Subgroups — #15863086

Order 3328.

922 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13114956	C₂	1	4	yes	maximal, complemented
4	#13114953	C₂	1	4	yes	maximal, complemented
5	#13114533	C₂	1	4	yes	maximal
6	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂³×#1055	C₂²	1	2	yes	—
11	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
13	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
14	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
15	C₁₃×#376	C₂²	1	2	yes	—
16	C₄³×C₁₃	C₂²	1	2	yes	—
17	#11718585	—	2	4	no	—
18	C₂×C₄⋊#1055	—	2	4	no	—
19	C₂×C₄⋊#1055	C₂²	1	2	yes	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	#11718116	C₂²	1	16	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×#117	—	2	4	no	—
28	C₁₃×#117	—	2	4	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	4	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6099	—	2	4	no	—
41	C₂×#6099	—	2	8	no	—
42	C₂²×#1055	—	2	4	no	—
43	C₂²×#1055	—	2	8	no	—
44	C₂²×#1055	C₂³	1	4	yes	—
45	C₂²×#1055	C₂³	1	4	yes	—
46	C₁₃×#95	C₂×C₄	1	4	yes	—
47	C₁₃×#95	C₂×C₄	1	4	yes	—
48	C₄⋊#1055	—	2	4	no	—
49	C₄⋊#1055	—	2	4	no	—
50	C₄⋊#1055	—	2	4	no	—
51	C₄⋊#1055	—	2	4	no	—
52	#20755	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	4	yes	—
67	C₂²×C₄×C₁₃	#41	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	—	2	8	no	—
70	C₁₃×C₂²⋊C₄	—	2	8	no	—
71	C₄²×C₁₃	—	2	8	no	—
72	C₄²×C₁₃	—	2	4	no	—
73	C₄²×C₁₃	—	2	4	no	—
74	C₄²×C₁₃	—	4	4	no	—
75	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
76	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
77	C₄²×C₁₃	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
81	#6099	—	4	8	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	8	no	—
85	C₂×#1055	—	2	8	no	—
86	C₂×#1055	—	4	8	no	—
87	C₂×#423	—	13	1	no	—
88	C₂²×C₄⋊C₈	—	13	1	no	—
89	C₄×#117	D₁₃	1	1	yes	characteristic
90	#1832	—	13	4	no	—
91	#1831	—	13	4	no	—
92	#1729	—	13	4	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₁₃	#117	1	4	yes	—
97	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	8	no	—
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	2	8	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃	#117	1	4	yes	—
112	C₂×C₄×C₁₃	#124	1	4	yes	—
113	C₂×C₄×C₁₃	#124	1	4	yes	—
114	C₂×C₄×C₁₃	#125	1	8	yes	—
115	C₂×C₄×C₁₃	#131	1	4	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
120	#1055	—	4	4	no	—
121	#1055	—	4	4	no	—
122	C₂²×M₁₆	—	13	2	no	—
123	C₂³×C₈	—	13	2	no	—
124	C₂×#117	C₂×D₁₃	1	2	yes	—
125	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
126	#423	—	26	4	no	—
127	C₂×C₄⋊C₈	—	13	2	no	—
128	C₂×C₄⋊C₈	—	13	2	no	—
129	C₂×C₄⋊C₈	—	13	2	no	—
130	C₂×C₄⋊C₈	—	13	2	no	—
131	C₂×C₄⋊C₈	—	26	4	no	—
132	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
133	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
134	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
135	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
136	#376	C₂×D₁₃	1	2	yes	—
137	C₄³	C₂×D₁₃	1	2	yes	—
138	#336	—	13	16	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	8	no	—
142	C₂²×C₁₃	#382	1	4	yes	—
143	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×#117	1	2	yes	—
145	C₂²×C₁₃	C₂×#131	1	2	yes	—
146	C₂²×C₁₃	#568	1	2	yes	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	4	8	no	—
151	C₄×C₁₃	#382	1	4	yes	—
152	C₄×C₁₃	#424	1	4	yes	—
153	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
154	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#125	1	2	yes	—
156	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
158	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
159	C₂×M₁₆	—	26	4	no	—
160	C₂×M₁₆	—	26	8	no	—
161	C₂²×C₈	—	13	4	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	—	26	4	no	—
164	C₂²×C₈	—	26	8	no	—
165	#117	—	2	4	no	—
166	#117	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
168	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
169	C₂×C₄²	Dic₂₆	1	4	yes	—
170	C₂×C₄²	D₅₂	1	4	yes	—
171	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
172	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
173	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
176	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊C₈	—	26	4	no	—
179	C₄⋊C₈	—	26	4	no	—
180	C₄⋊C₈	—	26	4	no	—
181	C₄⋊C₈	—	26	4	no	—
182	#95	C₂×Dic₁₃	1	4	yes	—
183	#95	C₂×Dic₁₃	1	4	yes	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
186	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
187	C₂×C₁₃	#2905	1	1	yes	characteristic
188	C₂×C₁₃	#2925	1	4	yes	—
189	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
193	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
194	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
195	C₂²×C₄	C₂×D₅₂	1	2	yes	—
196	C₂²×C₄	#6127	1	4	yes	—
197	C₂²×C₄	#6127	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
205	M₁₆	—	52	8	no	—
206	C₂×C₈	—	26	4	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	8	no	—
209	C₂×C₈	—	26	8	no	—
210	C₂×C₈	—	52	8	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₄²	—	2	8	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₄⋊Dic₁₃	1	4	yes	—
218	C₄²	C₄⋊Dic₁₃	1	4	yes	—
219	C₄²	C₂×Dic₂₆	1	2	yes	—
220	C₄²	C₂×Dic₂₆	1	2	yes	—
221	C₄²	C₂×D₅₂	1	2	yes	—
222	C₄²	C₂×D₅₂	1	2	yes	—
223	C₁₃	#20755	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
227	C₂³	#88180	1	4	yes	—
228	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	#88124	1	8	yes	—
238	C₂×C₄	#88129	1	4	yes	—
239	C₂×C₄	#88129	1	4	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
245	C₂×C₄	#88180	1	4	yes	—
246	C₂×C₄	#88202	1	4	yes	—
247	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
249	C₂×C₄	C₂×#6127	1	2	yes	—
250	C₂×C₄	C₂×#6127	1	2	yes	—
251	C₈	—	52	4	no	—
252	C₈	—	52	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	8	no	—
256	C₂²	#11718597	1	4	yes	—
257	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
258	C₂²	C₂×#88180	1	2	yes	—
259	C₂²	C₂×#88202	1	2	yes	—
260	C₂²	#11719524	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	8	no	—
265	C₄	#11718597	1	4	yes	—
266	C₄	#11718660	1	4	yes	—
267	C₄	C₂×#88124	1	2	yes	—
268	C₄	C₂×#88129	1	1	yes	characteristic
269	C₄	C₂×#88129	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	C₂×#11718597	1	1	yes	characteristic
272	C₂	C₂×#11718660	1	1	yes	characteristic
273	C₂	#13123400	1	1	yes	characteristic
274	C₂	#13123502	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×#117	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#20755
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #13114956	C₂	semidirect
4. #13114953	C₂	semidirect
223. C₁₃	#20755	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
C₂²×C₄⋊#1055	2
#13114956	2
#13114953	2
#13114533	2
C₄×C₁₃×#117	2
#20755	13