Subgroups — #15768107

Order 3328.

870 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₁₃	C₂	1	2	yes	maximal
2	C₂²×#88055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718150	C₂	1	2	yes	maximal
5	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	M₁₆×Dic₁₃	—	2	4	no	—
12	M₁₆×Dic₁₃	C₄	1	8	yes	—
13	C₂×#88055	—	2	4	no	—
14	C₂×#88055	—	2	2	no	—
15	C₂×#88055	C₄	1	4	yes	—
16	C₂×C₈×Dic₁₃	—	2	2	no	—
17	C₁₃×#341	C₄	1	4	yes	complemented
18	#11718150	C₄	1	8	yes	complemented
19	#11718144	C₄	1	4	yes	complemented
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₃	Q₈	1	2	yes	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₁₃	D₄	1	4	yes	—
38	C₂×#6099	—	2	2	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	C₂×C₄	1	4	yes	—
42	C₂×#6099	C₂×C₄	1	4	yes	—
43	C₂×#6099	C₂×C₄	1	4	yes	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	—	2	2	no	—
46	C₂²×#1055	—	2	1	no	—
47	C₂²×#1055	—	2	1	no	—
48	C₂²×#1055	C₂×C₄	1	4	yes	—
49	C₂²×#1055	C₂×C₄	1	2	yes	—
50	#88055	—	2	8	no	—
51	#88055	—	2	8	no	—
52	C₈×Dic₁₃	—	2	8	no	—
53	#10705	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	4	4	no	—
66	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
67	C₂²×Dic₁₃	C₄⋊C₄	1	8	yes	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	C₄²	1	8	yes	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	C₄²	1	4	yes	—
80	C₄×Dic₁₃	C₄²	1	4	yes	—
81	C₄×Dic₁₃	C₂²⋊C₄	1	4	yes	—
82	C₄×Dic₁₃	C₂²⋊C₄	1	4	yes	—
83	C₄×Dic₁₃	C₂²⋊C₄	1	4	yes	—
84	#6099	—	2	4	no	—
85	#6099	—	2	4	no	—
86	#6099	—	2	4	no	—
87	#6099	C₄²	1	8	yes	—
88	C₂×#1055	—	2	2	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	C₄²	1	4	yes	—
96	C₂×C₄×M₁₆	—	13	2	no	—
97	C₂²×#97	—	13	1	no	—
98	C₂×#341	—	13	2	no	—
99	C₂×#341	—	13	1	no	—
100	C₂×#341	D₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
105	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
106	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	2	yes	—
116	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#117	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×Dic₁₃	—	2	8	no	—
126	C₂×Dic₁₃	—	4	8	no	—
127	C₂×Dic₁₃	#95	1	8	yes	—
128	C₈×C₁₃	—	2	4	no	—
129	C₈×C₁₃	—	2	8	no	—
130	#1055	—	2	4	no	—
131	#1055	—	2	8	no	—
132	C₂²×M₁₆	—	13	2	no	—
133	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
134	C₂³×C₈	—	13	1	no	—
135	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂²×C₄²	—	13	1	no	—
137	C₄×M₁₆	—	13	8	no	—
138	C₄×M₁₆	—	26	4	no	—
139	C₂×#97	—	13	4	no	—
140	C₂×#97	—	26	4	no	—
141	C₂×#97	—	26	2	no	—
142	C₂×C₄×C₈	—	26	2	no	—
143	#341	—	13	8	no	—
144	#341	—	13	4	no	—
145	#341	Dic₁₃	1	4	yes	complemented
146	C₂²×C₁₃	—	2	4	no	—
147	C₂²×C₁₃	—	2	4	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
151	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
152	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
154	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₃	#430	1	4	yes	—
156	C₄×C₁₃	C₄³	1	1	yes	characteristic
157	C₄×C₁₃	#376	1	1	yes	characteristic
158	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
160	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
162	Dic₁₃	—	2	8	no	—
163	C₂³×C₄	—	13	2	no	—
164	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	13	4	no	—
169	C₂×M₁₆	—	13	4	no	—
170	C₂×M₁₆	—	13	4	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂×M₁₆	—	26	2	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
175	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
176	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	13	4	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	26	2	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	—	26	1	no	—
185	C₂²×C₈	—	26	1	no	—
186	C₂²×C₈	C₄×D₁₃	1	2	yes	—
187	C₂²×C₈	C₄×D₁₃	1	2	yes	—
188	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
189	C₂×C₄²	—	13	2	no	—
190	C₂×C₄²	—	13	4	no	—
191	C₂×C₄²	—	13	4	no	—
192	C₂×C₄²	—	13	2	no	—
193	#97	—	26	8	no	—
194	#97	—	26	8	no	—
195	C₄×C₈	—	26	8	no	—
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	2	2	no	—
198	C₂×C₁₃	C₄×#95	1	1	yes	characteristic
199	C₂×C₁₃	#1738	1	2	yes	—
200	C₂×C₁₃	#1778	1	2	yes	—
201	C₂×C₁₃	#1778	1	2	yes	—
202	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	—	13	4	no	—
204	C₂²×C₄	—	13	8	no	—
205	C₂²×C₄	—	52	4	no	—
206	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₁₃	1	2	yes	—
211	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
212	C₂²×C₄	#6129	1	2	yes	—
213	C₂²×C₄	#6129	1	1	yes	characteristic
214	C₂²×C₄	#6131	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	13	8	no	—
220	M₁₆	—	26	4	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	—	26	4	no	—
223	M₁₆	C₄×Dic₁₃	1	8	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	13	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	—	26	2	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
240	C₄²	—	13	4	no	—
241	C₄²	—	13	4	no	—
242	C₄²	—	13	4	no	—
243	C₄²	—	13	4	no	—
244	C₁₃	#10705	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	2	4	no	—
246	C₂³	#88141	1	2	yes	—
247	C₂³	#88141	1	1	yes	characteristic
248	C₂³	C₄⋊C₄×D₁₃	1	2	yes	—
249	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
250	C₂³	Q₈×Dic₁₃	1	1	yes	characteristic
251	C₂×C₄	—	13	8	no	—
252	C₂×C₄	—	26	8	no	—
253	C₂×C₄	—	52	8	no	—
254	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
255	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
256	C₂×C₄	C₄²×D₁₃	1	2	yes	—
257	C₂×C₄	#88131	1	1	yes	characteristic
258	C₂×C₄	#88131	1	2	yes	—
259	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂²⋊C₄×D₁₃	1	2	yes	—
261	C₂×C₄	#88135	1	2	yes	—
262	C₂×C₄	#88135	1	2	yes	—
263	C₂×C₄	#88135	1	2	yes	—
264	C₂×C₄	#88146	1	1	yes	characteristic
265	C₂×C₄	#88146	1	2	yes	—
266	C₂×C₄	#88147	1	1	yes	characteristic
267	C₂×C₄	#88147	1	2	yes	—
268	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
270	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	2	8	no	—
274	C₈	—	26	4	no	—
275	C₈	—	26	8	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	#11718310	1	2	yes	—
281	C₂²	#11718310	1	1	yes	characteristic
282	C₂²	D₁₃×#95	1	1	yes	characteristic
283	C₂²	#11718554	1	4	yes	—
284	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
285	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
286	C₄	—	26	8	no	—
287	C₄	#11718309	1	1	yes	characteristic
288	C₄	#11718327	1	1	yes	characteristic
289	C₄	#11718327	1	2	yes	—
290	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
291	C₄	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
292	C₄	C₂²⋊C₄×Dic₁₃	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	#95×Dic₁₃	1	1	yes	characteristic
296	C₂	#13114561	1	2	yes	—
297	C₂	#13114741	1	2	yes	—
298	C₂	#13114744	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₄³
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#341	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#341	D₁₃
hypocenter	C₁₃	#10705
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
17. C₁₃×#341	C₄	semidirect
18. #11718150	C₄	semidirect
19. #11718144	C₄	semidirect
145. #341	Dic₁₃	semidirect
244. C₁₃	#10705	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₁₃	2
C₂²×#88055	2
C₂×#11718144	2
C₂×#11718150	2
C₂×C₁₃×#341	2
#10705	13