Subgroups — #15863348

Order 3328.

1190 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718782	C₂	1	4	yes	maximal
2	#13117243	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718596	C₂	1	1	yes	maximal, characteristic
4	#13115004	C₂	1	4	yes	maximal, complemented
5	#13114996	C₂	1	2	yes	maximal, complemented
6	#13114993	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₁₃	—	2	2	no	—
12	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₄×C₁₃×M₁₆	C₄	1	8	yes	complemented
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	#11718873	—	2	2	no	—
17	#11718782	—	2	16	no	—
18	C₂×#88058	C₂²	1	8	yes	—
19	C₂×#88057	C₂²	1	8	yes	—
20	#11718597	C₂²	1	2	yes	—
21	#11718596	—	2	2	no	—
22	#11718596	C₂²	1	2	yes	—
23	#11718596	C₂²	1	2	yes	—
24	#11718596	C₂²	1	1	yes	characteristic
25	#11718596	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂³×C₄×C₁₃	D₄	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×Dic₁₃	—	2	2	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₁₃×M₁₆	D₄	1	8	yes	—
32	C₂×C₁₃×M₁₆	Q₈	1	8	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
40	#88180	—	2	8	no	—
41	C₂×C₄⋊Dic₁₃	—	2	8	no	—
42	C₂×C₄⋊Dic₁₃	—	2	4	no	—
43	C₂×C₄⋊Dic₁₃	—	2	4	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×C₄⋊Dic₁₃	—	4	8	no	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
51	C₂×C₄×Dic₁₃	—	2	2	no	—
52	C₁₃×#97	C₂×C₄	1	8	yes	complemented
53	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
54	#88071	—	2	4	no	—
55	#88058	—	2	16	no	—
56	#88057	—	2	16	no	—
57	#20800	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂²×Dic₁₃	—	2	2	no	—
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	—	4	8	no	—
74	C₂²×Dic₁₃	—	4	4	no	—
75	C₁₃×M₁₆	C₄⋊C₄	1	32	yes	complemented
76	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
78	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
79	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₂²⋊Dic₁₃	—	4	4	no	—
85	C₄⋊Dic₁₃	—	2	8	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	2	4	no	—
88	C₄⋊Dic₁₃	—	4	8	no	—
89	C₄⋊Dic₁₃	—	4	4	no	—
90	C₄⋊Dic₁₃	—	4	4	no	—
91	C₄⋊Dic₁₃	—	4	8	no	—
92	C₄⋊Dic₁₃	—	4	4	no	—
93	C₄×Dic₁₃	—	4	4	no	—
94	C₂×#428	—	13	4	no	—
95	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
96	#2285	—	13	1	no	—
97	C₂×#384	—	13	1	no	—
98	#1847	—	13	4	no	—
99	#1845	—	13	2	no	—
100	#1844	—	13	2	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₁₃	#127	1	2	yes	—
105	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	16	no	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₁₃	#127	1	2	yes	—
115	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
116	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
118	C₂×C₄×C₁₃	#137	1	4	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	4	8	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	8	8	no	—
130	C₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	complemented
131	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
132	C₂×#117	—	13	2	no	—
133	C₂²×C₄⋊C₄	—	13	2	no	—
134	C₂²×C₄⋊C₄	—	26	2	no	—
135	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
136	#428	—	26	16	no	—
137	C₂×#107	—	13	8	no	—
138	C₂×#106	—	13	8	no	—
139	C₄×M₁₆	Dic₁₃	1	8	yes	complemented
140	C₂×#97	C₂×D₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
142	#384	—	13	1	no	—
143	#384	—	13	2	no	—
144	#384	—	13	1	no	—
145	#384	—	13	2	no	—
146	#384	—	26	2	no	—
147	#382	—	13	2	no	—
148	#380	—	26	2	no	—
149	C₄×C₄⋊C₄	—	26	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	4	no	—
153	C₂²×C₁₃	—	2	4	no	—
154	C₂²×C₁₃	#384	1	4	yes	—
155	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₂²×C₁₃	C₂×#137	1	2	yes	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	#384	1	4	yes	—
162	C₄×C₁₃	#428	1	8	yes	complemented
163	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
164	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
165	Dic₁₃	—	8	2	no	—
166	Dic₁₃	—	8	4	no	—
167	Dic₁₃	—	8	2	no	—
168	C₂³×C₄	—	26	2	no	—
169	C₂³×C₄	D₅₂	1	2	yes	—
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×M₁₆	Dic₂₆	1	8	yes	—
172	C₂×M₁₆	D₅₂	1	8	yes	—
173	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
174	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
175	#117	—	26	8	no	—
176	C₂×C₄⋊C₄	—	13	4	no	—
177	C₂×C₄⋊C₄	—	13	2	no	—
178	C₂×C₄⋊C₄	—	13	2	no	—
179	C₂×C₄⋊C₄	—	26	8	no	—
180	C₂×C₄⋊C₄	—	26	2	no	—
181	C₂×C₄⋊C₄	—	26	4	no	—
182	C₂×C₄⋊C₄	—	26	4	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	—	26	2	no	—
185	C₂×C₄⋊C₄	—	52	8	no	—
186	C₂×C₂²⋊C₄	—	26	2	no	—
187	C₂×C₄²	—	26	2	no	—
188	C₂×C₄²	Dic₂₆	1	4	yes	—
189	C₂×C₄²	D₅₂	1	2	yes	—
190	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
191	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	#107	—	26	16	no	—
194	#106	—	26	16	no	—
195	#97	C₂×Dic₁₃	1	8	yes	complemented
196	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
197	#95	—	26	4	no	—
198	C₂×C₁₃	—	2	4	no	—
199	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
200	C₂×C₁₃	C₂×#428	1	2	yes	—
201	C₂×C₁₃	#3151	1	1	yes	characteristic
202	C₂×C₁₃	#3152	1	1	yes	characteristic
203	C₂×C₁₃	#3165	1	2	yes	—
204	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	26	2	no	—
208	C₂²×C₄	—	26	2	no	—
209	C₂²×C₄	—	26	4	no	—
210	C₂²×C₄	—	52	8	no	—
211	C₂²×C₄	—	52	4	no	—
212	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
213	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
214	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
215	C₂²×C₄	C₂×D₅₂	1	2	yes	—
216	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₅₂	1	4	yes	—
218	C₂²×C₄	C₂×D₅₂	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
220	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
221	M₁₆	C₄⋊Dic₁₃	1	32	yes	complemented
222	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
223	C₂×C₈	C₂×D₅₂	1	8	yes	—
224	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
225	C₄⋊C₄	—	26	4	no	—
226	C₄⋊C₄	—	26	8	no	—
227	C₄⋊C₄	—	26	4	no	—
228	C₄⋊C₄	—	52	8	no	—
229	C₄⋊C₄	—	52	4	no	—
230	C₄⋊C₄	—	52	8	no	—
231	C₄⋊C₄	—	52	4	no	—
232	C₄⋊C₄	—	52	4	no	—
233	C₂²⋊C₄	—	52	4	no	—
234	C₄²	—	52	4	no	—
235	C₄²	C₄⋊Dic₁₃	1	8	yes	—
236	C₄²	C₂×Dic₂₆	1	2	yes	—
237	C₄²	C₂×Dic₂₆	1	2	yes	—
238	C₄²	C₂²×Dic₁₃	1	2	yes	—
239	C₄²	C₂²×Dic₁₃	1	2	yes	—
240	C₁₃	#20800	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	#88123	1	2	yes	—
244	C₂³	#88128	1	2	yes	—
245	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
246	C₂³	C₂²×D₅₂	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	52	2	no	—
251	C₂×C₄	—	52	2	no	—
252	C₂×C₄	—	52	4	no	—
253	C₂×C₄	—	52	8	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	—	104	8	no	—
257	C₂×C₄	#88123	1	8	yes	—
258	C₂×C₄	#88123	1	2	yes	—
259	C₂×C₄	#88128	1	2	yes	—
260	C₂×C₄	#88162	1	4	yes	—
261	C₂×C₄	#88163	1	4	yes	—
262	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
266	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
268	C₂×C₄	C₂²×D₅₂	1	2	yes	—
269	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
270	C₈	C₂×C₄⋊Dic₁₃	1	16	yes	complemented
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	#11718596	1	4	yes	—
276	C₂²	C₂×#88123	1	1	yes	characteristic
277	C₂²	C₂×#88128	1	1	yes	characteristic
278	C₂²	C₂×#88162	1	2	yes	—
279	C₂²	C₂×#88163	1	2	yes	—
280	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	104	2	no	—
283	C₄	—	104	4	no	—
284	C₄	—	104	2	no	—
285	C₄	#11718596	1	4	yes	—
286	C₄	#11718782	1	8	yes	complemented
287	C₄	C₂×#88123	1	2	yes	—
288	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
289	C₂	—	2	4	no	—
290	C₂	C₂×#11718596	1	1	yes	characteristic
291	C₂	C₂×#11718782	1	2	yes	—
292	C₂	#13125288	1	1	yes	characteristic
293	C₂	#13125295	1	1	yes	characteristic
294	C₂	#13125382	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#20800
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #13115004	C₂	semidirect
5. #13114996	C₂	semidirect
6. #13114993	C₂	semidirect
13. C₄×C₁₃×M₁₆	C₄	semidirect
52. C₁₃×#97	C₂×C₄	semidirect
53. C₄×C₈×C₁₃	C₂×C₄	semidirect
75. C₁₃×M₁₆	C₄⋊C₄	semidirect
130. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
139. C₄×M₁₆	Dic₁₃	semidirect
162. C₄×C₁₃	#428	semidirect
195. #97	C₂×Dic₁₃	semidirect
196. C₄×C₈	C₂×Dic₁₃	semidirect
221. M₁₆	C₄⋊Dic₁₃	semidirect
240. C₁₃	#20800	semidirect
270. C₈	C₂×C₄⋊Dic₁₃	semidirect
286. C₄	#11718782	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718782	2
#13117243	2
C₂×#11718596	2
#13115004	2
#13114996	2
#13114993	2
C₂×C₄×C₁₃×M₁₆	2
#20800	13