Subgroups — #13123765

Order 1664.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719283	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718685	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718500	C₂	1	2	yes	maximal
5	#11718498	C₂	1	1	yes	maximal, characteristic
6	#11718492	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718489	C₂	1	2	yes	maximal, complemented
8	#11718350	C₂	1	2	yes	maximal
9	C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
10	C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
12	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	#88200	C₄	1	1	yes	characteristic
14	#88198	C₄	1	1	yes	characteristic
15	C₂×#6094	C₂²	1	2	yes	—
16	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
18	#88151	C₄	1	1	yes	characteristic
19	#88150	C₄	1	2	yes	—
20	#88143	C₄	1	2	yes	—
21	#88142	C₄	1	1	yes	characteristic
22	#88126	C₂²	1	2	yes	—
23	C₄²×D₁₃	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₆	C₂²	1	2	yes	—
25	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
27	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂²	1	2	yes	—
30	#88055	C₂²	1	2	yes	—
31	#88054	C₂²	1	2	yes	—
32	#88054	C₂²	1	2	yes	—
33	#88054	C₂²	1	2	yes	—
34	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃	C₂²	1	2	yes	—
37	C₄⋊#1055	C₂²	1	1	yes	characteristic
38	C₄⋊#1055	C₂²	1	2	yes	—
39	#88043	C₂²	1	2	yes	—
40	C₄×#1055	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂³	1	2	yes	—
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₄²×C₁₃	C₂³	1	2	yes	—
54	#6103	C₂×C₄	1	2	yes	—
55	#6103	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₃	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	#6101	C₂×C₄	1	2	yes	—
62	#6101	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂³	1	2	yes	—
67	C₂×#1055	C₂³	1	1	yes	characteristic
68	C₂×#1055	C₂³	1	2	yes	—
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	#6094	—	2	4	no	—
71	C₈×D₁₃	—	2	2	no	—
72	C₈×D₁₃	Q₈	1	4	yes	complemented
73	#2960	—	13	1	no	maximal, Sylow
74	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄×D₁₃	—	2	2	no	—
87	C₄×D₁₃	—	2	2	no	—
88	C₄×D₁₃	C₂×Q₈	1	2	yes	—
89	C₄×D₁₃	#41	1	2	yes	—
90	Dic₂₆	—	2	2	no	—
91	Dic₂₆	—	2	4	no	—
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	C₂×Q₈	1	2	yes	—
95	#1055	—	2	1	no	—
96	#1055	—	2	2	no	—
97	#1055	C₂×Q₈	1	2	yes	—
98	#533	—	13	1	no	—
99	#446	—	13	1	no	—
100	#446	—	13	2	no	—
101	#446	—	13	1	no	—
102	C₈×Q₈	—	13	2	no	—
103	C₈×Q₈	—	13	1	no	—
104	C₈×Q₈	D₁₃	1	1	yes	characteristic, complemented
105	#424	—	13	2	no	—
106	#423	—	13	1	no	—
107	#405	—	13	2	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×D₁₃	—	2	1	no	—
111	C₂×D₁₃	—	2	1	no	—
112	C₂×D₁₃	C₄×Q₈	1	2	yes	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	#131	1	2	yes	—
117	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	Dic₁₃	—	2	2	no	—
120	Dic₁₃	—	2	2	no	—
121	Dic₁₃	—	2	1	no	—
122	Dic₁₃	C₄×Q₈	1	2	yes	—
123	Dic₁₃	#131	1	2	yes	—
124	C₂×M₁₆	—	13	2	no	—
125	C₂²×C₈	—	13	1	no	—
126	C₂²×C₈	—	13	1	no	—
127	C₄⋊Q₈	—	13	1	no	—
128	C₄⋊Q₈	—	13	1	no	—
129	#125	—	13	2	no	—
130	C₂²⋊Q₈	—	13	1	no	—
131	C₂²⋊Q₈	—	13	2	no	—
132	C₂²⋊Q₈	—	13	1	no	—
133	C₄×Q₈	—	13	2	no	—
134	C₄×Q₈	—	13	1	no	—
135	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
136	#117	—	13	2	no	—
137	C₂×C₄²	—	13	1	no	—
138	C₄⋊C₈	—	13	2	no	—
139	C₄⋊C₈	—	13	2	no	—
140	C₄⋊C₈	—	13	2	no	—
141	C₄⋊C₈	—	13	1	no	—
142	C₄⋊C₈	—	13	2	no	—
143	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
145	#97	—	13	2	no	—
146	#97	—	13	2	no	—
147	C₄×C₈	—	13	1	no	—
148	C₄×C₈	—	13	1	no	—
149	C₄×C₈	—	13	2	no	—
150	C₄×C₈	—	13	1	no	—
151	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₄×C₈	C₂×D₁₃	1	2	yes	—
153	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
155	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
156	D₁₃	—	2	2	no	—
157	C₂×Q₈	—	13	2	no	—
158	C₂×Q₈	—	13	1	no	—
159	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	1	no	—
163	M₁₆	—	26	4	no	—
164	C₂×C₈	—	13	1	no	—
165	C₂×C₈	—	13	4	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	26	2	no	—
169	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₁₃	1	2	yes	—
171	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₄⋊C₄	—	13	2	no	—
173	C₄⋊C₄	—	13	2	no	—
174	C₄⋊C₄	—	13	2	no	—
175	C₄⋊C₄	—	13	2	no	—
176	C₄⋊C₄	—	13	2	no	—
177	C₄⋊C₄	—	13	2	no	—
178	C₄⋊C₄	—	13	1	no	—
179	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
180	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
181	C₂²⋊C₄	—	13	2	no	—
182	C₂²⋊C₄	—	13	2	no	—
183	C₄²	—	13	1	no	—
184	C₄²	—	13	1	no	—
185	C₄²	—	13	1	no	—
186	C₄²	—	13	2	no	—
187	C₄²	C₂²×D₁₃	1	1	yes	characteristic
188	C₄²	C₂²×D₁₃	1	2	yes	—
189	C₁₃	#2960	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	13	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	26	4	no	—
193	Q₈	—	26	2	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	1	no	—
199	C₂×C₄	—	13	2	no	—
200	C₂×C₄	—	13	2	no	—
201	C₂×C₄	—	26	2	no	—
202	C₂×C₄	—	26	2	no	—
203	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	13	2	no	—
211	C₈	—	26	1	no	—
212	C₈	—	26	2	no	—
213	C₈	Q₈×D₁₃	1	2	yes	—
214	C₂²	—	13	2	no	—
215	C₂²	—	26	1	no	—
216	C₂²	—	26	1	no	—
217	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	13	2	no	—
222	C₄	—	13	2	no	—
223	C₄	—	26	1	no	—
224	C₄	—	26	2	no	—
225	C₄	—	26	2	no	—
226	C₄	#88155	1	2	yes	—
227	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
228	C₄	#41×D₁₃	1	1	yes	characteristic
229	C₂	—	26	2	no	—
230	C₂	C₄×Q₈×D₁₃	1	1	yes	characteristic
231	C₂	C₂×#88155	1	1	yes	characteristic
232	C₂	D₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₈×Q₈×C₁₃	C₂
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₈×Q₈	D₁₃
hypocenter	C₁₃	#2960
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₁₃	C₂	semidirect
3. #11718685	C₂	semidirect
6. #11718492	C₂	semidirect
7. #11718489	C₂	semidirect
10. C₈×Dic₂₆	C₂	semidirect
11. C₈×Dic₂₆	C₂	semidirect
72. C₈×D₁₃	Q₈	semidirect
104. C₈×Q₈	D₁₃	semidirect
189. C₁₃	#2960	semidirect

Maximal subgroups

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

subgroup	index
#11719283	2
C₈×Q₈×C₁₃	2
#11718685	2
#11718500	2
#11718498	2
#11718492	2
#11718489	2
#11718350	2
C₄×C₈×D₁₃	2
C₈×Dic₂₆	2
#2960	13