Subgroups — #15781162

Order 3328.

482 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	#13125264	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718089	C₂	1	4	yes	maximal, complemented
3	C₂×#11718053	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88158	—	2	1	no	—
6	C₂×#88158	C₂²	1	2	yes	—
7	#11719182	—	2	1	no	—
8	C₂×C₁₃×#108	—	2	1	no	—
9	C₂×C₁₃×#108	C₂²	1	2	yes	—
10	C₁₃×#423	—	2	1	no	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	2	yes	—
13	#11718360	C₄	1	1	yes	characteristic
14	#11718357	—	2	1	no	—
15	#11718357	C₄	1	2	yes	—
16	#11718356	C₄	1	1	yes	characteristic
17	#11718354	—	2	1	no	—
18	#11718089	—	2	8	no	—
19	#11718053	—	2	1	no	—
20	#11718053	—	2	1	no	—
21	C₂×#6127	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	#88184	—	4	2	no	—
27	C₂×Dic₅₂	—	2	1	no	—
28	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
29	#88158	—	2	4	no	—
30	#88158	—	2	4	no	—
31	#88158	—	4	2	no	—
32	#88158	—	4	2	no	—
33	C₂×D₁₀₄	—	2	1	no	—
34	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
35	C₂×#6095	—	2	2	no	—
36	C₂×#6095	—	2	2	no	—
37	#88129	—	2	2	no	—
38	#88128	—	2	1	no	—
39	C₄×D₅₂	—	4	2	no	—
40	#88123	—	2	1	no	—
41	C₁₃×#108	—	2	4	no	—
42	C₁₃×#108	—	4	2	no	—
43	C₁₃×C₄⋊C₈	—	4	2	no	—
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₂×#6090	—	2	2	no	—
49	C₂×#6090	—	2	2	no	—
50	C₂×#6090	D₄	1	4	yes	complemented
51	#13209	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₃	—	4	2	no	—
55	#6127	—	4	2	no	—
56	#6127	—	4	2	no	—
57	#6127	—	4	2	no	—
58	#6127	—	4	2	no	—
59	C₂×D₅₂	—	2	2	no	—
60	C₂×D₅₂	—	4	2	no	—
61	C₂×C₄×D₁₃	—	4	2	no	—
62	C₂×Dic₂₆	—	2	2	no	—
63	C₁₃×M₁₆	—	4	2	no	—
64	C₁₃×M₁₆	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	D₈	1	2	yes	—
69	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	#41	1	2	yes	—
74	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₃	D₈	1	2	yes	—
77	#6103	—	4	4	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	Dic₅₂	—	2	4	no	—
80	Dic₅₂	—	4	2	no	—
81	D₁₀₄	—	2	4	no	—
82	D₁₀₄	—	4	2	no	—
83	#6095	—	4	4	no	—
84	#6095	—	4	4	no	—
85	#6090	—	2	4	no	—
86	#3147	—	13	1	no	—
87	C₂×#359	—	13	4	no	—
88	C₂×#346	—	13	1	no	—
89	#1848	D₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₂²×D₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	#102	1	2	yes	—
94	C₂×C₄×C₁₃	#102	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂²⋊D₁₃	—	8	2	no	—
102	C₂²⋊D₁₃	—	8	2	no	—
103	C₂×Dic₁₃	—	4	2	no	—
104	D₅₂	—	4	2	no	—
105	D₅₂	—	4	2	no	—
106	D₅₂	—	8	4	no	—
107	C₄×D₁₃	—	8	2	no	—
108	C₄×D₁₃	—	8	2	no	—
109	Dic₂₆	—	4	2	no	—
110	Dic₂₆	—	4	2	no	—
111	C₈×C₁₃	—	8	2	no	—
112	C₈×C₁₃	#102	1	2	yes	—
113	C₈×C₁₃	#102	1	2	yes	—
114	C₈×C₁₃	C₄×D₄	1	2	yes	—
115	C₈×C₁₃	#121	1	2	yes	—
116	C₂×#135	—	13	2	no	—
117	C₂×#135	—	26	1	no	—
118	#532	—	26	1	no	—
119	C₂²×C₁₆	—	13	2	no	—
120	#495	—	13	2	no	—
121	#495	—	26	1	no	—
122	#494	—	13	1	no	—
123	#493	—	13	1	no	—
124	#492	—	26	1	no	—
125	C₂×#108	—	2	1	no	—
126	C₂×#108	C₂×D₁₃	1	2	yes	—
127	#423	—	2	1	no	—
128	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
129	#359	—	26	8	no	—
130	#346	—	26	1	no	—
131	#346	—	26	1	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
134	C₂²×C₁₃	#486	1	1	yes	characteristic
135	C₂²×C₁₃	#492	1	1	yes	characteristic
136	C₂×D₁₃	—	8	4	no	—
137	C₂×D₁₃	—	8	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	#390	1	1	yes	characteristic
141	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
142	C₄×C₁₃	#486	1	1	yes	characteristic
143	C₄×C₁₃	#493	1	1	yes	characteristic
144	Dic₁₃	—	8	2	no	—
145	C₂×#41	—	26	2	no	—
146	#135	—	26	4	no	—
147	#135	—	26	4	no	—
148	#135	—	52	2	no	—
149	#135	—	52	2	no	—
150	C₂×Q₁₆	—	13	2	no	—
151	C₂×Q₁₆	—	26	1	no	—
152	C₂×SD₁₆	—	26	2	no	—
153	C₂×SD₁₆	—	26	2	no	—
154	C₂×D₈	—	13	2	no	—
155	C₂×D₈	—	26	1	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	26	1	no	—
160	#127	—	26	1	no	—
161	#124	—	26	2	no	—
162	#121	—	52	2	no	—
163	C₄×D₄	—	52	2	no	—
164	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂×C₁₆	—	13	4	no	—
166	C₂×C₁₆	—	26	2	no	—
167	C₂×C₁₆	—	26	2	no	—
168	#108	—	2	4	no	—
169	#108	—	4	2	no	—
170	C₄⋊C₈	—	4	2	no	—
171	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
172	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
173	C₄×C₈	D₅₂	1	1	yes	characteristic
174	C₄×C₈	D₅₂	1	1	yes	characteristic
175	C₂×C₁₃	—	2	1	no	—
176	C₂×C₁₃	—	2	1	no	—
177	C₂×C₁₃	#1957	1	1	yes	characteristic
178	C₂×C₁₃	#2143	1	2	yes	—
179	D₁₃	—	16	2	no	—
180	#41	—	52	2	no	—
181	#41	—	52	2	no	—
182	#41	—	52	2	no	—
183	#41	—	52	2	no	—
184	C₂×Q₈	—	26	2	no	—
185	C₂×D₄	—	26	2	no	—
186	C₂×D₄	—	52	2	no	—
187	C₂×D₄	—	52	2	no	—
188	C₂²×C₄	—	52	2	no	—
189	C₂²×C₄	#6106	1	2	yes	—
190	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
191	Q₁₆	—	26	4	no	—
192	Q₁₆	—	52	2	no	—
193	SD₁₆	—	52	4	no	—
194	SD₁₆	—	52	4	no	—
195	D₈	—	26	4	no	—
196	D₈	—	52	2	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	#6103	1	1	yes	characteristic
201	C₂×C₈	#6103	1	1	yes	characteristic
202	C₂×C₈	#6104	1	2	yes	—
203	C₂×C₈	#6106	1	2	yes	—
204	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
206	C₂×C₈	D₄×D₁₃	1	2	yes	—
207	C₂×C₈	#6131	1	2	yes	—
208	C₄⋊C₄	—	52	2	no	—
209	C₂²⋊C₄	—	52	4	no	—
210	C₄²	#6103	1	1	yes	characteristic
211	C₄²	#6103	1	1	yes	characteristic
212	C₄²	#6104	1	2	yes	—
213	C₁₆	—	26	4	no	—
214	C₁₃	#13209	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	52	2	no	—
216	C₂³	C₂×#6106	1	1	yes	characteristic
217	Q₈	—	52	2	no	—
218	Q₈	—	52	2	no	—
219	D₄	—	52	2	no	—
220	D₄	—	52	2	no	—
221	D₄	—	104	4	no	—
222	D₄	—	104	2	no	—
223	D₄	—	104	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	52	2	no	—
226	C₂×C₄	—	104	2	no	—
227	C₂×C₄	—	104	2	no	—
228	C₂×C₄	#88049	1	2	yes	—
229	C₂×C₄	#88049	1	2	yes	—
230	C₂×C₄	C₂×#6103	1	1	yes	characteristic
231	C₂×C₄	C₂×#6104	1	1	yes	characteristic
232	C₂×C₄	C₂×#6104	1	1	yes	characteristic
233	C₂×C₄	#88194	1	1	yes	characteristic
234	C₂×C₄	C₂×#6106	1	1	yes	characteristic
235	C₂×C₄	#88201	1	1	yes	characteristic
236	C₈	—	8	2	no	—
237	C₈	#88049	1	2	yes	—
238	C₈	#88049	1	2	yes	—
239	C₈	#88147	1	2	yes	—
240	C₈	#88149	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	104	4	no	—
243	C₂²	—	104	2	no	—
244	C₂²	C₂×#88049	1	1	yes	characteristic
245	C₂²	#11718746	1	1	yes	characteristic
246	C₂²	#11718749	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	104	2	no	—
250	C₄	C₂×#88049	1	1	yes	characteristic
251	C₄	#11718649	1	1	yes	characteristic
252	C₄	#11718734	1	1	yes	characteristic
253	C₄	#11718749	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	208	2	no	—
257	C₂	#13115641	1	1	yes	characteristic
258	C₂	#13116563	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	#13125264	C₂
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#13209
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
2. C₂×#11718089	C₂	semidirect
3. C₂×#11718053	C₂	semidirect
4. C₁₃×#1848	C₂	semidirect
50. C₂×#6090	D₄	semidirect
89. #1848	D₁₃	semidirect
214. C₁₃	#13209	semidirect

Maximal subgroups

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

subgroup	index
#13125264	2
C₂×#11718089	2
C₂×#11718053	2
C₁₃×#1848	2
#13209	13