Subgroups — #13123936

Order 1664.

330 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	#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	#11718501	C₂	1	2	yes	maximal
5	#11718494	C₂	1	1	yes	maximal, characteristic
6	#11718491	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718489	C₂	1	2	yes	maximal, complemented
8	#11718351	C₂	1	2	yes	maximal
9	#11718346	C₂	1	1	yes	maximal, characteristic
10	#11718340	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	#88151	C₄	1	1	yes	characteristic
16	#88150	C₄	1	2	yes	—
17	#88143	C₄	1	2	yes	—
18	#88142	C₄	1	1	yes	characteristic
19	#88126	C₂²	1	2	yes	—
20	C₄²×D₁₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₆	C₂²	1	2	yes	—
22	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂²	1	2	yes	—
26	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	#88060	C₂²	1	2	yes	—
28	#88060	C₂²	1	1	yes	characteristic
29	#88060	C₂²	1	1	yes	characteristic
30	#88055	C₂²	1	2	yes	—
31	#88054	C₂²	1	1	yes	characteristic
32	#88054	C₂²	1	2	yes	—
33	#88054	C₂²	1	1	yes	characteristic
34	#88054	C₂²	1	2	yes	—
35	C₈×Dic₁₃	C₂²	1	2	yes	—
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	1	yes	characteristic
69	C₂×#1055	C₂³	1	2	yes	—
70	#2986	—	13	1	no	maximal, Sylow
71	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₁₃	—	2	2	no	—
84	C₄×D₁₃	—	2	1	no	—
85	C₄×D₁₃	—	2	1	no	—
86	C₄×D₁₃	—	2	2	no	—
87	Dic₂₆	—	2	2	no	—
88	Dic₂₆	—	2	4	no	—
89	C₈×C₁₃	—	2	1	no	—
90	C₈×C₁₃	—	2	2	no	—
91	C₈×C₁₃	—	2	1	no	—
92	#1055	—	2	1	no	—
93	#1055	—	2	2	no	—
94	#1055	—	2	1	no	—
95	#533	—	13	1	no	—
96	#446	—	13	1	no	—
97	#446	—	13	2	no	—
98	#446	—	13	1	no	—
99	C₈×Q₈	—	13	1	no	—
100	C₈×Q₈	—	13	2	no	—
101	C₈×Q₈	D₁₃	1	1	yes	characteristic, complemented
102	#433	—	13	2	no	—
103	#433	—	13	2	no	—
104	#431	—	13	1	no	—
105	#431	—	13	1	no	—
106	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₁₃	—	2	1	no	—
108	C₂×D₁₃	—	2	1	no	—
109	C₂×D₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	#131	1	2	yes	—
114	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
116	Dic₁₃	—	2	1	no	—
117	Dic₁₃	—	2	2	no	—
118	Dic₁₃	—	2	2	no	—
119	Dic₁₃	—	2	1	no	—
120	Dic₁₃	#131	1	2	yes	—
121	C₄⋊Q₈	—	13	1	no	—
122	C₄⋊Q₈	—	13	1	no	—
123	#125	—	13	2	no	—
124	C₂²⋊Q₈	—	13	1	no	—
125	C₂²⋊Q₈	—	13	2	no	—
126	C₂²⋊Q₈	—	13	1	no	—
127	C₄×Q₈	—	13	2	no	—
128	C₄×Q₈	—	13	1	no	—
129	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
130	#117	—	13	2	no	—
131	C₂×C₄²	—	13	1	no	—
132	C₄⋊C₈	—	13	1	no	—
133	C₄⋊C₈	—	13	1	no	—
134	C₄⋊C₈	—	13	2	no	—
135	C₄⋊C₈	—	13	2	no	—
136	C₄⋊C₈	—	13	1	no	—
137	C₄⋊C₈	—	13	2	no	—
138	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
140	C₂²⋊C₈	—	13	1	no	—
141	C₂²⋊C₈	—	13	2	no	—
142	C₂²⋊C₈	—	13	1	no	—
143	#97	—	13	2	no	—
144	#97	—	13	2	no	—
145	C₄×C₈	—	13	2	no	—
146	C₄×C₈	—	13	2	no	—
147	C₄×C₈	—	13	1	no	—
148	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
149	C₄×C₈	C₂×D₁₃	1	2	yes	—
150	C₂×C₁₃	#519	1	1	yes	characteristic
151	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
152	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
153	D₁₃	—	4	1	no	—
154	C₂×Q₈	—	13	2	no	—
155	C₂×Q₈	—	13	1	no	—
156	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂×C₈	—	13	1	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	1	no	—
163	C₂×C₈	C₂²×D₁₃	1	2	yes	—
164	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	C₄⋊C₄	—	13	2	no	—
167	C₄⋊C₄	—	13	2	no	—
168	C₄⋊C₄	—	13	2	no	—
169	C₄⋊C₄	—	13	2	no	—
170	C₄⋊C₄	—	13	2	no	—
171	C₄⋊C₄	—	13	2	no	—
172	C₄⋊C₄	—	13	1	no	—
173	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
174	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
175	C₂²⋊C₄	—	13	2	no	—
176	C₂²⋊C₄	—	13	2	no	—
177	C₄²	—	13	1	no	—
178	C₄²	—	13	1	no	—
179	C₄²	—	13	1	no	—
180	C₄²	—	13	2	no	—
181	C₄²	C₂²×D₁₃	1	1	yes	characteristic
182	C₄²	C₂²×D₁₃	1	2	yes	—
183	C₁₃	#2986	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	13	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	26	4	no	—
187	Q₈	—	26	2	no	—
188	C₂×C₄	—	13	1	no	—
189	C₂×C₄	—	13	1	no	—
190	C₂×C₄	—	13	1	no	—
191	C₂×C₄	—	13	2	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	26	2	no	—
194	C₂×C₄	—	26	2	no	—
195	C₂×C₄	—	26	1	no	—
196	C₂×C₄	—	26	1	no	—
197	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	1	no	—
205	C₈	—	26	1	no	—
206	C₈	—	26	2	no	—
207	C₈	—	26	1	no	—
208	C₂²	—	26	1	no	—
209	C₂²	—	26	1	no	—
210	C₂²	—	26	1	no	—
211	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	13	2	no	—
216	C₄	—	26	1	no	—
217	C₄	—	26	2	no	—
218	C₄	—	26	2	no	—
219	C₄	—	26	1	no	—
220	C₄	#88155	1	2	yes	—
221	C₄	#88254	1	1	yes	characteristic
222	C₄	#88257	1	1	yes	characteristic
223	C₂	—	52	1	no	—
224	C₂	#11719277	1	1	yes	characteristic
225	C₂	C₂×#88155	1	1	yes	characteristic
226	C₂	D₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₈×Q₈×C₁₃	C₂
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₈×Q₈	D₁₃
hypocenter	C₁₃	#2986
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ 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. #11718491	C₂	semidirect
7. #11718489	C₂	semidirect
10. #11718340	C₂	semidirect
11. C₈×Dic₂₆	C₂	semidirect
101. C₈×Q₈	D₁₃	semidirect
183. C₁₃	#2986	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
#11718501	2
#11718494	2
#11718491	2
#11718489	2
#11718351	2
#11718346	2
#11718340	2
C₈×Dic₂₆	2
#2986	13