Subgroups — #16807382

Order 3328.

444 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#13131204	C₂	1	2	yes	maximal
6	#13131203	C₂	1	2	yes	maximal
7	#13131200	C₂	1	1	yes	maximal, characteristic
8	#13131199	C₂	1	1	yes	maximal, characteristic
9	#13123767	C₂	1	1	yes	maximal, characteristic
10	#11719283	C₄	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	1	yes	characteristic
12	Q₈×#1055	C₂²	1	1	yes	characteristic
13	#11718500	C₄	1	2	yes	—
14	#11718498	C₄	1	1	yes	characteristic
15	#11718491	C₂²	1	2	yes	—
16	#11718491	C₂²	1	1	yes	characteristic
17	#11718369	C₂²	1	2	yes	—
18	#11718362	C₄	1	2	yes	—
19	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
20	#11718340	C₄	1	1	yes	characteristic
21	#11718260	C₂²	1	2	yes	—
22	#11718260	C₂²	1	2	yes	—
23	#11718260	C₂²	1	2	yes	—
24	#11718259	C₂²	1	1	yes	characteristic
25	#11718259	C₂²	1	1	yes	characteristic
26	#11718259	C₂²	1	2	yes	—
27	#11718258	C₂²	1	1	yes	characteristic
28	#11718258	C₂²	1	1	yes	characteristic
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	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₁₃×#97	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	#88055	C₂×C₄	1	2	yes	—
60	#88054	C₂×C₄	1	2	yes	—
61	#88054	C₂×C₄	1	2	yes	—
62	#88054	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₁₃	C₂³	1	2	yes	—
65	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
66	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
67	C₄⋊#1055	C₂×C₄	1	2	yes	—
68	C₄×#1055	C₂³	1	2	yes	—
69	C₄×#1055	C₂³	1	1	yes	characteristic
70	#33501	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
76	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
84	#6103	C₂×C₈	1	2	yes	—
85	#6103	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
87	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
88	#6101	C₂×C₈	1	2	yes	—
89	#6101	C₂×C₈	1	2	yes	—
90	#6101	C₂×C₈	1	2	yes	—
91	#6101	C₂×C₈	1	2	yes	—
92	#6101	C₂×C₈	1	2	yes	—
93	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
97	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
98	C₂×#1055	C₂²×C₄	1	2	yes	—
99	C₂×#1055	C₂⁴	1	1	yes	characteristic
100	#6094	—	4	2	no	—
101	C₈×D₁₃	—	2	2	no	—
102	C₈×D₁₃	—	4	1	no	—
103	#6092	—	2	2	no	—
104	#6092	—	2	2	no	—
105	#6092	—	2	2	no	—
106	#6092	—	2	2	no	—
107	#2960	—	13	1	no	—
108	#2183	—	13	2	no	—
109	#2183	—	13	2	no	—
110	#2183	—	13	2	no	—
111	#2183	—	13	2	no	—
112	#2164	—	13	2	no	—
113	#2164	—	13	2	no	—
114	#2163	—	13	1	no	—
115	#2163	—	13	1	no	—
116	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
117	Q₈×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
128	C₄×D₁₃	—	2	1	no	—
129	C₄×D₁₃	—	2	1	no	—
130	C₄×D₁₃	—	4	1	no	—
131	C₄×D₁₃	—	4	1	no	—
132	Dic₂₆	—	4	2	no	—
133	Dic₂₆	—	4	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	2	no	—
137	#1055	—	2	1	no	—
138	#1055	—	2	2	no	—
139	#1055	2₋¹⁺⁴	1	2	yes	—
140	#533	—	13	1	no	—
141	#446	—	13	2	no	—
142	#446	—	13	1	no	—
143	#446	#261	1	1	yes	characteristic
144	C₈×Q₈	—	13	1	no	—
145	C₈×Q₈	—	13	1	no	—
146	C₈×Q₈	—	13	2	no	—
147	#424	—	13	2	no	—
148	#423	—	13	1	no	—
149	#405	—	13	2	no	—
150	C₂×C₄×C₈	—	13	1	no	—
151	C₄⋊C₁₆	—	13	2	no	—
152	C₄⋊C₁₆	—	13	2	no	—
153	C₄⋊C₁₆	—	13	2	no	—
154	C₄⋊C₁₆	—	13	2	no	—
155	C₄⋊C₁₆	—	13	2	no	—
156	C₄⋊C₁₆	—	13	2	no	—
157	C₂²⋊C₁₆	—	13	1	no	—
158	C₂²⋊C₁₆	—	13	1	no	—
159	C₂²⋊C₁₆	—	13	2	no	—
160	#346	—	13	2	no	—
161	#346	—	13	2	no	—
162	#346	—	13	1	no	—
163	#346	—	13	1	no	—
164	#346	—	13	2	no	—
165	C₄×C₁₆	—	13	2	no	—
166	C₄×C₁₆	—	13	2	no	—
167	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
168	C₂×D₁₃	—	2	1	no	—
169	C₂×D₁₃	—	2	1	no	—
170	C₂×D₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	#519	1	1	yes	characteristic
176	C₄×C₁₃	#568	1	1	yes	characteristic
177	Dic₁₃	—	2	1	no	—
178	Dic₁₃	—	2	2	no	—
179	Dic₁₃	—	2	2	no	—
180	Dic₁₃	—	2	1	no	—
181	Dic₁₃	—	2	1	no	—
182	C₂×M₁₆	—	13	2	no	—
183	C₂²×C₈	—	13	1	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₄⋊Q₈	—	13	1	no	—
186	C₄⋊Q₈	—	13	1	no	—
187	#125	—	13	2	no	—
188	C₂²⋊Q₈	—	13	1	no	—
189	C₂²⋊Q₈	—	13	2	no	—
190	C₂²⋊Q₈	—	13	1	no	—
191	C₄×Q₈	—	13	2	no	—
192	C₄×Q₈	—	13	1	no	—
193	C₄×Q₈	C₂×#261	1	1	yes	characteristic
194	#117	—	13	2	no	—
195	C₂×C₄²	—	13	1	no	—
196	C₂×C₁₆	—	13	2	no	—
197	C₂×C₁₆	—	13	2	no	—
198	C₂×C₁₆	—	13	2	no	—
199	C₂×C₁₆	—	13	2	no	—
200	C₄⋊C₈	—	13	2	no	—
201	C₄⋊C₈	—	13	2	no	—
202	C₄⋊C₈	—	13	2	no	—
203	C₄⋊C₈	—	13	1	no	—
204	C₄⋊C₈	—	13	2	no	—
205	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
206	C₄⋊C₈	C₂×#261	1	2	yes	—
207	#97	—	13	2	no	—
208	#97	C₂×#261	1	2	yes	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	—	13	1	no	—
211	C₄×C₈	—	13	1	no	—
212	C₄×C₈	—	13	2	no	—
213	C₄×C₈	—	13	1	no	—
214	C₄×C₈	C₂×#261	1	1	yes	characteristic
215	C₂×C₁₃	#2982	1	1	yes	characteristic
216	C₂×C₁₃	#3407	1	1	yes	characteristic
217	C₂×C₁₃	#3407	1	1	yes	characteristic
218	D₁₃	—	4	1	no	—
219	C₂×Q₈	—	13	2	no	—
220	C₂×Q₈	—	13	1	no	—
221	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	1	no	—
224	C₂²×C₄	—	13	1	no	—
225	M₁₆	—	52	2	no	—
226	C₂×C₈	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	52	1	no	—
231	C₂×C₈	C₂²×#261	1	1	yes	characteristic
232	C₂×C₈	C₂²×#261	1	1	yes	characteristic
233	C₂×C₈	C₂²×#261	1	2	yes	—
234	C₄⋊C₄	—	13	2	no	—
235	C₄⋊C₄	—	13	2	no	—
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	13	2	no	—
239	C₄⋊C₄	—	13	2	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
242	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
243	C₂²⋊C₄	—	13	2	no	—
244	C₂²⋊C₄	—	13	2	no	—
245	C₄²	—	13	1	no	—
246	C₄²	—	13	1	no	—
247	C₄²	—	13	1	no	—
248	C₄²	—	13	2	no	—
249	C₄²	C₂²×#261	1	1	yes	characteristic
250	C₄²	C₂²×#261	1	2	yes	—
251	C₁₆	—	26	2	no	—
252	C₁₆	—	26	2	no	—
253	C₁₆	—	26	2	no	—
254	C₁₆	—	26	2	no	—
255	C₁₃	#33501	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	13	1	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	52	2	no	—
259	Q₈	—	52	1	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	1	no	—
263	C₂×C₄	—	13	2	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	26	1	no	—
266	C₂×C₄	—	26	1	no	—
267	C₂×C₄	—	52	1	no	—
268	C₂×C₄	—	52	1	no	—
269	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
271	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
272	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
273	C₂×C₄	C₂³×#261	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	2	no	—
277	C₈	—	13	2	no	—
278	C₈	—	26	1	no	—
279	C₈	—	26	2	no	—
280	C₂²	—	26	1	no	—
281	C₂²	—	26	1	no	—
282	C₂²	—	26	1	no	—
283	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	26	1	no	—
289	C₄	—	26	2	no	—
290	C₄	—	26	1	no	—
291	C₄	—	26	2	no	—
292	C₄	—	26	1	no	—
293	C₄	#11719627	1	1	yes	characteristic
294	C₄	#11719629	1	1	yes	characteristic
295	C₂	—	52	1	no	—
296	C₂	#13131730	1	1	yes	characteristic
297	C₂	#13131804	1	1	yes	characteristic
298	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	#33501
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
255. C₁₃	#33501	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
#13131204	2
#13131203	2
#13131200	2
#13131199	2
#13123767	2
#33501	13