Subgroups — #16807258

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	2	yes	maximal, complemented
2	#13131253	C₂	1	2	yes	maximal, complemented
3	#13131252	C₂	1	2	yes	maximal, complemented
4	#13131251	C₂	1	2	yes	maximal, complemented
5	#13131172	C₂	1	2	yes	maximal
6	#13131169	C₂	1	1	yes	maximal, characteristic
7	#13131046	C₂	1	2	yes	maximal
8	#13131037	C₂	1	1	yes	maximal, characteristic
9	#13123767	C₂	1	1	yes	maximal, characteristic
10	#11719283	C₄	1	1	yes	characteristic
11	C₂×#88098	C₂²	1	2	yes	—
12	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×#446	C₄	1	1	yes	characteristic
15	Q₈×#1055	C₂²	1	1	yes	characteristic
16	#11718500	C₄	1	2	yes	—
17	#11718498	C₄	1	1	yes	characteristic
18	#11718491	C₂²	1	2	yes	—
19	#11718491	C₂²	1	1	yes	characteristic
20	#11718369	C₂²	1	2	yes	—
21	#11718362	C₄	1	2	yes	—
22	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	#11718340	C₄	1	1	yes	characteristic
24	#11718260	C₂²	1	1	yes	characteristic
25	#11718260	C₂²	1	1	yes	characteristic
26	#11718260	C₂²	1	2	yes	—
27	#11718260	C₂²	1	2	yes	—
28	#11718258	C₂²	1	2	yes	—
29	#11718258	C₂²	1	2	yes	—
30	#11718257	C₂²	1	2	yes	—
31	#11718231	C₂²	1	2	yes	—
32	#11718231	C₂²	1	2	yes	—
33	C₄⋊#6092	C₂²	1	2	yes	—
34	C₄⋊#6092	C₂²	1	2	yes	—
35	C₄⋊#6092	C₂²	1	2	yes	—
36	C₄×#6092	C₂²	1	2	yes	—
37	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88200	C₈	1	1	yes	characteristic
39	#88198	C₈	1	1	yes	characteristic
40	C₂×#6094	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
43	#88151	C₈	1	1	yes	characteristic
44	#88150	C₈	1	2	yes	—
45	#88143	C₈	1	2	yes	—
46	#88142	C₈	1	1	yes	characteristic
47	#88126	C₂×C₄	1	2	yes	—
48	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
51	C₂×#6092	C₂³	1	2	yes	—
52	C₂×#6092	C₂³	1	2	yes	—
53	C₂×#6092	C₂³	1	2	yes	—
54	C₂×#6092	C₂³	1	2	yes	—
55	#88098	—	2	4	no	—
56	D₁₃⋊C₁₆	—	2	2	no	—
57	D₁₃⋊C₁₆	Q₈	1	4	yes	complemented
58	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₁₃×#97	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	#88055	C₂×C₄	1	2	yes	—
63	#88054	C₂×C₄	1	2	yes	—
64	#88054	C₂×C₄	1	2	yes	—
65	#88054	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₁₃	C₂³	1	2	yes	—
68	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
70	C₄⋊#1055	C₂×C₄	1	2	yes	—
71	C₄×#1055	C₂³	1	2	yes	—
72	C₄×#1055	C₂³	1	1	yes	characteristic
73	#33475	—	13	1	no	maximal, Sylow
74	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
79	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
87	#6103	C₂×C₈	1	2	yes	—
88	#6103	C₂×C₈	1	2	yes	—
89	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
90	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
91	#6101	C₂×C₈	1	2	yes	—
92	#6101	C₂×C₈	1	2	yes	—
93	#6101	C₂×C₈	1	2	yes	—
94	#6101	C₂×C₈	1	2	yes	—
95	#6101	C₂×C₈	1	2	yes	—
96	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
101	C₂×#1055	C₂²×C₄	1	2	yes	—
102	C₂×#1055	C₂⁴	1	1	yes	characteristic
103	#6094	—	2	4	no	—
104	C₈×D₁₃	—	2	2	no	—
105	C₈×D₁₃	C₂×Q₈	1	2	yes	—
106	C₈×D₁₃	#41	1	2	yes	—
107	#6092	—	2	2	no	—
108	#6092	—	2	2	no	—
109	#6092	—	2	2	no	—
110	#6092	C₂×Q₈	1	4	yes	—
111	#2960	—	13	1	no	—
112	#2183	—	13	2	no	—
113	#2183	—	13	2	no	—
114	#2183	—	13	2	no	—
115	#2183	—	13	2	no	—
116	#2151	—	13	2	no	—
117	#2150	—	13	1	no	—
118	#2108	—	13	2	no	—
119	C₂×#346	—	13	1	no	—
120	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
121	Q₈×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
130	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
131	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₄×D₁₃	—	4	1	no	—
133	C₄×D₁₃	—	4	1	no	—
134	C₄×D₁₃	C₄×Q₈	1	2	yes	—
135	C₄×D₁₃	#131	1	2	yes	—
136	Dic₂₆	—	4	2	no	—
137	Dic₂₆	—	4	1	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	C₄×Q₈	1	2	yes	—
141	#1055	—	2	1	no	—
142	#1055	—	2	2	no	—
143	#1055	C₂²×Q₈	1	1	yes	characteristic
144	#1055	C₂×#41	1	1	yes	characteristic
145	#533	—	13	1	no	—
146	C₂×M₃₂	—	13	2	no	—
147	C₂²×C₁₆	—	13	1	no	—
148	C₂²×C₁₆	—	13	1	no	—
149	#446	—	13	2	no	—
150	#446	—	13	1	no	—
151	#446	#261	1	1	yes	characteristic
152	C₈×Q₈	—	13	1	no	—
153	C₈×Q₈	—	13	1	no	—
154	C₈×Q₈	—	13	2	no	—
155	#424	—	13	2	no	—
156	#423	—	13	1	no	—
157	#405	—	13	2	no	—
158	C₂×C₄×C₈	—	13	1	no	—
159	C₄⋊C₁₆	—	13	2	no	—
160	C₄⋊C₁₆	—	13	2	no	—
161	C₄⋊C₁₆	—	13	2	no	—
162	C₄⋊C₁₆	—	13	1	no	—
163	C₄⋊C₁₆	—	13	1	no	—
164	C₄⋊C₁₆	—	13	2	no	—
165	C₄⋊C₁₆	—	13	2	no	—
166	#346	—	13	2	no	—
167	#346	—	13	2	no	—
168	#346	—	13	2	no	—
169	#346	—	13	2	no	—
170	C₄×C₁₆	—	13	2	no	—
171	C₄×C₁₆	—	13	2	no	—
172	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
173	C₂×D₁₃	—	2	1	no	—
174	C₂×D₁₃	—	2	1	no	—
175	C₂×D₁₃	C₈×Q₈	1	2	yes	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
182	Dic₁₃	—	2	1	no	—
183	Dic₁₃	—	2	2	no	—
184	Dic₁₃	—	2	2	no	—
185	Dic₁₃	—	2	1	no	—
186	Dic₁₃	C₈×Q₈	1	2	yes	—
187	C₂×M₁₆	—	13	2	no	—
188	C₂²×C₈	—	13	1	no	—
189	C₂²×C₈	—	13	1	no	—
190	C₄⋊Q₈	—	13	1	no	—
191	C₄⋊Q₈	—	13	1	no	—
192	#125	—	13	2	no	—
193	C₂²⋊Q₈	—	13	1	no	—
194	C₂²⋊Q₈	—	13	2	no	—
195	C₂²⋊Q₈	—	13	1	no	—
196	C₄×Q₈	—	13	2	no	—
197	C₄×Q₈	—	13	1	no	—
198	C₄×Q₈	C₂×#261	1	1	yes	characteristic
199	#117	—	13	2	no	—
200	C₂×C₄²	—	13	1	no	—
201	M₃₂	—	26	4	no	—
202	C₂×C₁₆	—	13	4	no	—
203	C₂×C₁₆	—	13	2	no	—
204	C₂×C₁₆	—	13	2	no	—
205	C₂×C₁₆	—	13	2	no	—
206	C₂×C₁₆	—	13	2	no	—
207	C₂×C₁₆	—	26	2	no	—
208	C₄⋊C₈	—	13	2	no	—
209	C₄⋊C₈	—	13	2	no	—
210	C₄⋊C₈	—	13	2	no	—
211	C₄⋊C₈	—	13	1	no	—
212	C₄⋊C₈	—	13	2	no	—
213	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
214	C₄⋊C₈	C₂×#261	1	2	yes	—
215	#97	—	13	2	no	—
216	#97	C₂×#261	1	2	yes	—
217	C₄×C₈	—	13	2	no	—
218	C₄×C₈	—	13	1	no	—
219	C₄×C₈	—	13	1	no	—
220	C₄×C₈	—	13	2	no	—
221	C₄×C₈	—	13	1	no	—
222	C₄×C₈	C₂×#261	1	1	yes	characteristic
223	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
224	C₂×C₁₃	#3407	1	1	yes	characteristic
225	C₂×C₁₃	#3407	1	1	yes	characteristic
226	D₁₃	—	2	2	no	—
227	C₂×Q₈	—	13	2	no	—
228	C₂×Q₈	—	13	1	no	—
229	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
230	C₂²×C₄	—	13	1	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	13	1	no	—
233	M₁₆	—	26	4	no	—
234	C₂×C₈	—	13	1	no	—
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	1	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	26	2	no	—
240	C₂×C₈	C₂²×#261	1	1	yes	characteristic
241	C₂×C₈	C₂²×#261	1	2	yes	—
242	C₂×C₈	C₂²×#261	1	1	yes	characteristic
243	C₄⋊C₄	—	13	2	no	—
244	C₄⋊C₄	—	13	2	no	—
245	C₄⋊C₄	—	13	2	no	—
246	C₄⋊C₄	—	13	2	no	—
247	C₄⋊C₄	—	13	2	no	—
248	C₄⋊C₄	—	13	2	no	—
249	C₄⋊C₄	—	13	1	no	—
250	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
251	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
252	C₂²⋊C₄	—	13	2	no	—
253	C₂²⋊C₄	—	13	2	no	—
254	C₄²	—	13	1	no	—
255	C₄²	—	13	1	no	—
256	C₄²	—	13	1	no	—
257	C₄²	—	13	2	no	—
258	C₄²	C₂²×#261	1	1	yes	characteristic
259	C₄²	C₂²×#261	1	2	yes	—
260	C₁₆	—	13	4	no	—
261	C₁₆	—	26	2	no	—
262	C₁₆	—	26	2	no	—
263	C₁₆	—	26	2	no	—
264	C₁₃	#33475	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	13	1	no	—
266	Q₈	—	4	1	no	—
267	Q₈	—	52	2	no	—
268	Q₈	—	52	1	no	—
269	C₂×C₄	—	13	1	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	13	1	no	—
273	C₂×C₄	—	13	1	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	2	no	—
276	C₂×C₄	—	52	1	no	—
277	C₂×C₄	—	52	1	no	—
278	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
281	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
282	C₂×C₄	C₂³×#261	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	2	2	no	—
285	C₈	—	13	1	no	—
286	C₈	—	13	1	no	—
287	C₈	—	26	1	no	—
288	C₈	—	26	2	no	—
289	C₈	Q₈×#261	1	2	yes	—
290	C₂²	—	13	2	no	—
291	C₂²	—	26	1	no	—
292	C₂²	—	26	1	no	—
293	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	13	2	no	—
299	C₄	—	26	1	no	—
300	C₄	—	26	2	no	—
301	C₄	—	26	1	no	—
302	C₄	—	26	2	no	—
303	C₄	C₂×Q₈×#261	1	1	yes	characteristic
304	C₄	D₁₃⋊#131	1	1	yes	characteristic
305	C₂	—	26	2	no	—
306	C₂	Q₈×D₁₃⋊C₈	1	1	yes	characteristic
307	C₂	#13131804	1	1	yes	characteristic
308	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#446	C₄
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#446	#261
hypocenter	C₁₃	#33475
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131253	C₂	semidirect
3. #13131252	C₂	semidirect
4. #13131251	C₂	semidirect
57. D₁₃⋊C₁₆	Q₈	semidirect
264. C₁₃	#33475	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131252	2
#13131251	2
#13131172	2
#13131169	2
#13131046	2
#13131037	2
#13123767	2
#33475	13