Subgroups — #15881825

Order 3328.

876 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1055	C₂	1	2	yes	maximal
2	#13115555	C₂	1	6	yes	maximal
3	#13114953	C₂	1	3	yes	maximal
4	C₄×C₄⋊#1055	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
10	C₄×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
11	C₄×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
12	C₄³×C₁₃	C₂²	1	3	yes	—
13	Q₈×#1055	—	2	8	no	—
14	C₂×C₄⋊#1055	C₂²	1	6	yes	—
15	C₂×C₄⋊#1055	C₂²	1	6	yes	—
16	C₂×C₄×#1055	C₂²	1	6	yes	—
17	#11718116	C₂²	1	1	yes	characteristic
18	#11718116	C₂²	1	3	yes	—
19	#11718116	C₂²	1	6	yes	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	—	2	4	no	—
22	C₄×Q₈×C₁₃	C₈	1	8	yes	complemented
23	C₄×Q₈×C₁₃	D₄	1	4	yes	—
24	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
25	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
26	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
35	C₂²×#1055	C₂³	1	6	yes	—
36	C₂²×#1055	C₂³	1	2	yes	—
37	C₁₃×#95	C₂×C₄	1	6	yes	—
38	C₄⋊#1055	—	2	12	no	—
39	C₄⋊#1055	—	2	12	no	—
40	C₄×#1055	—	2	12	no	—
41	#22734	—	13	1	no	maximal, Sylow
42	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
43	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
44	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	6	no	—
53	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
54	C₁₃×C₄⋊C₄	C₂×C₈	1	12	yes	—
55	C₄²×C₁₃	—	2	6	no	—
56	C₄²×C₁₃	—	2	6	no	—
57	C₄²×C₁₃	—	2	6	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	4	3	no	—
60	C₄²×C₁₃	C₂×C₈	1	12	yes	—
61	C₄²×C₁₃	C₂×D₄	1	6	yes	—
62	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
63	C₂×#1055	—	2	2	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	6	no	—
66	C₂×#1055	—	2	6	no	—
67	C₂×#1055	—	2	6	no	—
68	C₂×#1055	#41	1	4	yes	—
69	C₂×C₈×Q₈	—	13	2	no	—
70	C₄²×Q₈	D₁₃	1	1	yes	characteristic
71	#1939	—	13	6	no	—
72	#1831	—	13	3	no	—
73	C₄×C₄⋊C₈	—	13	3	no	—
74	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₁₃	C₄⋊C₈	1	16	yes	complemented
76	C₂×C₄×C₁₃	—	2	3	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	3	no	—
81	C₂×C₄×C₁₃	—	2	3	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	3	no	—
84	C₂×C₄×C₁₃	—	2	3	no	—
85	C₂×C₄×C₁₃	—	2	3	no	—
86	C₂×C₄×C₁₃	—	2	3	no	—
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
88	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
89	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
92	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
93	C₂×C₄×C₁₃	#131	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
97	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
98	#1055	—	2	4	no	—
99	#1055	—	4	6	no	—
100	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
101	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
102	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
103	C₈×Q₈	—	26	8	no	—
104	C₂×C₄⋊C₈	—	13	6	no	—
105	C₂×C₄⋊C₈	—	13	6	no	—
106	C₂×C₄×C₈	—	13	6	no	—
107	C₄×C₄⋊C₄	Dic₁₃	1	3	yes	—
108	C₄×C₄⋊C₄	Dic₁₃	1	3	yes	—
109	C₄×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
110	C₄³	C₂×D₁₃	1	3	yes	—
111	#336	—	13	3	no	—
112	#336	—	13	6	no	—
113	#336	—	13	1	no	—
114	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
116	C₂²×C₁₃	#519	1	1	yes	characteristic
117	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
119	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
120	C₂²×C₁₃	#568	1	1	yes	characteristic
121	C₄×C₁₃	—	2	6	no	—
122	C₄×C₁₃	—	2	6	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	4	3	no	—
126	C₄×C₁₃	C₂×C₄⋊C₈	1	12	yes	—
127	C₄×C₁₃	#548	1	2	yes	—
128	C₄×C₁₃	#557	1	2	yes	—
129	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
130	C₂²×C₈	—	13	2	no	—
131	C₂²×C₈	—	13	6	no	—
132	C₄×Q₈	—	2	4	no	—
133	C₄×Q₈	#1055	1	8	yes	complemented
134	C₄×Q₈	Dic₂₆	1	4	yes	—
135	C₄×Q₈	D₅₂	1	4	yes	—
136	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
137	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
138	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
139	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
140	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
141	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
142	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
143	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
145	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
146	C₄⋊C₈	—	26	12	no	—
147	C₄⋊C₈	—	26	12	no	—
148	C₄×C₈	—	26	12	no	—
149	#95	C₂×Dic₁₃	1	6	yes	—
150	C₂×C₁₃	#2351	1	1	yes	characteristic
151	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₁₃	#2906	1	1	yes	characteristic
153	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
154	C₂×C₁₃	#2971	1	1	yes	characteristic
155	C₂×C₁₃	#2982	1	1	yes	characteristic
156	C₂×C₁₃	#2984	1	1	yes	characteristic
157	C₂×Q₈	C₂×#1055	1	4	yes	—
158	C₂×Q₈	#6099	1	4	yes	—
159	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
160	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
161	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
162	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
163	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
164	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
166	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
167	C₂×C₈	—	13	4	no	—
168	C₂×C₈	—	26	2	no	—
169	C₂×C₈	—	26	2	no	—
170	C₂×C₈	—	26	6	no	—
171	C₂×C₈	—	26	6	no	—
172	C₂×C₈	—	26	6	no	—
173	C₄⋊C₄	—	2	6	no	—
174	C₄⋊C₄	C₂×#1055	1	12	yes	—
175	C₄⋊C₄	C₄⋊Dic₁₃	1	12	yes	—
176	C₄²	—	2	6	no	—
177	C₄²	—	2	6	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	6	no	—
180	C₄²	—	4	3	no	—
181	C₄²	C₂×#1055	1	12	yes	—
182	C₄²	C₂×Dic₂₆	1	6	yes	—
183	C₄²	C₂×D₅₂	1	6	yes	—
184	C₁₃	#22734	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
186	Q₈	C₄⋊#1055	1	16	yes	complemented
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	C₂²×#1055	1	2	yes	—
199	C₂×C₄	C₂²×#1055	1	6	yes	—
200	C₂×C₄	C₂²×#1055	1	6	yes	—
201	C₂×C₄	C₂×#6099	1	6	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
204	C₂×C₄	#88202	1	2	yes	—
205	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
207	C₂×C₄	#88254	1	2	yes	—
208	C₂×C₄	#41×D₁₃	1	2	yes	—
209	C₈	—	26	4	no	—
210	C₈	—	52	6	no	—
211	C₂²	C₂³×#1055	1	1	yes	characteristic
212	C₂²	C₂²×#6099	1	1	yes	characteristic
213	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
214	C₂²	#11719517	1	1	yes	characteristic
215	C₂²	C₂×#88202	1	1	yes	characteristic
216	C₂²	#11719524	1	1	yes	characteristic
217	C₂²	#41×Dic₁₃	1	1	yes	characteristic
218	C₄	—	2	6	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	6	no	—
222	C₄	—	4	3	no	—
223	C₄	C₂×C₄⋊#1055	1	12	yes	—
224	C₄	#11719274	1	2	yes	—
225	C₄	#11719281	1	2	yes	—
226	C₂	#13117757	1	1	yes	characteristic
227	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
228	C₂	#13123404	1	1	yes	characteristic
229	C₂	#41×#1055	1	1	yes	characteristic
230	C₂	#13123856	1	1	yes	characteristic
231	C₂	#13123917	1	1	yes	characteristic
232	C₂	#13123925	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Fitting	C₄²×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×Q₈	D₁₃
hypocenter	C₁₃	#22734
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
22. C₄×Q₈×C₁₃	C₈	semidirect
75. Q₈×C₁₃	C₄⋊C₈	semidirect
133. C₄×Q₈	#1055	semidirect
184. C₁₃	#22734	semidirect
186. Q₈	C₄⋊#1055	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1055	2
#13115555	2
#13114953	2
C₄×C₄⋊#1055	2
C₄²×Q₈×C₁₃	2
#22734	13