Subgroups — #15773991

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
2	#13114865	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112845	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#121	—	2	1	no	—
8	C₁₃×#466	—	2	1	no	—
9	C₁₃×#466	—	2	1	no	—
10	C₁₃×#466	C₄	1	2	yes	complemented
11	C₁₃×#466	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₄	1	1	yes	characteristic, complemented
13	C₂×C₁₃×#102	—	2	1	no	—
14	#11718873	—	2	1	no	—
15	C₂×#88058	—	2	1	no	—
16	C₂²×#6090	—	2	1	no	—
17	#11718143	C₂²	1	2	yes	complemented
18	C₂²⋊#6090	—	2	1	no	—
19	C₂²⋊#6090	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₃	—	2	1	no	—
21	C₂²×D₄×C₁₃	—	4	1	no	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₁₃×D₈	—	2	1	no	—
24	C₂×C₁₃×D₈	—	4	2	no	—
25	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	—	4	2	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#121	—	2	2	no	—
34	C₁₃×#121	—	2	2	no	—
35	C₁₃×#121	—	4	1	no	—
36	C₁₃×#121	—	4	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
39	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₁₃×#107	—	2	1	no	—
42	C₁₃×#107	C₂×C₄	1	2	yes	—
43	C₁₃×#102	—	2	2	no	—
44	C₁₃×#102	—	2	2	no	—
45	#88071	—	4	1	no	—
46	#88058	—	4	2	no	—
47	C₂×#6090	—	2	2	no	—
48	C₂×#6090	—	4	2	no	—
49	#11886	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	—	4	1	no	—
51	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	4	2	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	8	2	no	—
59	C₂×D₄×C₁₃	—	8	2	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	4	2	no	—
70	C₁₃×D₈	—	4	2	no	—
71	C₁₃×D₈	—	4	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	4	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	C₁₃×C₂²⋊C₄	—	4	2	no	—
85	C₁₃×C₂²⋊C₄	—	4	2	no	—
86	C₂²⋊Dic₁₃	—	8	2	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	C₄⋊Dic₁₃	—	4	2	no	—
89	#6090	—	4	2	no	—
90	C₂×#466	D₁₃	1	1	yes	characteristic
91	C₂×C₂²⋊C₁₆	—	13	1	no	—
92	#1808	—	13	1	no	—
93	#1347	—	13	4	no	—
94	C₂³×C₁₃	—	4	1	no	—
95	C₂³×C₁₃	—	4	1	no	—
96	C₂³×C₁₃	—	4	2	no	—
97	C₂³×C₁₃	—	8	2	no	—
98	C₂³×C₁₃	—	8	2	no	—
99	C₂³×C₁₃	—	8	2	no	—
100	C₂³×C₁₃	#102	1	1	yes	characteristic
101	C₂³×C₁₃	#102	1	1	yes	characteristic
102	C₂³×C₁₃	D₁₆	1	2	yes	—
103	C₂³×C₁₃	SD₃₂	1	2	yes	—
104	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	8	2	no	—
108	D₄×C₁₃	—	8	4	no	—
109	D₄×C₁₃	—	8	2	no	—
110	D₄×C₁₃	—	8	2	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×Dic₁₃	—	8	2	no	—
127	C₂×Dic₁₃	—	8	2	no	—
128	C₂×Dic₁₃	—	8	2	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	2	1	no	—
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	#99	1	2	yes	—
133	C₂²×D₈	Dic₁₃	1	1	yes	characteristic, complemented
134	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
135	C₂×#121	—	2	1	no	—
136	C₂²×C₁₆	—	26	1	no	—
137	#466	—	2	1	no	—
138	#466	—	2	1	no	—
139	#466	Dic₁₃	1	2	yes	complemented
140	#466	C₂×D₁₃	1	2	yes	—
141	C₂×#107	—	26	1	no	—
142	C₂×#107	Dic₁₃	1	1	yes	characteristic, complemented
143	C₂×#102	—	2	1	no	—
144	#380	—	26	1	no	—
145	C₂²⋊C₁₆	—	13	2	no	—
146	C₂²⋊C₁₆	—	26	1	no	—
147	#340	—	13	2	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	8	2	no	—
159	C₂²×C₁₃	—	16	4	no	—
160	C₂²×C₁₃	#357	1	2	yes	—
161	C₂²×C₁₃	#357	1	2	yes	—
162	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
170	C₄×C₁₃	#421	1	1	yes	characteristic
171	Dic₁₃	—	16	2	no	—
172	C₂²×D₄	—	2	1	no	—
173	C₂²×D₄	—	4	1	no	—
174	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
178	C₂²×C₈	—	4	2	no	—
179	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	#121	—	2	2	no	—
186	#121	—	2	2	no	—
187	#121	—	4	1	no	—
188	#121	—	4	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	26	2	no	—
191	C₂×C₂²⋊C₄	—	4	1	no	—
192	C₂×C₂²⋊C₄	—	52	1	no	—
193	C₂×C₁₆	—	26	2	no	—
194	C₂×C₁₆	—	52	2	no	—
195	#107	—	2	1	no	—
196	#107	—	52	2	no	—
197	#107	C₂×Dic₁₃	1	2	yes	—
198	#102	—	2	2	no	—
199	#102	—	2	2	no	—
200	#95	—	52	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	2	1	no	—
205	C₂×C₁₃	—	16	2	no	—
206	C₂×C₁₃	#1519	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
208	C₂×C₁₃	#2145	1	1	yes	characteristic
209	C₂⁴	—	4	1	no	—
210	C₂⁴	#6104	1	1	yes	characteristic
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	8	2	no	—
218	C₂×D₄	—	8	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	#6104	1	1	yes	characteristic
224	C₂²×C₄	#6105	1	1	yes	characteristic
225	C₂²×C₄	#6105	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	D₈	—	4	2	no	—
230	D₈	—	4	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	52	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	104	2	no	—
248	C₁₆	—	52	2	no	—
249	C₁₃	#11886	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	#88066	1	2	yes	—
257	C₂³	#88067	1	2	yes	—
258	C₂³	#88072	1	1	yes	characteristic
259	C₂³	#88072	1	1	yes	characteristic
260	C₂³	C₂×#6104	1	1	yes	characteristic
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	4	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	104	2	no	—
279	C₂×C₄	—	104	2	no	—
280	C₂×C₄	—	104	2	no	—
281	C₂×C₄	#88072	1	1	yes	characteristic
282	C₂×C₄	#88072	1	1	yes	characteristic
283	C₂×C₄	C₂×#6105	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	#88074	1	2	yes	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	16	4	no	—
301	C₂²	#11718154	1	2	yes	—
302	C₂²	#11718154	1	2	yes	—
303	C₂²	C₂×#88066	1	1	yes	characteristic
304	C₂²	C₂×#88067	1	1	yes	characteristic
305	C₂²	C₂×#88072	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	208	2	no	—
311	C₄	C₂×#88074	1	1	yes	characteristic
312	C₄	#11718897	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#13113404	1	1	yes	characteristic
319	C₂	C₂×#11718154	1	1	yes	characteristic
320	C₂	#13116572	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting, prime radical	C₂×C₁₃×#466	C₂
socle	C₂²×C₁₃	C₂×#102
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#466	D₁₃
hypocenter	C₁₃	#11886
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊#6090	C₂	semidirect
2. #13114865	C₂	semidirect
3. #13112845	C₂	semidirect
5. C₂²×C₁₃×D₈	C₄	semidirect
10. C₁₃×#466	C₄	semidirect
12. C₂×C₁₃×#107	C₄	semidirect
17. #11718143	C₂²	semidirect
19. C₂²⋊#6090	C₂²	semidirect
133. C₂²×D₈	Dic₁₃	semidirect
139. #466	Dic₁₃	semidirect
142. C₂×#107	Dic₁₃	semidirect
249. C₁₃	#11886	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#6090	2
#13114865	2
#13112845	2
C₂×C₁₃×#466	2
#11886	13