Subgroups — #15772694

Order 3328.

450 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112803	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₁₃	—	2	1	no	—
7	C₄×C₁₃×D₈	—	2	1	no	—
8	C₄×C₁₃×D₈	—	2	1	no	—
9	C₄×C₁₃×D₈	C₄	1	2	yes	—
10	C₄×C₁₃×D₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#102	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88052	—	2	1	no	—
15	#11718586	—	2	1	no	—
16	#11718053	C₂²	1	2	yes	complemented
17	C₄×#6090	—	2	1	no	—
18	#11718050	—	2	1	no	—
19	#11718049	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	—	4	1	no	—
22	C₂×C₁₃×D₈	—	2	1	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	C₈	1	4	yes	complemented
25	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×D₄×C₁₃	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	4	1	no	—
31	C₄×D₄×C₁₃	—	4	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₁₃×#107	—	2	1	no	—
36	C₁₃×#107	C₂×C₄	1	2	yes	—
37	C₁₃×#102	—	2	2	no	—
38	C₁₃×#102	—	2	2	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²⋊#1055	—	4	1	no	—
44	#88052	—	2	4	no	—
45	C₂×#6090	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄×#1055	—	4	1	no	—
48	#11599	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	—	4	1	no	—
50	C₂×D₄×C₁₃	—	2	4	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	4	1	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	—	8	4	no	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
61	C₁₃×D₈	—	2	4	no	—
62	C₁₃×D₈	—	2	2	no	—
63	C₁₃×D₈	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	4	1	no	—
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₁₃×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₃	D₈	1	1	yes	characteristic
77	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×#1055	—	4	2	no	—
80	#6090	—	2	4	no	—
81	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
82	C₄×M₃₂	—	13	1	no	—
83	#1564	—	13	1	no	—
84	#1333	—	13	4	no	—
85	C₂³×C₁₃	—	4	4	no	—
86	C₂³×C₁₃	—	4	2	no	—
87	C₂³×C₁₃	—	8	2	no	—
88	C₂³×C₁₃	—	8	2	no	—
89	C₂³×C₁₃	#102	1	1	yes	characteristic
90	D₄×C₁₃	—	2	4	no	—
91	D₄×C₁₃	—	4	4	no	—
92	D₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	4	1	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	8	4	no	—
97	C₂×C₄×C₁₃	—	8	2	no	—
98	C₂×C₄×C₁₃	—	8	4	no	—
99	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#104	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₈×C₁₃	—	2	1	no	—
108	C₈×C₁₃	—	2	1	no	—
109	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
110	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
112	#1055	—	8	2	no	—
113	C₂²×D₈	Dic₁₃	1	1	yes	characteristic
114	C₂×C₄×D₄	—	2	1	no	—
115	C₂×M₃₂	—	26	1	no	—
116	C₄×D₈	—	2	1	no	—
117	C₄×D₈	—	2	1	no	—
118	C₄×D₈	Dic₁₃	1	2	yes	—
119	C₄×D₈	C₂×D₁₃	1	2	yes	—
120	#431	—	26	1	no	—
121	C₂×#107	Dic₁₃	1	1	yes	characteristic
122	C₂×#102	—	2	1	no	—
123	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
124	#346	—	13	2	no	—
125	C₄×C₁₆	—	26	1	no	—
126	#335	—	26	1	no	—
127	#334	—	13	2	no	—
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	4	4	no	—
131	C₂²×C₁₃	—	8	8	no	—
132	C₂²×C₁₃	—	8	2	no	—
133	C₂²×C₁₃	—	8	2	no	—
134	C₂²×C₁₃	D₄⋊C₈	1	2	yes	—
135	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	8	2	no	—
140	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
141	C₄×C₁₃	#361	1	2	yes	—
142	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
144	C₂²×D₄	—	2	1	no	—
145	C₂³×C₄	—	4	1	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×D₈	—	2	1	no	—
148	C₂×D₈	#1055	1	4	yes	complemented
149	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
150	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₄×D₄	—	4	1	no	—
155	C₄×D₄	—	4	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
159	M₃₂	—	26	4	no	—
160	C₂×C₁₆	—	26	2	no	—
161	#107	—	2	1	no	—
162	#107	C₂×Dic₁₃	1	2	yes	—
163	C₄⋊C₈	—	26	2	no	—
164	#102	—	2	2	no	—
165	#102	—	2	2	no	—
166	C₂²⋊C₈	—	52	1	no	—
167	C₄×C₈	—	52	1	no	—
168	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×C₁₃	—	2	2	no	—
173	C₂×C₁₃	—	8	4	no	—
174	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
175	C₂×C₁₃	#2145	1	1	yes	characteristic
176	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
177	C₂⁴	—	4	1	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	8	4	no	—
186	C₂²×C₄	#6104	1	1	yes	characteristic
187	C₂²×C₄	#6105	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
189	D₈	—	2	4	no	—
190	D₈	—	2	2	no	—
191	D₈	C₂×#1055	1	4	yes	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	52	2	no	—
194	C₂×C₈	C₂×#1055	1	2	yes	—
195	C₂×C₈	#6099	1	2	yes	—
196	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
197	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	#6104	1	1	yes	characteristic
205	C₄²	#6105	1	1	yes	characteristic
206	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₁₆	—	26	4	no	—
209	C₁₃	#11599	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	4	2	no	—
211	C₂³	—	4	4	no	—
212	C₂³	—	8	2	no	—
213	C₂³	—	8	2	no	—
214	C₂³	#88072	1	1	yes	characteristic
215	D₄	—	2	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	C₂²⋊#1055	1	2	yes	—
225	C₂×C₄	#88072	1	1	yes	characteristic
226	C₂×C₄	#88072	1	1	yes	characteristic
227	C₂×C₄	#88072	1	1	yes	characteristic
228	C₂×C₄	#88077	1	2	yes	—
229	C₂×C₄	C₂×#6104	1	1	yes	characteristic
230	C₂×C₄	C₂×#6105	1	1	yes	characteristic
231	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	104	2	no	—
235	C₈	C₂²⋊#1055	1	2	yes	—
236	C₈	C₂²×#1055	1	1	yes	characteristic
237	C₈	C₂×#6099	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	8	8	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	2	no	—
244	C₂²	D₄⋊#1055	1	2	yes	—
245	C₂²	C₂×#88072	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	8	2	no	—
250	C₄	D₄⋊#1055	1	2	yes	—
251	C₄	#11718155	1	2	yes	—
252	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
253	C₄	C₂×#88077	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	8	4	no	—
256	C₂	C₂×D₄⋊#1055	1	1	yes	characteristic
257	C₂	#13116572	1	1	yes	characteristic
258	C₂	C₂×#11718155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88072
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂²×C₁₃×D₈	C₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	#11599
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₄×#88052	C₂	semidirect
2. #13113584	C₂	semidirect
3. #13112803	C₂	semidirect
16. #11718053	C₂²	semidirect
19. #11718049	C₂²	semidirect
24. C₂×C₁₃×D₈	C₈	semidirect
148. C₂×D₈	#1055	semidirect
209. C₁₃	#11599	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
#13113584	2
#13112803	2
C₂×C₄×C₁₃×D₈	2
#11599	13