Subgroups — #13124008

Order 1664.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719184	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718501	C₂	1	2	yes	maximal, complemented
5	#11718500	C₂	1	2	yes	maximal, complemented
6	#11718390	C₂	1	2	yes	maximal, complemented
7	#11718389	C₂	1	2	yes	maximal, complemented
8	#11718367	C₂	1	2	yes	maximal, complemented
9	#11718362	C₂	1	2	yes	maximal, complemented
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	#88184	C₄	1	1	yes	characteristic
12	#88183	C₄	1	2	yes	—
13	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
14	#88178	C₄	1	1	yes	characteristic
15	C₂×#6099	C₂²	1	1	yes	characteristic
16	C₂×#6099	C₂²	1	1	yes	characteristic
17	C₂×#6094	C₂²	1	2	yes	—
18	C₂×C₈×D₁₃	C₂²	1	2	yes	—
19	#88130	C₄	1	2	yes	—
20	#88129	C₄	1	1	yes	characteristic
21	C₄×D₅₂	C₂²	1	1	yes	characteristic
22	#88126	C₂²	1	2	yes	—
23	#88124	C₄	1	1	yes	characteristic
24	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₁₃×#97	C₂²	1	2	yes	—
29	#88060	C₂²	1	2	yes	—
30	#88060	C₂²	1	2	yes	—
31	#88055	C₂²	1	2	yes	—
32	#88054	C₂²	1	2	yes	—
33	#88054	C₂²	1	2	yes	—
34	C₈×Dic₁₃	C₂²	1	2	yes	—
35	C₄⋊#1055	C₂²	1	2	yes	—
36	C₄⋊#1055	C₂²	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
41	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃	C₂³	1	2	yes	—
43	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂³	1	2	yes	—
45	C₂×C₈×C₁₃	C₂³	1	2	yes	—
46	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
51	#6103	C₂×C₄	1	2	yes	—
52	#6103	C₂×C₄	1	2	yes	—
53	#6103	C₂×C₄	1	2	yes	—
54	#6103	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
57	#6101	C₂×C₄	1	2	yes	—
58	#6101	C₂×C₄	1	2	yes	—
59	#6101	C₂×C₄	1	2	yes	—
60	#6101	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₃	C₂³	1	2	yes	—
62	#6099	—	4	1	no	—
63	#6099	—	4	1	no	—
64	C₂×#1055	C₂³	1	2	yes	—
65	C₂×#1055	C₂³	1	2	yes	—
66	#6094	—	4	2	no	—
67	C₈×D₁₃	—	4	2	no	—
68	#2995	—	13	1	no	maximal, Sylow
69	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	4	1	no	—
74	C₂×C₄×C₁₃	—	4	1	no	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₁₃	—	4	2	no	—
82	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
85	D₅₂	—	4	1	no	—
86	C₄×D₁₃	—	2	2	no	—
87	C₄×D₁₃	—	2	2	no	—
88	Dic₂₆	—	4	1	no	—
89	C₈×C₁₃	—	2	2	no	—
90	C₈×C₁₃	—	2	2	no	—
91	#1055	—	2	2	no	—
92	#1055	—	2	2	no	—
93	#529	—	13	1	no	—
94	#446	—	13	2	no	—
95	#436	—	13	2	no	—
96	#435	—	13	2	no	—
97	#435	—	13	2	no	—
98	#433	—	13	2	no	—
99	#432	D₁₃	1	1	yes	characteristic, complemented
100	#424	—	13	2	no	—
101	#423	—	13	1	no	—
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₁₃	—	2	2	no	—
107	C₂×D₁₃	—	2	2	no	—
108	C₂×D₁₃	—	2	2	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	2	1	no	—
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
116	Dic₁₃	—	2	2	no	—
117	Dic₁₃	—	2	2	no	—
118	Dic₁₃	—	2	2	no	—
119	C₂×M₁₆	—	13	1	no	—
120	C₂×M₁₆	—	13	1	no	—
121	C₂×M₁₆	—	13	2	no	—
122	C₂²×C₈	—	13	2	no	—
123	#126	—	13	2	no	—
124	#125	—	13	1	no	—
125	#124	—	13	1	no	—
126	#123	—	13	2	no	—
127	C₂²⋊Q₈	—	13	1	no	—
128	#121	—	13	1	no	—
129	C₄×Q₈	—	13	1	no	—
130	C₄×D₄	—	13	2	no	—
131	C₄×D₄	—	13	1	no	—
132	#117	—	13	2	no	—
133	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
134	C₄⋊C₈	—	13	2	no	—
135	C₄⋊C₈	—	13	2	no	—
136	C₄⋊C₈	—	13	2	no	—
137	C₄⋊C₈	—	13	2	no	—
138	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂²⋊C₈	—	13	2	no	—
141	C₂²⋊C₈	—	13	2	no	—
142	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
143	#97	—	13	2	no	—
144	#97	C₂×D₁₃	1	2	yes	—
145	C₄×C₈	—	13	2	no	—
146	C₂×C₁₃	—	4	1	no	—
147	C₂×C₁₃	#520	1	1	yes	characteristic
148	C₂×C₁₃	#568	1	1	yes	characteristic
149	C₂×C₁₃	#568	1	1	yes	characteristic
150	D₁₃	—	4	2	no	—
151	C₂×Q₈	—	13	1	no	—
152	C₂×D₄	—	13	2	no	—
153	C₂×D₄	—	13	1	no	—
154	C₂²×C₄	—	13	2	no	—
155	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
158	M₁₆	—	52	2	no	—
159	M₁₆	—	52	1	no	—
160	M₁₆	—	52	1	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	52	2	no	—
164	C₂×C₈	C₂²×D₁₃	1	2	yes	—
165	C₂×C₈	C₂²×D₁₃	1	2	yes	—
166	C₄⋊C₄	—	13	2	no	—
167	C₄⋊C₄	—	13	2	no	—
168	C₄⋊C₄	—	13	1	no	—
169	C₄⋊C₄	—	13	2	no	—
170	C₄⋊C₄	—	13	1	no	—
171	C₄⋊C₄	—	13	2	no	—
172	C₂²⋊C₄	—	13	2	no	—
173	C₂²⋊C₄	—	13	2	no	—
174	C₂²⋊C₄	—	13	2	no	—
175	C₂²⋊C₄	—	13	2	no	—
176	C₂²⋊C₄	—	13	2	no	—
177	C₄²	—	13	2	no	—
178	C₄²	C₄×D₁₃	1	1	yes	characteristic
179	C₄²	C₄×D₁₃	1	1	yes	characteristic
180	C₄²	C₂²×D₁₃	1	1	yes	characteristic
181	C₄²	C₂²×D₁₃	1	1	yes	characteristic
182	C₁₃	#2995	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	13	2	no	—
184	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
185	Q₈	—	52	1	no	—
186	D₄	—	52	1	no	—
187	D₄	—	52	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	13	2	no	—
194	C₂×C₄	—	13	2	no	—
195	C₂×C₄	—	26	2	no	—
196	C₂×C₄	—	26	2	no	—
197	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	26	2	no	—
206	C₈	—	26	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	26	2	no	—
211	C₂²	—	26	2	no	—
212	C₂²	—	26	2	no	—
213	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	26	2	no	—
220	C₄	—	26	2	no	—
221	C₄	—	26	2	no	—
222	C₄	#88256	1	1	yes	characteristic
223	C₄	#88257	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	52	2	no	—
226	C₂	#11719228	1	1	yes	characteristic
227	C₂	#11719450	1	1	yes	characteristic
228	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#432	C₂
prime radical	#88184	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#432	D₁₃
hypocenter	C₁₃	#2995
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. C₁₃×#432	C₂	semidirect
3. #11718585	C₂	semidirect
4. #11718501	C₂	semidirect
5. #11718500	C₂	semidirect
6. #11718390	C₂	semidirect
7. #11718389	C₂	semidirect
8. #11718367	C₂	semidirect
9. #11718362	C₂	semidirect
99. #432	D₁₃	semidirect
182. C₁₃	#2995	semidirect

Maximal subgroups

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

subgroup	index
#11719184	2
C₁₃×#432	2
#11718585	2
#11718501	2
#11718500	2
#11718390	2
#11718389	2
#11718367	2
#11718362	2
#2995	13