Subgroups — #15772255

Order 3328.

342 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×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	#13116635	C₂	1	1	yes	maximal, characteristic
3	#13112775	C₂	1	4	yes	maximal
4	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88054	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
9	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
10	C₈×Dic₂₆	—	2	1	no	—
11	C₈×Dic₂₆	—	2	1	no	—
12	C₈×Dic₂₆	—	2	1	no	—
13	C₈×Dic₂₆	—	2	1	no	—
14	C₁₃×C₄⋊C₁₆	—	2	1	no	—
15	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
16	C₄×C₁₃×C₁₆	—	2	1	no	—
17	C₂²⋊#6090	C₂²	1	2	yes	—
18	C₄⋊#6090	—	2	1	no	—
19	C₄×#6090	—	2	1	no	—
20	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
25	C₂×#6101	C₈	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	2	yes	—
28	C₄×Dic₂₆	—	2	1	no	—
29	C₄×Dic₂₆	—	2	1	no	—
30	C₄×Dic₂₆	—	2	1	no	—
31	C₄×Dic₂₆	—	2	1	no	—
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
37	#88054	—	2	2	no	—
38	#88054	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	C₈×Dic₁₃	—	2	2	no	—
41	C₂×#6090	—	2	2	no	—
42	C₄⋊#1055	—	2	1	no	—
43	C₄⋊#1055	—	2	1	no	—
44	#11493	—	13	1	no	maximal, Sylow
45	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
49	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
50	C₂×Dic₂₆	—	2	1	no	—
51	C₂×Dic₂₆	—	2	1	no	—
52	C₂×Dic₂₆	—	2	2	no	—
53	C₂×Dic₂₆	C₁₆	1	4	yes	complemented
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
63	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
64	C₄⋊Dic₁₃	—	2	1	no	—
65	C₄⋊Dic₁₃	—	2	1	no	—
66	#6101	—	2	2	no	—
67	#6101	—	2	2	no	—
68	C₄×Dic₁₃	—	2	2	no	—
69	C₄×Dic₁₃	—	2	2	no	—
70	C₂×#1055	—	2	2	no	—
71	C₂×#1055	—	4	2	no	—
72	C₁₃×C₁₆	—	4	2	no	—
73	#6090	—	4	2	no	—
74	C₂×C₈×Q₈	—	13	1	no	—
75	#2162	—	13	1	no	—
76	#2162	D₁₃	1	1	yes	characteristic
77	#1322	—	13	4	no	—
78	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	4	1	no	—
82	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	—	4	2	no	—
94	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
95	Dic₂₆	—	2	4	no	—
96	Dic₂₆	—	2	4	no	—
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	#101	1	2	yes	—
101	#1055	—	4	2	no	—
102	C₂×C₄×Q₈	—	13	1	no	—
103	C₈×Q₈	—	26	1	no	—
104	C₈×Q₈	—	26	1	no	—
105	C₈×Q₈	—	26	1	no	—
106	C₈×Q₈	—	26	1	no	—
107	C₂×C₄⋊C₈	—	13	2	no	—
108	C₂×C₄⋊C₈	—	13	1	no	—
109	C₂×C₄×C₈	—	13	2	no	—
110	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
111	C₄⋊C₁₆	—	2	1	no	—
112	C₄⋊C₁₆	—	26	1	no	—
113	C₂²⋊C₁₆	—	13	2	no	—
114	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
115	C₄×C₁₆	—	2	1	no	—
116	C₄×C₁₆	—	26	1	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
126	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
127	C₄×C₁₃	#410	1	1	yes	characteristic
128	C₄×C₁₃	C₂×#101	1	1	yes	characteristic
129	Dic₁₃	—	2	4	no	—
130	Dic₁₃	—	4	2	no	—
131	C₂²×Q₈	—	13	1	no	—
132	C₂²×C₈	—	13	2	no	—
133	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
135	C₄×Q₈	—	26	1	no	—
136	C₄×Q₈	—	26	1	no	—
137	C₄×Q₈	—	26	1	no	—
138	C₄×Q₈	—	26	1	no	—
139	C₂×C₄⋊C₄	—	13	2	no	—
140	C₂×C₄⋊C₄	—	13	1	no	—
141	C₂×C₄²	—	13	2	no	—
142	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	26	2	no	—
145	C₄⋊C₈	—	26	2	no	—
146	C₄⋊C₈	—	26	2	no	—
147	C₄⋊C₈	—	26	1	no	—
148	C₄⋊C₈	—	26	1	no	—
149	C₄×C₈	—	26	2	no	—
150	C₄×C₈	—	26	2	no	—
151	C₄×C₈	D₅₂	1	1	yes	characteristic
152	C₄×C₈	D₅₂	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
155	C₂×C₁₃	—	2	1	no	—
156	C₂×C₁₃	—	2	1	no	—
157	C₂×C₁₃	#1461	1	1	yes	characteristic
158	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
159	C₂×C₁₃	#2115	1	1	yes	characteristic
160	C₂×Q₈	—	13	4	no	—
161	C₂×Q₈	—	26	2	no	—
162	C₂×Q₈	—	26	1	no	—
163	C₂×Q₈	—	26	1	no	—
164	C₂²×C₄	—	13	2	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	26	2	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	#6103	1	1	yes	characteristic
174	C₂×C₈	#6103	1	1	yes	characteristic
175	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
177	C₄⋊C₄	—	26	2	no	—
178	C₄⋊C₄	—	26	2	no	—
179	C₄⋊C₄	—	26	1	no	—
180	C₄⋊C₄	—	26	1	no	—
181	C₄²	—	26	2	no	—
182	C₄²	—	26	2	no	—
183	C₄²	#6094	1	1	yes	characteristic
184	C₄²	#6094	1	1	yes	characteristic
185	C₄²	#6103	1	1	yes	characteristic
186	C₄²	#6103	1	1	yes	characteristic
187	C₁₆	—	4	2	no	—
188	C₁₆	—	52	2	no	—
189	C₁₃	#11493	1	1	yes	characteristic, complemented, Sylow
190	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
191	Q₈	—	26	4	no	—
192	Q₈	—	26	4	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	13	4	no	—
197	C₂×C₄	—	26	2	no	—
198	C₂×C₄	—	26	2	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	52	2	no	—
201	C₂×C₄	C₁₆×D₁₃	1	2	yes	—
202	C₂×C₄	#88038	1	2	yes	—
203	C₂×C₄	#88060	1	1	yes	characteristic
204	C₂×C₄	#88060	1	1	yes	characteristic
205	C₂×C₄	#88060	1	1	yes	characteristic
206	C₂×C₄	#88060	1	1	yes	characteristic
207	C₂×C₄	C₂×#6094	1	1	yes	characteristic
208	C₂×C₄	C₂×#6103	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	52	2	no	—
213	C₈	#88064	1	2	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂×C₁₆×D₁₃	1	1	yes	characteristic
217	C₂²	C₂×#88038	1	1	yes	characteristic
218	C₂²	C₂×#88060	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	26	4	no	—
223	C₄	—	52	2	no	—
224	C₄	#11718099	1	2	yes	—
225	C₄	#11718099	1	2	yes	—
226	C₄	#11718667	1	1	yes	characteristic
227	C₄	C₂×#88064	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#13113159	1	1	yes	characteristic
231	C₂	C₂×#11718099	1	1	yes	characteristic
232	C₂	#13116456	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#6103
commutator	C₂²×C₁₃	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2162	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2162	D₁₃
hypocenter	C₁₃	#11493
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
53. C₂×Dic₂₆	C₁₆	semidirect
189. C₁₃	#11493	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₆	2
#13116635	2
#13112775	2
C₁₃×#2162	2
#11493	13