C₄	—	13	2	no	—
155	C₂²×C₄	—	13	2	no	—
156	C₂²×C₄	—	26	2	no	—
157	C₂²×C₄	—	26	2	no	—
158	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
160	M₁₆	—	2	2	no	—
161	M₁₆	—	26	4	no	—
162	M₁₆	—	26	4	no	—
163	M₁₆	—	26	4	no	—
164	M₁₆	—	26	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	13	4	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	26	4	no	—
171	C₂×C₈	C₂²×D₁₃	1	2	yes	—
172	C₂×C₈	C₂²×D₁₃	1	2	yes	—
173	C₄⋊C₄	—	13	2	no	—
174	C₄⋊C₄	—	13	4	no	—
175	C₂²⋊C₄	—	13	2	no	—
176	C₂²⋊C₄	—	13	4	no	—
177	C₂²⋊C₄	—	13	4	no	—
178	C₄²	—	13	2	no	—
179	C₁₃	#2930	1	1	yes	characteristic, complemented, Sylow
180	C₂³	—	13	2	no	—
181	C₂³	—	26	2	no	—
182	C₂³	—	52	2	no	—
183	C₂³	C₂×C₄×D₁₃	1	2	yes	—
184	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	52	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	13	4	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	26	2	no	—
194	C₂×C₄	—	26	4	no	—
195	C₂×C₄	—	26	2	no	—
196	C₂×C₄	—	26	2	no	—
197	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
199	C₂×C₄	D₄×D₁₃	1	2	yes	—
200	C₂×C₄	#6129	1	2	yes	—
201	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	13	2	no	—
205	C₈	—	13	2	no	—
206	C₈	—	26	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	26	2	no	—
211	C₂²	—	26	2	no	—
212	C₂²	—	26	2	no	—
213	C₂²	—	52	4	no	—
214	C₂²	#88135	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	26	2	no	—
218	C₄	—	26	4	no	—
219	C₄	#88135	1	2	yes	—
220	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
221	C₄	C₂×#6129	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	52	2	no	—
225	C₂	C₂×#88135	1	1	yes	characteristic
226	C₂	#11719455	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#408	C₂
prime radical	#11719533	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#408	D₁₃
hypocenter	C₁₃	#2930
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂²×#6094	C₂	semidirect
4. C₁₃×#408	C₂	semidirect
5. #11718787	C₂	semidirect
6. #11718783	C₂	semidirect
7. #11718763	C₂	semidirect
8. #11718391	C₂	semidirect
9. #11718390	C₂	semidirect
22. C₂×#6099	C₂²	semidirect
24. C₂²×#1055	C₂²	semidirect
28. C₁₃×C₂²⋊C₈	C₂²	semidirect
30. #88055	C₂²	semidirect
31. #88054	C₂²	semidirect
32. C₈×Dic₁₃	C₂²	semidirect
60. C₂×#1055	D₄	semidirect
100. #408	D₁₃	semidirect
138. C₂²⋊C₈	C₂×D₁₃	semidirect
179. C₁₃	#2930	semidirect

Maximal subgroups

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

subgroup	index
#11719533	2
C₂×#88202	2
C₂²×#6094	2
C₁₃×#408	2
#11718787	2
#11718783	2
#11718763	2
#11718391	2
#11718390	2
#2930	13