Subgroups — #13123959

Order 1664.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6129	C₂	1	1	yes	maximal, characteristic
2	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718772	C₂	1	2	yes	maximal, complemented
4	C₂×#88054	C₂	1	2	yes	maximal, complemented
5	D₄×#1055	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718498	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718492	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718391	C₂	1	2	yes	maximal, complemented
9	#11718380	C₂	1	2	yes	maximal, complemented
10	#11718346	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6129	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	D₄×Dic₁₃	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
18	C₂²×#1055	C₂²	1	2	yes	—
19	C₂×#6094	C₂²	1	2	yes	—
20	#88146	C₄	1	1	yes	characteristic
21	#88141	C₄	1	1	yes	characteristic
22	#88135	C₄	1	2	yes	—
23	#88131	C₄	1	2	yes	—
24	C₄²×D₁₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₂²⋊#1055	C₂²	1	2	yes	—
30	#88060	C₂²	1	1	yes	characteristic
31	#88060	C₂²	1	1	yes	characteristic
32	#88055	C₂²	1	2	yes	—
33	#88055	C₂²	1	2	yes	—
34	#88054	C₂²	1	2	yes	—
35	#88054	C₂²	1	2	yes	—
36	#88054	C₂²	1	1	yes	characteristic
37	#88054	C₂²	1	2	yes	—
38	#88054	C₂²	1	1	yes	characteristic
39	C₄⋊#1055	C₂²	1	1	yes	characteristic
40	C₄×#1055	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
46	#6129	—	2	4	no	—
47	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	—	4	2	no	—
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂³	1	2	yes	—
55	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
59	#6103	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
66	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₃	C₂³	1	2	yes	—
68	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#1055	—	4	2	no	—
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	C₂×#1055	C₂³	1	2	yes	—
72	C₂×#1055	C₂³	1	1	yes	characteristic
73	#6094	—	4	2	no	—
74	#2990	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₁₃	M₁₆	1	4	yes	complemented
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₁₃	—	2	4	no	—
85	C₂×Dic₁₃	—	2	4	no	—
86	C₂×Dic₁₃	—	2	4	no	—
87	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
91	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₁₃	—	2	2	no	—
93	C₄×D₁₃	—	2	1	no	—
94	C₄×D₁₃	—	2	1	no	—
95	C₄×D₁₃	—	2	2	no	—
96	Dic₂₆	—	2	2	no	—
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	1	no	—
100	#1055	—	2	2	no	—
101	#1055	—	2	1	no	—
102	#1055	—	2	1	no	—
103	C₄×#41	—	13	1	no	—
104	#446	—	13	1	no	—
105	#446	—	13	1	no	—
106	#435	—	13	2	no	—
107	#435	—	13	2	no	—
108	C₈×D₄	—	13	1	no	—
109	C₈×D₄	D₁₃	1	1	yes	characteristic, complemented
110	#432	—	13	2	no	—
111	#431	—	13	1	no	—
112	#423	—	13	1	no	—
113	C₂×C₄⋊C₈	—	13	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	C₂×M₁₆	1	4	yes	—
117	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×D₁₃	—	2	1	no	—
119	C₂×D₁₃	—	2	1	no	—
120	C₂×D₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
124	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	—	2	2	no	—
128	Dic₁₃	—	2	1	no	—
129	Dic₁₃	—	2	2	no	—
130	Dic₁₃	#131	1	2	yes	—
131	C₂×#41	—	13	1	no	—
132	C₂×M₁₆	—	13	2	no	—
133	C₂²×C₈	—	13	2	no	—
134	C₂²×C₈	C₂×D₁₃	1	2	yes	—
135	C₄×Q₈	—	13	1	no	—
136	C₄×Q₈	—	13	1	no	—
137	C₄×D₄	—	13	1	no	—
138	C₄×D₄	—	13	2	no	—
139	C₄×D₄	—	13	2	no	—
140	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
141	#117	—	13	1	no	—
142	#117	—	13	2	no	—
143	C₂×C₄²	—	13	2	no	—
144	C₂×C₄²	—	13	1	no	—
145	C₄⋊C₈	—	13	1	no	—
146	C₄⋊C₈	—	13	2	no	—
147	C₄⋊C₈	—	13	2	no	—
148	C₄⋊C₈	—	13	2	no	—
149	C₄⋊C₈	—	13	1	no	—
150	C₄⋊C₈	—	13	1	no	—
151	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₂²⋊C₈	—	13	2	no	—
153	C₂²⋊C₈	—	13	1	no	—
154	C₂²⋊C₈	—	13	1	no	—
155	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
156	#97	—	13	2	no	—
157	#97	—	13	2	no	—
158	C₄×C₈	—	13	1	no	—
159	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
160	C₂×C₁₃	—	2	4	no	—
161	C₂×C₁₃	#520	1	1	yes	characteristic
162	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
164	D₁₃	—	4	1	no	—
165	#41	—	26	4	no	—
166	C₂×Q₈	—	13	1	no	—
167	C₂×D₄	—	13	2	no	—
168	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	—	13	2	no	—
170	C₂²×C₄	—	13	2	no	—
171	C₂²×C₄	—	13	1	no	—
172	C₂²×C₄	—	13	1	no	—
173	C₂²×C₄	—	13	1	no	—
174	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
175	M₁₆	—	52	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	13	1	no	—
178	C₂×C₈	—	13	1	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	52	2	no	—
181	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₃	1	2	yes	—
184	C₄⋊C₄	—	13	2	no	—
185	C₄⋊C₄	—	13	2	no	—
186	C₄⋊C₄	—	13	1	no	—
187	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
188	C₂²⋊C₄	—	13	2	no	—
189	C₂²⋊C₄	—	13	2	no	—
190	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
191	C₄²	—	13	2	no	—
192	C₄²	—	13	1	no	—
193	C₄²	—	13	1	no	—
194	C₄²	—	13	2	no	—
195	C₄²	—	13	2	no	—
196	C₄²	—	13	1	no	—
197	C₄²	C₂²×D₁₃	1	1	yes	characteristic
198	C₁₃	#2990	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	13	1	no	—
200	C₂³	C₂×C₄×D₁₃	1	2	yes	—
201	Q₈	—	26	2	no	—
202	D₄	—	26	4	no	—
203	D₄	#6094	1	4	yes	complemented
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	13	1	no	—
207	C₂×C₄	—	13	1	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	13	1	no	—
210	C₂×C₄	—	13	2	no	—
211	C₂×C₄	—	26	2	no	—
212	C₂×C₄	—	26	4	no	—
213	C₂×C₄	—	26	4	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	1	no	—
216	C₂×C₄	—	26	1	no	—
217	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	26	1	no	—
225	C₈	—	26	2	no	—
226	C₈	—	26	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	26	1	no	—
230	C₂²	—	26	1	no	—
231	C₂²	—	26	1	no	—
232	C₂²	C₂×#6094	1	4	yes	—
233	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	13	2	no	—
237	C₄	—	26	1	no	—
238	C₄	—	26	2	no	—
239	C₄	—	26	2	no	—
240	C₄	—	26	1	no	—
241	C₄	C₂×#6094	1	2	yes	—
242	C₄	#88251	1	1	yes	characteristic
243	C₄	#88257	1	1	yes	characteristic
244	C₂	—	2	4	no	—
245	C₂	—	52	1	no	—
246	C₂	#11719251	1	1	yes	characteristic
247	C₂	C₂²×#6094	1	1	yes	characteristic
248	C₂	D₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₈×D₄×C₁₃	C₂
prime radical	C₂×#6129	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₈×D₄	D₁₃
hypocenter	C₁₃	#2990
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×D₄×C₁₃	C₂	semidirect
3. #11718772	C₂	semidirect
4. C₂×#88054	C₂	semidirect
5. D₄×#1055	C₂	semidirect
6. #11718498	C₂	semidirect
7. #11718492	C₂	semidirect
8. #11718391	C₂	semidirect
9. #11718380	C₂	semidirect
10. #11718346	C₂	semidirect
11. #11718340	C₂	semidirect
77. D₄×C₁₃	M₁₆	semidirect
109. C₈×D₄	D₁₃	semidirect
198. C₁₃	#2990	semidirect
203. D₄	#6094	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6129	2
C₈×D₄×C₁₃	2
#11718772	2
C₂×#88054	2
D₄×#1055	2
#11718498	2
#11718492	2
#11718391	2
#11718380	2
#11718346	2
#11718340	2
#2990	13