Subgroups — #15863191

Order 3328.

684 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123254	C₂	1	1	yes	maximal, characteristic
2	#13114986	C₂	1	8	yes	maximal, complemented
3	C₂×#11718151	C₂	1	2	yes	maximal
4	#13114547	C₂	1	2	yes	maximal
5	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂×#88180	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	1	yes	characteristic
12	#11718783	C₄	1	4	yes	complemented
13	M₁₆×Dic₁₃	C₂²	1	4	yes	—
14	#11718762	C₄	1	4	yes	complemented
15	C₁₃×#338	C₂²	1	4	yes	—
16	#11718151	C₂²	1	8	yes	—
17	#11718149	C₂²	1	4	yes	—
18	#11718147	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
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	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
29	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
33	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
34	#88180	—	2	1	no	—
35	#88180	—	2	1	no	—
36	#88180	—	2	2	no	—
37	#88180	—	2	2	no	—
38	#88180	—	2	2	no	—
39	#88180	C₂×C₄	1	2	yes	—
40	#88180	C₂×C₄	1	2	yes	—
41	#88180	C₂³	1	4	yes	—
42	C₂×C₄⋊Dic₁₃	—	2	1	no	—
43	C₂×C₄⋊Dic₁₃	—	2	2	no	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	C₂×C₄	1	2	yes	—
49	C₂×#6099	C₂³	1	2	yes	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	#88055	C₂×C₄	1	8	yes	complemented
52	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
53	#20776	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	#41	1	2	yes	—
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂²×C₄×C₁₃	#41	1	2	yes	—
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	2	no	—
66	C₂²×Dic₁₃	—	4	2	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	4	2	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂²⋊Dic₁₃	—	2	2	no	—
80	C₂²⋊Dic₁₃	—	2	2	no	—
81	C₂²⋊Dic₁₃	—	2	4	no	—
82	C₂²⋊Dic₁₃	—	4	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	—	4	2	no	—
87	C₄×Dic₁₃	—	2	2	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	C₄×Dic₁₃	—	4	2	no	—
90	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
91	#6099	—	2	2	no	—
92	#6099	—	2	2	no	—
93	#6099	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	C₂²×C₄	1	4	yes	—
96	#2873	—	13	1	no	—
97	#1841	—	13	8	no	—
98	C₂×#344	—	13	2	no	—
99	#1733	—	13	2	no	—
100	C₂×#338	D₁₃	1	1	yes	characteristic
101	C₂×H(3,ℤ/4)	—	13	1	no	—
102	C₂³×C₁₃	—	8	1	no	—
103	C₂³×C₁₃	#117	1	1	yes	characteristic
104	C₂³×C₁₃	#117	1	1	yes	characteristic
105	C₂³×C₁₃	#126	1	2	yes	—
106	C₂³×C₁₃	#126	1	2	yes	—
107	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	2	yes	—
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#126	1	2	yes	—
117	C₂×C₄×C₁₃	#126	1	2	yes	—
118	C₂×C₄×C₁₃	#126	1	2	yes	—
119	C₂×C₄×C₁₃	#126	1	2	yes	—
120	C₂×C₄×C₁₃	#126	1	2	yes	—
121	C₂×C₄×C₁₃	#126	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₁₃	—	2	4	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	8	2	no	—
129	C₂×Dic₁₃	—	8	4	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	4	4	no	—
132	#1055	—	2	4	no	—
133	C₂²×M₁₆	—	13	1	no	—
134	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
135	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
136	C₂×#117	—	13	1	no	—
137	C₂×#117	—	13	2	no	—
138	#405	—	13	4	no	—
139	#405	—	13	4	no	—
140	C₄×M₁₆	—	13	4	no	—
141	#344	—	13	8	no	—
142	#339	—	13	8	no	—
143	#338	C₂×D₁₃	1	4	yes	—
144	H(3,ℤ/4)	—	13	4	no	—
145	C₂²×C₁₃	—	4	2	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	—	4	2	no	—
150	C₂²×C₁₃	#383	1	2	yes	—
151	C₂²×C₁₃	#383	1	1	yes	characteristic
152	C₂²×C₁₃	#383	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×#126	1	2	yes	—
155	C₄×C₁₃	#383	1	2	yes	—
156	C₄×C₁₃	#383	1	1	yes	characteristic
157	C₄×C₁₃	#383	1	1	yes	characteristic
158	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
159	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
160	C₄×C₁₃	C₂×#126	1	2	yes	—
161	Dic₁₃	—	4	4	no	—
162	Dic₁₃	—	8	4	no	—
163	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	13	2	no	—
172	C₂×M₁₆	—	13	2	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
176	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
177	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	C₄×D₁₃	1	2	yes	—
181	#117	—	13	2	no	—
182	#117	—	13	2	no	—
183	#117	—	13	4	no	—
184	#117	—	26	1	no	—
185	#117	—	26	1	no	—
186	#117	—	26	2	no	—
187	#117	—	26	2	no	—
188	#117	—	26	2	no	—
189	C₂×C₄⋊C₄	—	26	1	no	—
190	C₂×C₄⋊C₄	—	26	2	no	—
191	C₂×C₂²⋊C₄	—	26	1	no	—
192	C₂×C₂²⋊C₄	—	26	2	no	—
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	—	26	2	no	—
195	#97	—	13	8	no	—
196	C₄×C₈	—	13	8	no	—
197	C₂×C₁₃	—	2	1	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	C₂×#383	1	1	yes	characteristic
203	C₂⁴	#6131	1	1	yes	characteristic
204	C₂²×C₄	—	26	2	no	—
205	C₂²×C₄	—	52	2	no	—
206	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	#6127	1	2	yes	—
209	C₂²×C₄	#6127	1	2	yes	—
210	C₂²×C₄	#6129	1	2	yes	—
211	C₂²×C₄	#6129	1	2	yes	—
212	C₂²×C₄	#6129	1	1	yes	characteristic
213	C₂²×C₄	#6129	1	1	yes	characteristic
214	C₂²×C₄	#6131	1	1	yes	characteristic
215	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	26	2	no	—
223	M₁₆	—	26	2	no	—
224	M₁₆	—	26	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	13	4	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
233	C₄⋊C₄	—	26	2	no	—
234	C₄⋊C₄	—	26	2	no	—
235	C₄⋊C₄	—	26	4	no	—
236	C₄⋊C₄	—	52	2	no	—
237	C₂²⋊C₄	—	26	2	no	—
238	C₂²⋊C₄	—	26	2	no	—
239	C₂²⋊C₄	—	26	4	no	—
240	C₂²⋊C₄	—	52	2	no	—
241	C₄²	—	13	4	no	—
242	C₄²	—	26	2	no	—
243	C₄²	—	26	4	no	—
244	C₄²	—	52	2	no	—
245	C₁₃	#20776	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	8	1	no	—
247	C₂³	#88146	1	1	yes	characteristic
248	C₂³	#88146	1	1	yes	characteristic
249	C₂³	#88152	1	2	yes	—
250	C₂³	#88152	1	2	yes	—
251	C₂³	C₂×#6131	1	1	yes	characteristic
252	C₂×C₄	—	26	4	no	—
253	C₂×C₄	—	52	4	no	—
254	C₂×C₄	—	104	2	no	—
255	C₂×C₄	—	104	4	no	—
256	C₂×C₄	#88126	1	2	yes	—
257	C₂×C₄	#88126	1	1	yes	characteristic
258	C₂×C₄	#88126	1	1	yes	characteristic
259	C₂×C₄	#88131	1	1	yes	characteristic
260	C₂×C₄	#88131	1	1	yes	characteristic
261	C₂×C₄	#88131	1	2	yes	—
262	C₂×C₄	#88133	1	2	yes	—
263	C₂×C₄	#88133	1	2	yes	—
264	C₂×C₄	#88133	1	2	yes	—
265	C₂×C₄	#88133	1	2	yes	—
266	C₂×C₄	#88146	1	1	yes	characteristic
267	C₂×C₄	#88146	1	1	yes	characteristic
268	C₂×C₄	#88152	1	2	yes	—
269	C₂×C₄	#88152	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#6127	1	2	yes	—
272	C₂×C₄	C₂×#6129	1	2	yes	—
273	C₂×C₄	C₂×#6129	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	4	4	no	—
276	C₈	—	26	4	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#11718312	1	2	yes	—
283	C₂²	#11718312	1	1	yes	characteristic
284	C₂²	#11718312	1	1	yes	characteristic
285	C₂²	C₂×#88146	1	1	yes	characteristic
286	C₂²	C₂×#88152	1	2	yes	—
287	C₄	—	52	4	no	—
288	C₄	—	104	4	no	—
289	C₄	#11718312	1	2	yes	—
290	C₄	#11718312	1	1	yes	characteristic
291	C₄	#11718312	1	1	yes	characteristic
292	C₄	C₂×#88126	1	1	yes	characteristic
293	C₄	C₂×#88131	1	1	yes	characteristic
294	C₄	C₂×#88133	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	C₂×#11718312	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88126
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#338	C₂
socle	C₂×C₁₃	C₂×#383
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#338	D₁₃
hypocenter	C₁₃	#20776
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
2. #13114986	C₂	semidirect
12. #11718783	C₄	semidirect
14. #11718762	C₄	semidirect
51. #88055	C₂×C₄	semidirect
52. C₈×Dic₁₃	C₂×C₄	semidirect
245. C₁₃	#20776	semidirect

Maximal subgroups

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

subgroup	index
#13123254	2
#13114986	2
C₂×#11718151	2
#13114547	2
C₂×#11718149	2
C₂×C₁₃×#338	2
#20776	13