Subgroups — #13123788

Order 1664.

388 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718685	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718498	C₂	1	2	yes	maximal
5	#11718498	C₂	1	1	yes	maximal, characteristic
6	#11718491	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718489	C₂	1	2	yes	maximal, complemented
8	#11718368	C₂	1	2	yes	maximal
9	#11718362	C₂	1	2	yes	maximal, complemented
10	C₄×#6094	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	Q₈×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×#6094	C₂²	1	1	yes	characteristic
16	C₂×#6094	C₂²	1	2	yes	—
17	C₂×#6094	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
19	C₄⋊C₄×D₁₃	C₄	1	2	yes	—
20	#88141	C₄	1	2	yes	—
21	#88141	C₄	1	1	yes	characteristic
22	C₄²×D₁₃	C₂²	1	1	yes	characteristic
23	C₄²×D₁₃	C₂²	1	2	yes	—
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	C₁₃×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	#88055	C₂²	1	1	yes	characteristic
31	#88055	C₂²	1	1	yes	characteristic
32	#88054	C₂²	1	2	yes	—
33	#88054	C₂²	1	2	yes	—
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	Q₈×D₁₃	—	2	4	no	—
43	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₄²×C₁₃	C₂³	1	2	yes	—
57	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	#6101	C₂×C₄	1	2	yes	—
60	#6101	C₂×C₄	1	2	yes	—
61	#6101	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂³	1	2	yes	—
67	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
68	C₂×#1055	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	2	yes	—
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	#6094	—	2	2	no	—
72	#6094	—	2	4	no	—
73	#6094	Q₈	1	4	yes	complemented
74	Q₈×M₁₆	—	13	1	no	maximal, Sylow
75	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₄×D₁₃	—	2	2	no	—
88	C₄×D₁₃	—	2	4	no	—
89	C₄×D₁₃	—	4	1	no	—
90	C₄×D₁₃	—	4	2	no	—
91	C₄×D₁₃	C₂×Q₈	1	2	yes	—
92	C₄×D₁₃	#41	1	2	yes	—
93	Dic₂₆	—	2	4	no	—
94	Dic₂₆	M₁₆	1	4	yes	complemented
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	C₂×Q₈	1	2	yes	complemented
98	#1055	—	2	1	no	—
99	#1055	—	2	2	no	—
100	#1055	C₂×Q₈	1	2	yes	complemented
101	C₂×C₄×Q₈	—	13	1	no	—
102	#446	—	13	1	no	—
103	#446	—	13	2	no	—
104	#446	—	13	2	no	—
105	#446	D₁₃	1	1	yes	characteristic, complemented
106	C₈×Q₈	—	13	1	no	—
107	C₈×Q₈	—	13	1	no	—
108	#423	—	13	2	no	—
109	#423	—	13	1	no	—
110	C₄×M₁₆	—	13	2	no	—
111	C₄×M₁₆	—	13	1	no	—
112	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×D₁₃	—	2	1	no	—
114	C₂×D₁₃	—	2	1	no	—
115	C₂×D₁₃	C₄×Q₈	1	2	yes	—
116	C₄×C₁₃	—	2	2	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
120	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
121	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
122	Dic₁₃	—	2	1	no	—
123	Dic₁₃	—	2	2	no	—
124	Dic₁₃	—	2	1	no	—
125	Dic₁₃	C₄×Q₈	1	2	yes	—
126	Dic₁₃	C₂×M₁₆	1	4	yes	—
127	C₂²×Q₈	—	13	1	no	—
128	C₂×M₁₆	—	13	1	no	—
129	C₂×M₁₆	—	13	2	no	—
130	C₂×M₁₆	—	13	1	no	—
131	C₄×Q₈	—	13	1	no	—
132	C₄×Q₈	—	13	1	no	—
133	C₄×Q₈	—	13	2	no	—
134	C₄×Q₈	—	13	1	no	—
135	C₄×Q₈	—	13	2	no	—
136	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	13	2	no	—
138	C₂×C₄⋊C₄	—	13	1	no	—
139	C₂×C₄²	—	13	2	no	—
140	C₂×C₄²	—	13	1	no	—
141	C₄⋊C₈	—	13	2	no	—
142	C₄⋊C₈	—	13	2	no	—
143	C₄⋊C₈	—	13	2	no	—
144	C₄⋊C₈	—	13	1	no	—
145	C₄⋊C₈	—	13	2	no	—
146	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
148	#97	—	13	1	no	—
149	#97	—	13	1	no	—
150	#97	—	13	2	no	—
151	#97	C₂×D₁₃	1	2	yes	—
152	C₄×C₈	—	13	2	no	—
153	C₄×C₈	—	13	2	no	—
154	C₄×C₈	—	13	1	no	—
155	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₂×C₁₃	#568	1	1	yes	characteristic
159	D₁₃	—	2	2	no	—
160	C₂×Q₈	—	13	1	no	—
161	C₂×Q₈	—	13	2	no	—
162	C₂×Q₈	—	26	4	no	—
163	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	2	no	—
168	C₂²×C₄	—	13	1	no	—
169	M₁₆	—	13	4	no	—
170	M₁₆	—	26	4	no	—
171	M₁₆	—	26	2	no	—
172	C₂×C₈	—	13	1	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	13	1	no	—
175	C₂×C₈	C₂²×D₁₃	1	2	yes	—
176	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊C₄	—	13	2	no	—
179	C₄⋊C₄	—	13	2	no	—
180	C₄⋊C₄	—	13	2	no	—
181	C₄⋊C₄	—	13	2	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
184	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
185	C₄²	—	13	2	no	—
186	C₄²	—	13	2	no	—
187	C₄²	—	13	1	no	—
188	C₄²	—	13	1	no	—
189	C₄²	—	13	2	no	—
190	C₄²	—	13	1	no	—
191	C₄²	C₂²×D₁₃	1	1	yes	characteristic
192	C₄²	C₂²×D₁₃	1	2	yes	—
193	C₁₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
194	C₂³	—	13	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	13	4	no	—
197	Q₈	—	26	4	no	—
198	C₂×C₄	—	13	1	no	—
199	C₂×C₄	—	13	2	no	—
200	C₂×C₄	—	13	2	no	—
201	C₂×C₄	—	13	2	no	—
202	C₂×C₄	—	13	2	no	—
203	C₂×C₄	—	13	1	no	—
204	C₂×C₄	—	13	1	no	—
205	C₂×C₄	—	26	4	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	52	1	no	—
208	C₂×C₄	—	52	2	no	—
209	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	13	2	no	—
217	C₈	—	26	1	no	—
218	C₈	—	26	2	no	—
219	C₈	Q₈×D₁₃	1	2	yes	complemented
220	C₂²	—	13	2	no	—
221	C₂²	—	26	1	no	—
222	C₂²	—	26	1	no	—
223	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	13	4	no	—
228	C₄	—	13	2	no	—
229	C₄	—	26	1	no	—
230	C₄	—	26	2	no	—
231	C₄	—	26	1	no	—
232	C₄	M₁₆×D₁₃	1	2	yes	—
233	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
234	C₄	#41×D₁₃	1	1	yes	characteristic
235	C₂	—	26	2	no	—
236	C₂	C₄×Q₈×D₁₃	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
238	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#446	C₂
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#446	D₁₃
hypocenter	C₁₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₁₃×#446	C₂	semidirect
3. #11718685	C₂	semidirect
6. #11718491	C₂	semidirect
7. #11718489	C₂	semidirect
9. #11718362	C₂	semidirect
11. C₈×Dic₂₆	C₂	semidirect
73. #6094	Q₈	semidirect
94. Dic₂₆	M₁₆	semidirect
97. C₈×C₁₃	C₂×Q₈	semidirect
100. #1055	C₂×Q₈	semidirect
105. #446	D₁₃	semidirect
193. C₁₃	Q₈×M₁₆	semidirect
219. C₈	Q₈×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₁₃	2
C₁₃×#446	2
#11718685	2
#11718498	2
#11718491	2
#11718489	2
#11718368	2
#11718362	2
C₄×#6094	2
C₈×Dic₂₆	2
Q₈×M₁₆	13