Subgroups — #15863530

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123469	C₂	1	2	yes	maximal
2	#13115008	C₂	1	2	yes	maximal, complemented
3	#13114974	C₂	1	2	yes	maximal, complemented
4	#13114874	C₂	1	4	yes	maximal
5	C₄×#88059	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718868	C₂²	1	4	yes	—
15	C₂×#88059	C₂²	1	8	yes	—
16	#11718587	—	2	4	no	—
17	#11718586	—	2	4	no	—
18	#11718585	C₂²	1	4	yes	—
19	C₄×#6099	C₂²	1	4	yes	—
20	#11718144	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	—	2	8	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	D₄	1	2	yes	—
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6099	C₂³	1	8	yes	—
35	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₂²⋊#1055	—	2	8	no	—
41	#88059	—	2	16	no	—
42	C₄⋊#1055	—	2	8	no	—
43	#88043	—	2	4	no	—
44	C₄×#1055	—	2	4	no	—
45	#20826	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	8	no	—
48	C₂²×C₄×C₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	2	16	no	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₁₃	#41	1	8	yes	—
64	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂×D₄	1	2	yes	—
73	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
74	#6099	—	2	16	no	—
75	C₂×#1055	—	2	8	no	—
76	#2920	—	13	2	no	—
77	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
78	#1848	—	13	2	no	—
79	#1837	—	13	2	no	—
80	#1810	—	13	4	no	—
81	C₄×#108	—	13	4	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	—	2	4	no	—
84	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	#108	1	8	yes	complemented
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
103	C₈×C₁₃	—	2	4	no	—
104	C₈×C₁₃	#117	1	8	yes	—
105	#1055	—	4	8	no	—
106	C₂³×C₈	C₂×D₁₃	1	2	yes	—
107	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
108	#432	—	26	4	no	—
109	#431	—	26	4	no	—
110	C₂×#108	—	13	8	no	—
111	#423	—	13	4	no	—
112	#407	—	13	4	no	—
113	C₄×M₁₆	—	13	4	no	—
114	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
115	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
116	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
119	#341	—	13	8	no	—
120	C₂²×C₁₃	—	2	8	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	C₂×#108	1	4	yes	—
125	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₃	#533	1	2	yes	—
127	C₄×C₁₃	—	2	2	no	—
128	C₄×C₁₃	—	2	2	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	C₂×#108	1	4	yes	—
131	C₄×C₁₃	C₂×#108	1	4	yes	—
132	C₄×C₁₃	C₂×#117	1	2	yes	—
133	C₄×C₁₃	#532	1	2	yes	—
134	C₂³×C₄	Dic₂₆	1	2	yes	—
135	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂×M₁₆	—	13	8	no	—
137	C₂²×C₈	—	2	8	no	—
138	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
139	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
140	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
141	C₂×C₄²	Dic₂₆	1	2	yes	—
142	C₂×C₄²	D₅₂	1	2	yes	—
143	C₂×C₄²	D₅₂	1	2	yes	—
144	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
145	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
146	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
148	#108	—	26	16	no	—
149	C₄⋊C₈	—	26	8	no	—
150	C₂²⋊C₈	—	26	8	no	—
151	#97	—	26	4	no	—
152	C₄×C₈	—	26	4	no	—
153	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
154	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
155	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
156	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
157	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
158	C₂×C₁₃	—	2	2	no	—
159	C₂×C₁₃	—	2	4	no	—
160	C₂×C₁₃	#2285	1	1	yes	characteristic
161	C₂×C₁₃	C₂²×#108	1	2	yes	—
162	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
166	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
167	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
169	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
170	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
171	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
172	C₂²×C₄	C₂×D₅₂	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
174	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
175	M₁₆	—	26	16	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	16	no	—
178	C₂×C₈	—	26	8	no	—
179	C₂×C₈	#6127	1	8	yes	—
180	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
181	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
182	C₄²	C₄⋊Dic₁₃	1	2	yes	—
183	C₄²	C₄⋊Dic₁₃	1	2	yes	—
184	C₄²	C₄⋊Dic₁₃	1	2	yes	—
185	C₄²	C₄⋊Dic₁₃	1	2	yes	—
186	C₄²	C₂×D₅₂	1	1	yes	characteristic
187	C₄²	C₂×D₅₂	1	2	yes	—
188	C₄²	C₂×D₅₂	1	1	yes	characteristic
189	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
190	C₄²	C₂²×Dic₁₃	1	2	yes	—
191	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
192	C₁₃	#20826	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	2	no	—
194	C₂³	—	2	4	no	—
195	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
196	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	#88059	1	8	yes	complemented
206	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
211	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
212	C₂×C₄	C₂×#6127	1	4	yes	—
213	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₈	—	52	8	no	—
216	C₈	#88180	1	8	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	C₂×#88059	1	4	yes	—
222	C₂²	#11719175	1	2	yes	—
223	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	C₂×#88059	1	4	yes	—
228	C₄	C₂×#88059	1	4	yes	—
229	C₄	#11719182	1	2	yes	—
230	C₄	C₂×#88180	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	#13117243	1	1	yes	characteristic
234	C₂	C₂²×#88059	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#20826
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13115008	C₂	semidirect
3. #13114974	C₂	semidirect
5. C₄×#88059	C₂	semidirect
94. C₂×C₄×C₁₃	#108	semidirect
192. C₁₃	#20826	semidirect
205. C₂×C₄	#88059	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
#13115008	2
#13114974	2
#13114874	2
C₄×#88059	2
C₂²×C₄×C₈×C₁₃	2
#20826	13