Subgroups — #13131365

Order 1664.

552 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#261	C₂	1	2	yes	maximal
2	D₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	(C₂×#261)⋊Q₈	C₂	1	2	yes	maximal
4	#11719083	C₂	1	2	yes	maximal
5	#11719082	C₂	1	4	yes	maximal
6	C₄⋊C₄×#261	C₂	1	2	yes	maximal
7	C₄×C₄⋊#261	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊#261	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×D₁₃	C₂²	1	2	yes	—
10	Q₈×#261	—	2	8	no	—
11	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#261	C₂²	1	4	yes	—
14	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
16	C₂×C₄×#261	C₂²	1	2	yes	—
17	C₂×C₄×#261	C₂²	1	2	yes	—
18	C₂×C₄×#261	C₂²	1	4	yes	—
19	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
20	#88198	C₄	1	2	yes	complemented
21	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
22	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
23	#88142	C₄	1	2	yes	complemented
24	#88142	C₄	1	2	yes	complemented
25	C₄²×D₁₃	C₂²	1	1	yes	characteristic
26	#88123	C₄	1	1	yes	characteristic, complemented
27	(C₂×#261)⋊C₄	C₂²	1	2	yes	—
28	(C₂×#261)⋊C₄	C₂²	1	2	yes	—
29	(C₂×#261)⋊C₄	C₂²	1	4	yes	—
30	(C₂×#261)⋊C₄	C₂²	1	4	yes	—
31	C₂²×#261	C₂³	1	2	yes	—
32	C₂²×#261	C₂³	1	2	yes	—
33	C₂²×#261	C₂³	1	2	yes	—
34	C₂²×#261	C₂³	1	2	yes	—
35	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
36	Q₈×D₁₃	—	2	4	no	—
37	Q₈×D₁₃	—	2	4	no	—
38	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₁₃	C₂³	1	4	yes	—
40	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
43	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
44	C₄⋊#261	—	2	8	no	—
45	C₄⋊#261	—	2	2	no	—
46	C₄⋊#261	—	2	2	no	—
47	C₄⋊#261	D₄	1	4	yes	—
48	C₄⋊#261	Q₈	1	4	yes	complemented
49	C₄×#261	—	2	8	no	—
50	C₄×#261	—	2	4	no	—
51	C₄×#261	—	2	4	no	—
52	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
57	#6101	C₂×C₄	1	4	yes	—
58	#6101	C₂×C₄	1	4	yes	—
59	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	#2365	—	13	1	no	maximal, Sylow
63	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
64	C₂×#261	—	2	2	no	—
65	C₂×#261	—	2	2	no	—
66	C₂×#261	—	2	2	no	—
67	C₂×#261	—	2	2	no	—
68	C₂×#261	—	2	2	no	—
69	C₂×#261	—	2	2	no	—
70	C₂×#261	—	2	2	no	—
71	C₂×#261	—	2	2	no	—
72	C₂×#261	—	2	2	no	—
73	C₂×#261	—	2	2	no	—
74	C₂×#261	C₂×D₄	1	4	yes	—
75	C₂×#261	C₂×Q₈	1	4	yes	—
76	Q₈×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
83	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₁₃	—	2	4	no	—
86	C₄×D₁₃	—	2	4	no	—
87	C₄×D₁₃	—	2	1	no	—
88	C₄×D₁₃	—	2	1	no	—
89	C₄×D₁₃	C₂×D₄	1	2	yes	—
90	C₄×D₁₃	C₂×Q₈	1	2	yes	—
91	C₄×D₁₃	#41	1	2	yes	—
92	C₄×D₁₃	#41	1	2	yes	—
93	Dic₂₆	—	2	4	no	—
94	Dic₂₆	—	2	8	no	—
95	C₂×C₄⋊Q₈	—	13	1	no	—
96	C₂×C₄×Q₈	—	13	2	no	—
97	#391	—	13	2	no	—
98	#389	—	13	2	no	—
99	#387	—	13	4	no	—
100	C₄×C₄⋊C₄	—	13	1	no	—
101	C₄×C₄⋊C₄	—	13	1	no	—
102	C₄×C₄⋊C₄	—	13	2	no	—
103	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×D₁₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×D₁₃	C₂×#41	1	1	yes	characteristic
107	C₂×D₁₃	C₂×#41	1	1	yes	characteristic
108	C₂×D₁₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×D₁₃	2₋¹⁺⁴	1	1	yes	characteristic
110	#261	—	2	4	no	—
111	#261	—	2	4	no	—
112	#261	—	4	2	no	—
113	#261	—	4	2	no	—
114	C₄×C₁₃	—	2	4	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₄×C₁₃	C₄×Q₈	1	2	yes	complemented
118	Dic₁₃	—	2	4	no	—
119	Dic₁₃	—	2	1	no	—
120	Dic₁₃	C₄×D₄	1	2	yes	—
121	Dic₁₃	C₄×Q₈	1	2	yes	complemented
122	C₂²×Q₈	—	13	2	no	—
123	C₄⋊Q₈	—	13	2	no	—
124	C₄⋊Q₈	—	13	2	no	—
125	C₄⋊Q₈	—	13	2	no	—
126	C₄⋊Q₈	—	13	1	no	—
127	C₄⋊Q₈	#261	1	1	yes	characteristic, complemented
128	C₄×Q₈	—	26	8	no	—
129	C₂×C₄⋊C₄	—	13	1	no	—
130	C₂×C₄⋊C₄	—	13	1	no	—
131	C₂×C₄⋊C₄	—	13	2	no	—
132	C₂×C₄⋊C₄	—	13	2	no	—
133	C₂×C₄⋊C₄	—	13	4	no	—
134	C₂×C₄⋊C₄	—	13	1	no	—
135	C₂×C₄⋊C₄	—	13	1	no	—
136	C₂×C₄²	—	13	1	no	—
137	C₂×C₄²	—	13	4	no	—
138	C₂×C₄²	—	13	2	no	—
139	C₂×C₄²	—	13	2	no	—
140	#95	—	13	2	no	—
141	#95	—	13	2	no	—
142	#95	—	13	4	no	—
143	#95	—	13	4	no	—
144	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₂×C₁₃	#519	1	1	yes	characteristic
147	D₁₃	#547	1	1	yes	characteristic
148	D₁₃	D₄×Q₈	1	1	yes	characteristic
149	D₁₃	#556	1	1	yes	characteristic
150	D₁₃	#557	1	1	yes	characteristic
151	C₂×Q₈	—	13	4	no	—
152	C₂×Q₈	—	13	2	no	—
153	C₂×Q₈	—	26	4	no	—
154	C₂×Q₈	—	26	4	no	—
155	C₂×Q₈	C₂×#261	1	2	yes	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	13	2	no	—
159	C₂²×C₄	—	13	2	no	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	4	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	2	no	—
164	C₄⋊C₄	—	13	4	no	—
165	C₄⋊C₄	—	13	4	no	—
166	C₄⋊C₄	—	13	4	no	—
167	C₄⋊C₄	—	13	4	no	—
168	C₄⋊C₄	—	13	2	no	—
169	C₄⋊C₄	—	13	2	no	—
170	C₄⋊C₄	—	26	8	no	—
171	C₄⋊C₄	—	26	2	no	—
172	C₄⋊C₄	—	26	2	no	—
173	C₄⋊C₄	C₂×#261	1	2	yes	—
174	C₄⋊C₄	C₂×#261	1	2	yes	—
175	C₄²	—	13	1	no	—
176	C₄²	—	13	1	no	—
177	C₄²	—	13	1	no	—
178	C₄²	—	26	4	no	—
179	C₄²	—	26	8	no	—
180	C₄²	—	26	4	no	—
181	C₄²	C₂×#261	1	1	yes	characteristic
182	C₁₃	#2365	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	13	1	no	—
184	Q₈	—	2	4	no	—
185	Q₈	—	26	8	no	—
186	Q₈	—	26	4	no	—
187	C₂×C₄	—	13	1	no	—
188	C₂×C₄	—	13	1	no	—
189	C₂×C₄	—	13	2	no	—
190	C₂×C₄	—	13	2	no	—
191	C₂×C₄	—	13	2	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	13	4	no	—
194	C₂×C₄	—	13	4	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	4	no	—
197	C₂×C₄	—	26	4	no	—
198	C₂×C₄	—	26	1	no	—
199	C₂×C₄	—	26	1	no	—
200	C₂×C₄	—	26	4	no	—
201	C₂×C₄	—	26	2	no	—
202	C₂×C₄	—	26	2	no	—
203	C₂×C₄	—	26	2	no	—
204	C₂×C₄	—	26	2	no	—
205	C₂×C₄	—	26	2	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	C₂²×#261	1	1	yes	characteristic
212	C₂×C₄	C₂²×#261	1	4	yes	—
213	C₂×C₄	C₂²×#261	1	1	yes	characteristic
214	C₂×C₄	C₂²×#261	1	1	yes	characteristic
215	C₂²	—	13	1	no	—
216	C₂²	—	13	1	no	—
217	C₂²	—	13	1	no	—
218	C₂²	—	13	1	no	—
219	C₂²	—	13	1	no	—
220	C₂²	—	13	1	no	—
221	C₂²	C₂³×#261	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	2	1	no	—
224	C₄	—	13	2	no	—
225	C₄	—	13	2	no	—
226	C₄	—	26	4	no	—
227	C₄	—	26	4	no	—
228	C₄	—	26	4	no	—
229	C₄	—	26	1	no	—
230	C₄	—	52	2	no	—
231	C₄	—	52	2	no	—
232	C₄	D₄×#261	1	2	yes	—
233	C₄	Q₈×#261	1	2	yes	complemented
234	C₂	—	13	1	no	—
235	C₂	—	13	1	no	—
236	C₂	—	13	1	no	—
237	C₂	—	13	1	no	—
238	C₂	C₂×D₄×#261	1	1	yes	characteristic
239	C₂	C₂×Q₈×#261	1	1	yes	characteristic
240	C₂	#11719627	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×C₄⋊Q₈	C₄
prime radical, cosocle	C₂²×D₁₃	C₂⁴
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#2365
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
19. C₁₃×C₄⋊Q₈	C₄	semidirect
20. #88198	C₄	semidirect
23. #88142	C₄	semidirect
24. #88142	C₄	semidirect
26. #88123	C₄	semidirect
48. C₄⋊#261	Q₈	semidirect
117. C₄×C₁₃	C₄×Q₈	semidirect
121. Dic₁₃	C₄×Q₈	semidirect
127. C₄⋊Q₈	#261	semidirect
182. C₁₃	#2365	semidirect
233. C₄	Q₈×#261	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#261	2
D₁₃×C₄⋊Q₈	2
(C₂×#261)⋊Q₈	2
#11719083	2
#11719082	2
C₄⋊C₄×#261	2
C₄×C₄⋊#261	2
#2365	13