Subgroups — #16018129

Order 3328.

544 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123705	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117024	C₂	1	2	yes	maximal, complemented
3	#13116997	C₂	1	2	yes	maximal, complemented
4	#13116873	C₂	1	2	yes	maximal, complemented
5	#13116752	C₂	1	2	yes	maximal, complemented
6	#13116700	C₂	1	2	yes	maximal, complemented
7	C₄×#88067	C₂	1	2	yes	maximal, complemented
8	#13116643	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3176	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#540	—	2	1	no	—
11	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
12	#11719221	—	2	1	no	—
13	C₁₃×#499	C₂²	1	1	yes	characteristic
14	C₁₃×#493	C₂²	1	1	yes	characteristic
15	C₁₃×#490	—	2	1	no	—
16	C₁₃×#469	—	2	1	no	—
17	C₁₃×#466	C₂²	1	2	yes	—
18	C₁₃×#442	—	2	1	no	—
19	C₄×C₁₃×D₈	C₂²	1	2	yes	—
20	C₁₃×#405	C₂²	1	1	yes	characteristic
21	C₂×#88067	C₂²	1	2	yes	—
22	C₂×#88067	C₂²	1	2	yes	—
23	#11718781	—	2	1	no	—
24	#11718767	C₂²	1	1	yes	characteristic
25	#11718377	—	2	1	no	—
26	#11718377	—	2	1	no	—
27	C₄×Dic₅₂	C₂²	1	2	yes	—
28	C₄×Dic₅₂	C₂²	1	2	yes	—
29	#11718154	C₂²	1	2	yes	—
30	#11718154	C₂²	1	2	yes	—
31	C₂²⋊#6090	C₂²	1	2	yes	—
32	#11718084	C₂²	1	2	yes	—
33	#11718084	C₂²	1	2	yes	—
34	#11718082	C₂²	1	2	yes	—
35	#11718081	C₂²	1	2	yes	—
36	C₄⋊#6090	C₂²	1	2	yes	—
37	#11718053	C₂²	1	1	yes	characteristic
38	C₄×#6090	C₂²	1	1	yes	characteristic
39	C₂²×Dic₂₆	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#127	—	2	1	no	—
45	C₁₃×#125	—	2	1	no	—
46	C₁₃×#123	—	4	2	no	—
47	C₁₃×#121	—	2	2	no	—
48	C₁₃×#121	—	4	2	no	—
49	C₄×D₄×C₁₃	—	2	2	no	—
50	C₁₃×#117	D₄	1	1	yes	characteristic
51	#88180	—	2	1	no	—
52	C₂×Dic₅₂	—	2	1	no	—
53	C₂×Dic₅₂	—	2	4	no	—
54	C₂×Dic₅₂	C₂³	1	2	yes	—
55	#88141	—	4	2	no	—
56	#88131	—	4	2	no	—
57	C₄×Dic₂₆	—	2	2	no	—
58	C₄×Dic₂₆	—	2	2	no	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#106	—	2	1	no	—
61	C₁₃×#102	—	2	2	no	—
62	C₁₃×#102	—	2	2	no	—
63	C₁₃×#97	D₄	1	1	yes	characteristic
64	C₁₃×#97	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	#88067	—	2	4	no	—
68	#88067	—	2	4	no	—
69	#88058	C₂³	1	2	yes	—
70	#88057	—	2	1	no	—
71	#88056	—	2	2	no	—
72	#88056	—	2	2	no	—
73	C₂×#6090	C₂³	1	2	yes	—
74	C₂×#6090	C₂³	1	2	yes	—
75	#33686	—	13	1	no	maximal, Sylow
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	—	4	2	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₁₃	—	4	1	no	—
82	C₂×Dic₂₆	—	2	4	no	—
83	C₂×Dic₂₆	—	2	2	no	—
84	C₂×Dic₂₆	—	4	2	no	—
85	C₂×Dic₂₆	—	4	1	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₁₃×D₈	—	4	2	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂²⋊Dic₁₃	—	4	1	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	#6101	—	4	2	no	—
106	#6101	—	4	2	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	C₄×Dic₁₃	—	4	1	no	—
110	Dic₅₂	—	2	4	no	—
111	Dic₅₂	—	2	2	no	—
112	Dic₅₂	#41	1	4	yes	—
113	#6090	—	2	2	no	—
114	#6090	—	2	2	no	—
115	#3176	D₁₃	1	1	yes	characteristic
116	#2950	—	13	1	no	—
117	#2234	—	13	2	no	—
118	#2229	—	13	2	no	—
119	#2207	—	13	2	no	—
120	#2185	—	13	2	no	—
121	#2175	—	13	2	no	—
122	C₄×SD₃₂	—	13	2	no	—
123	#2164	—	13	1	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	16	2	no	—
129	D₄×C₁₃	—	16	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	16	2	no	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₁₃	—	4	4	no	—
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	Dic₂₆	—	2	4	no	—
142	Dic₂₆	—	4	4	no	—
143	Dic₂₆	—	4	2	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	2	yes	—
148	C₂²×Q₁₆	—	13	1	no	—
149	#540	—	2	1	no	—
150	#519	—	26	1	no	—
151	C₂×SD₃₂	—	13	2	no	—
152	C₂×SD₃₂	—	13	2	no	—
153	#499	C₂×D₁₃	1	1	yes	characteristic
154	#493	C₂×D₁₃	1	1	yes	characteristic
155	#490	—	2	1	no	—
156	#469	—	2	1	no	—
157	#466	C₂×D₁₃	1	2	yes	—
158	#442	—	2	1	no	—
159	#441	—	26	1	no	—
160	#441	—	26	1	no	—
161	C₄×Q₁₆	—	13	2	no	—
162	C₄×Q₁₆	—	13	2	no	—
163	C₄×D₈	C₂×D₁₃	1	2	yes	—
164	#427	—	13	1	no	—
165	#419	—	26	1	no	—
166	#405	C₂×D₁₃	1	1	yes	characteristic
167	#367	—	13	2	no	—
168	#366	—	13	2	no	—
169	C₄⋊C₁₆	—	13	2	no	—
170	#358	—	13	2	no	—
171	#358	—	13	2	no	—
172	#357	—	13	2	no	—
173	#357	—	13	2	no	—
174	C₂²⋊C₁₆	—	13	2	no	—
175	#346	—	13	1	no	—
176	C₄×C₁₆	—	13	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	#525	1	1	yes	characteristic
188	C₄×C₁₃	#575	1	1	yes	characteristic
189	Dic₁₃	—	4	4	no	—
190	Dic₁₃	—	8	2	no	—
191	C₂²×Q₈	—	26	1	no	—
192	C₂×Q₁₆	—	13	2	no	—
193	C₂×Q₁₆	—	26	4	no	—
194	C₂×Q₁₆	—	26	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	C₂²×D₁₃	1	2	yes	—
197	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
199	#127	—	2	1	no	—
200	#125	—	2	1	no	—
201	#123	—	4	2	no	—
202	#121	—	2	2	no	—
203	#121	—	4	2	no	—
204	C₄×Q₈	—	26	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×Q₈	—	52	2	no	—
207	C₄×D₄	—	2	2	no	—
208	#117	—	26	1	no	—
209	#117	—	52	2	no	—
210	#117	C₂²⋊D₁₃	1	1	yes	characteristic
211	SD₃₂	—	26	4	no	—
212	SD₃₂	—	26	4	no	—
213	C₂×C₁₆	—	13	2	no	—
214	C₂×C₁₆	—	13	2	no	—
215	#107	—	13	2	no	—
216	#107	C₂²×D₁₃	1	2	yes	—
217	#106	—	2	1	no	—
218	#106	—	26	1	no	—
219	#103	—	26	2	no	—
220	#103	—	26	2	no	—
221	#102	—	2	2	no	—
222	#102	—	2	2	no	—
223	#97	C₂²⋊D₁₃	1	1	yes	characteristic
224	#97	C₂²⋊D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×C₁₃	—	4	1	no	—
228	C₂×C₁₃	—	16	2	no	—
229	C₂×C₁₃	#3100	1	1	yes	characteristic
230	C₂×C₁₃	#3416	1	2	yes	—
231	C₂×Q₈	—	26	4	no	—
232	C₂×Q₈	—	26	2	no	—
233	C₂×Q₈	—	52	2	no	—
234	C₂×Q₈	—	52	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	52	1	no	—
240	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
241	Q₁₆	—	13	4	no	—
242	Q₁₆	—	26	4	no	—
243	Q₁₆	—	26	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	4	2	no	—
247	M₁₆	—	4	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	26	2	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	#6104	1	2	yes	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	52	1	no	—
262	C₂²⋊C₄	#6104	1	2	yes	—
263	C₄²	—	52	1	no	—
264	C₄²	—	52	2	no	—
265	C₄²	—	52	2	no	—
266	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₁₆	—	26	2	no	—
269	C₁₆	—	26	2	no	—
270	C₁₃	#33686	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	4	2	no	—
272	C₂³	C₂×#6104	1	1	yes	characteristic
273	Q₈	—	26	4	no	—
274	Q₈	—	52	4	no	—
275	Q₈	—	52	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	16	2	no	—
279	D₄	—	16	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	16	2	no	—
284	C₂×C₄	—	52	4	no	—
285	C₂×C₄	—	52	2	no	—
286	C₂×C₄	—	52	2	no	—
287	C₂×C₄	C₂×#6104	1	2	yes	—
288	C₂×C₄	C₂×#6104	1	2	yes	—
289	C₂×C₄	C₂×#6104	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	#41×D₁₃	1	2	yes	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	C₂²×#6104	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	2	no	—
305	C₄	—	52	4	no	—
306	C₄	—	104	2	no	—
307	C₄	#11719527	1	1	yes	characteristic
308	C₄	#11719533	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	16	2	no	—
311	C₂	#13124832	1	1	yes	characteristic
312	C₂	#13127983	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3176	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3176	D₁₃
hypocenter	C₁₃	#33686
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

Complements

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

N	H ≅ G/N	type
1. #13123705	C₂	semidirect
2. #13117024	C₂	semidirect
3. #13116997	C₂	semidirect
4. #13116873	C₂	semidirect
5. #13116752	C₂	semidirect
6. #13116700	C₂	semidirect
7. C₄×#88067	C₂	semidirect
8. #13116643	C₂	semidirect
270. C₁₃	#33686	semidirect

Maximal subgroups

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

subgroup	index
#13123705	2
#13117024	2
#13116997	2
#13116873	2
#13116752	2
#13116700	2
C₄×#88067	2
#13116643	2
C₁₃×#3176	2
#33686	13