Subgroups — #13123823

Order 1664.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#529	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	2	yes	maximal, complemented
3	#11718673	C₂	1	2	yes	maximal, complemented
4	#11718672	C₂	1	2	yes	maximal, complemented
5	#11718672	C₂	1	2	yes	maximal, complemented
6	#11718661	C₂	1	2	yes	maximal, complemented
7	#11718658	C₂	1	2	yes	maximal, complemented
8	#11718587	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#126	C₄	1	2	yes	—
11	C₁₃×#125	C₄	1	1	yes	characteristic
12	C₁₃×#124	C₄	1	1	yes	characteristic
13	C₁₃×#123	C₄	1	2	yes	—
14	C₁₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
15	C₁₃×#121	C₄	1	1	yes	characteristic
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	2	yes	—
19	C₁₃×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×#6099	C₂²	1	2	yes	—
22	C₂×#6099	C₂²	1	2	yes	—
23	C₂²×#1055	C₂²	1	2	yes	—
24	C₂²⋊#1055	C₂²	1	2	yes	—
25	C₂²⋊#1055	C₂²	1	2	yes	—
26	C₂²⋊#1055	C₂²	1	1	yes	characteristic
27	C₂²⋊#1055	C₂²	1	1	yes	characteristic
28	C₄⋊#1055	C₂²	1	2	yes	—
29	C₄⋊#1055	C₂²	1	2	yes	—
30	C₄⋊#1055	C₂²	1	2	yes	—
31	#88043	C₂²	1	2	yes	—
32	#88043	C₂²	1	2	yes	—
33	#88043	C₂²	1	2	yes	—
34	C₄×#1055	C₂²	1	1	yes	characteristic
35	C₄×#1055	C₂²	1	2	yes	—
36	C₄×#1055	C₂²	1	1	yes	characteristic
37	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×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	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃	C₂³	1	2	yes	—
60	#6099	—	2	4	no	—
61	#6099	—	2	4	no	—
62	C₂×#1055	—	2	4	no	—
63	C₂×#1055	C₂³	1	2	yes	—
64	C₂×#1055	C₂³	1	2	yes	—
65	C₂×#1055	C₂³	1	2	yes	—
66	C₂×#1055	C₂³	1	2	yes	—
67	#2967	—	13	1	no	maximal, Sylow
68	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	Q₈×C₁₃	—	4	1	no	—
71	D₄×C₁₃	—	4	1	no	—
72	D₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	4	1	no	—
77	C₂×C₄×C₁₃	—	4	1	no	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	#1055	—	2	2	no	—
88	#1055	—	2	2	no	—
89	#1055	—	2	2	no	—
90	#1055	#41	1	4	yes	—
91	#529	D₁₃	1	1	yes	characteristic
92	#446	—	13	2	no	—
93	#436	—	13	2	no	—
94	#435	—	13	2	no	—
95	#435	—	13	2	no	—
96	#433	—	13	2	no	—
97	#432	—	13	1	no	—
98	#405	—	13	2	no	—
99	C₄×M₁₆	—	13	1	no	—
100	C₂²×C₁₃	—	2	2	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	2	1	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	2	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	C₂×#41	1	2	yes	—
116	C₂×M₁₆	—	13	2	no	—
117	C₂×M₁₆	—	13	2	no	—
118	C₂²×C₈	—	13	2	no	—
119	#126	Dic₁₃	1	2	yes	—
120	#125	Dic₁₃	1	1	yes	characteristic
121	#124	Dic₁₃	1	1	yes	characteristic
122	#123	Dic₁₃	1	2	yes	—
123	C₂²⋊Q₈	Dic₁₃	1	1	yes	characteristic
124	#121	Dic₁₃	1	1	yes	characteristic
125	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
126	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
127	C₄×D₄	C₂×D₁₃	1	2	yes	—
128	#117	C₂×D₁₃	1	2	yes	—
129	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
130	C₄⋊C₈	—	13	2	no	—
131	C₄⋊C₈	—	13	2	no	—
132	C₄⋊C₈	—	13	2	no	—
133	C₂²⋊C₈	—	13	1	no	—
134	C₂²⋊C₈	—	13	1	no	—
135	C₂²⋊C₈	—	13	2	no	—
136	C₂²⋊C₈	—	13	2	no	—
137	#97	—	13	2	no	—
138	#97	—	13	2	no	—
139	#97	—	13	2	no	—
140	C₄×C₈	—	13	1	no	—
141	C₄×C₈	—	13	1	no	—
142	C₄×C₈	—	13	2	no	—
143	C₂×C₁₃	—	4	1	no	—
144	C₂×C₁₃	—	4	2	no	—
145	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
146	C₂×C₁₃	#568	1	1	yes	characteristic
147	C₂×C₁₃	#568	1	1	yes	characteristic
148	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
149	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
150	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
155	M₁₆	—	26	4	no	—
156	M₁₆	—	26	4	no	—
157	C₂×C₈	—	13	2	no	—
158	C₂×C₈	—	13	2	no	—
159	C₂×C₈	—	13	2	no	—
160	C₂×C₈	—	13	2	no	—
161	C₂×C₈	—	26	4	no	—
162	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
163	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
164	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
165	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
166	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
167	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
168	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
172	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
173	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
174	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
175	C₄²	C₂²×D₁₃	1	1	yes	characteristic
176	C₄²	C₂²×D₁₃	1	2	yes	—
177	C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₁₃	#2967	1	1	yes	characteristic, complemented, Sylow
179	C₂³	C₂²×Dic₁₃	1	2	yes	—
180	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
181	Q₈	—	4	1	no	—
182	D₄	—	4	1	no	—
183	D₄	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
191	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
193	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
195	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
196	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
197	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
198	C₈	—	13	4	no	—
199	C₈	—	26	2	no	—
200	C₈	—	26	2	no	—
201	C₈	—	26	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	#41×D₁₃	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	4	2	no	—
220	C₂	#11719524	1	1	yes	characteristic
221	C₂	#11719524	1	1	yes	characteristic
222	C₂	#41×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#529	C₂
prime radical	C₁₃×#121	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#529	D₁₃
hypocenter	C₁₃	#2967
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
3. #11718673	C₂	semidirect
4. #11718672	C₂	semidirect
5. #11718672	C₂	semidirect
6. #11718661	C₂	semidirect
7. #11718658	C₂	semidirect
8. #11718587	C₂	semidirect
9. C₄×#6099	C₂	semidirect
178. C₁₃	#2967	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#529	2
#11718685	2
#11718673	2
#11718672	2
#11718661	2
#11718658	2
#11718587	2
C₄×#6099	2
#2967	13