Subgroups — #15768682

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718148	C₂	1	2	yes	maximal
3	#13114553	C₂	1	1	yes	maximal, characteristic
4	#13114547	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	C₂×#88180	C₂²	1	2	yes	—
9	C₂×#88180	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	1	yes	characteristic
11	#11718762	—	2	2	no	—
12	#11718762	C₄	1	4	yes	complemented
13	C₂×#88055	—	2	1	no	—
14	C₂×C₈×Dic₁₃	—	2	1	no	—
15	C₁₃×#341	—	2	1	no	—
16	C₁₃×#341	—	2	1	no	—
17	#11718148	C₄	1	8	yes	complemented
18	#11718147	—	2	4	no	—
19	#11718143	—	2	2	no	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
35	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
36	#88180	—	2	1	no	—
37	#88180	—	2	1	no	—
38	#88180	—	2	2	no	—
39	#88180	—	2	2	no	—
40	#88180	—	2	2	no	—
41	#88180	C₂×C₄	1	4	yes	—
42	#88180	C₂×C₄	1	2	yes	—
43	#88180	D₄	1	2	yes	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄⋊Dic₁₃	Q₈	1	2	yes	—
46	C₂×C₄×Dic₁₃	—	2	1	no	—
47	C₂×C₄×Dic₁₃	—	2	2	no	—
48	C₂×#6099	—	2	1	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	C₂×C₄	1	4	yes	—
51	C₂×#6099	C₂×C₄	1	2	yes	—
52	C₂²×#1055	—	2	1	no	—
53	#88055	—	2	4	no	—
54	C₈×Dic₁₃	—	2	4	no	—
55	#10828	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×Dic₁₃	—	2	2	no	—
68	C₂²×Dic₁₃	—	4	2	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	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
83	C₂²⋊Dic₁₃	—	2	2	no	—
84	C₂²⋊Dic₁₃	—	2	2	no	—
85	C₂²⋊Dic₁₃	—	2	4	no	—
86	C₂²⋊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	C₄⋊C₄	1	4	yes	—
91	C₄×Dic₁₃	—	2	2	no	—
92	C₄×Dic₁₃	—	2	2	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	4	2	no	—
95	#6099	—	2	2	no	—
96	#6099	—	2	4	no	—
97	#6099	C₄²	1	4	yes	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#405	—	13	1	no	—
101	C₂×#344	—	13	2	no	—
102	C₂×#341	D₁₃	1	1	yes	characteristic
103	#1735	—	13	1	no	—
104	#1733	—	13	2	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	#117	1	1	yes	characteristic
107	C₂³×C₁₃	#117	1	2	yes	—
108	C₂³×C₁₃	#121	1	1	yes	characteristic
109	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂³×C₁₃	#126	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
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×Q₈	1	2	yes	—
122	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#123	1	2	yes	—
125	C₂×C₄×C₁₃	#123	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#125	1	2	yes	—
128	C₂×C₄×C₁₃	#126	1	2	yes	—
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	4	4	no	—
132	C₂×Dic₁₃	—	8	4	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	4	4	no	—
136	#1055	—	2	4	no	—
137	C₂²×M₁₆	—	13	1	no	—
138	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
139	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂×#117	—	13	1	no	—
141	C₂×#117	—	13	2	no	—
142	#405	—	13	4	no	—
143	#405	—	26	2	no	—
144	C₂×#97	—	26	1	no	—
145	C₂×C₄×C₈	—	26	1	no	—
146	#344	—	13	8	no	—
147	#341	—	2	1	no	—
148	#341	—	2	1	no	—
149	#340	—	26	2	no	—
150	#339	—	26	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	#376	1	1	yes	characteristic
157	C₂²×C₁₃	#380	1	1	yes	characteristic
158	C₂²×C₁₃	#383	1	1	yes	characteristic
159	C₂²×C₁₃	#387	1	2	yes	—
160	C₂²×C₁₃	#388	1	2	yes	—
161	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₁₃	#381	1	1	yes	characteristic
164	C₄×C₁₃	#382	1	1	yes	characteristic
165	C₄×C₁₃	#387	1	2	yes	—
166	C₄×C₁₃	#388	1	2	yes	—
167	Dic₁₃	—	4	4	no	—
168	Dic₁₃	—	8	4	no	—
169	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	13	4	no	—
176	C₂×M₁₆	—	13	2	no	—
177	C₂×M₁₆	—	26	1	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	26	1	no	—
184	C₂²×C₈	C₄×D₁₃	1	2	yes	—
185	C₂²×C₈	D₅₂	1	2	yes	—
186	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
188	#117	—	13	2	no	—
189	#117	—	13	2	no	—
190	#117	—	13	4	no	—
191	#117	—	26	1	no	—
192	#117	—	26	1	no	—
193	#117	—	26	2	no	—
194	#117	—	26	2	no	—
195	#117	—	26	2	no	—
196	C₂×C₄⋊C₄	—	13	2	no	—
197	C₂×C₄⋊C₄	—	26	2	no	—
198	C₂×C₂²⋊C₄	—	26	1	no	—
199	C₂×C₂²⋊C₄	—	26	2	no	—
200	C₂×C₄²	—	26	1	no	—
201	C₂×C₄²	—	26	2	no	—
202	#97	—	26	4	no	—
203	C₄×C₈	—	26	4	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	—	2	1	no	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	#1434	1	1	yes	characteristic
209	C₂×C₁₃	#1814	1	2	yes	—
210	C₂⁴	#6131	1	1	yes	characteristic
211	C₂²×C₄	—	26	2	no	—
212	C₂²×C₄	—	52	2	no	—
213	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
215	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
216	C₂²×C₄	#6127	1	2	yes	—
217	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	#6129	1	1	yes	characteristic
219	C₂²×C₄	#6129	1	2	yes	—
220	C₂²×C₄	#6129	1	2	yes	—
221	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	13	4	no	—
228	M₁₆	—	26	2	no	—
229	M₁₆	—	26	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	26	2	no	—
239	C₂×C₈	—	26	4	no	—
240	C₂×C₈	#6103	1	2	yes	—
241	C₂×C₈	#6103	1	2	yes	—
242	C₄⋊C₄	—	13	4	no	—
243	C₄⋊C₄	—	26	2	no	—
244	C₄⋊C₄	—	26	4	no	—
245	C₄⋊C₄	—	52	2	no	—
246	C₂²⋊C₄	—	26	2	no	—
247	C₂²⋊C₄	—	26	2	no	—
248	C₂²⋊C₄	—	26	4	no	—
249	C₂²⋊C₄	—	52	2	no	—
250	C₄²	—	26	2	no	—
251	C₄²	—	26	2	no	—
252	C₄²	—	26	4	no	—
253	C₄²	—	52	2	no	—
254	C₁₃	#10828	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	4	2	no	—
256	C₂³	#88146	1	1	yes	characteristic
257	C₂³	#88146	1	2	yes	—
258	C₂³	#88149	1	1	yes	characteristic
259	C₂³	#88152	1	2	yes	—
260	C₂³	#88200	1	1	yes	characteristic
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	52	4	no	—
263	C₂×C₄	—	52	4	no	—
264	C₂×C₄	—	104	4	no	—
265	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
266	C₂×C₄	#88126	1	1	yes	characteristic
267	C₂×C₄	C₄×D₅₂	1	2	yes	—
268	C₂×C₄	#88131	1	1	yes	characteristic
269	C₂×C₄	#88131	1	1	yes	characteristic
270	C₂×C₄	#88131	1	2	yes	—
271	C₂×C₄	#88133	1	2	yes	—
272	C₂×C₄	#88135	1	2	yes	—
273	C₂×C₄	#88139	1	2	yes	—
274	C₂×C₄	#88140	1	2	yes	—
275	C₂×C₄	#88141	1	2	yes	—
276	C₂×C₄	#88143	1	2	yes	—
277	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
278	C₂×C₄	#88151	1	1	yes	characteristic
279	C₂×C₄	C₂×#6103	1	1	yes	characteristic
280	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
281	C₂×C₄	#88189	1	2	yes	—
282	C₂×C₄	#88192	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	4	4	no	—
286	C₈	—	26	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	#11718309	1	1	yes	characteristic
293	C₂²	#11718312	1	1	yes	characteristic
294	C₂²	#11718331	1	2	yes	—
295	C₂²	#11718639	1	2	yes	—
296	C₂²	#11718648	1	1	yes	characteristic
297	C₄	—	52	4	no	—
298	C₄	—	104	4	no	—
299	C₄	#11718310	1	1	yes	characteristic
300	C₄	#11718311	1	1	yes	characteristic
301	C₄	#11718314	1	2	yes	—
302	C₄	C₄×#6103	1	1	yes	characteristic
303	C₄	#11718604	1	1	yes	characteristic
304	C₄	#11718605	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	#13113060	1	1	yes	characteristic
310	C₂	#13114891	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#341	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#341	D₁₃
hypocenter	C₁₃	#10828
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
12. #11718762	C₄	semidirect
17. #11718148	C₄	semidirect
254. C₁₃	#10828	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718762	2
C₂×#11718148	2
#13114553	2
#13114547	2
C₂×C₁₃×#341	2
#10828	13