Subgroups — #16016294

Order 3328.

646 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	C₂×D₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	#13123699	C₂	1	2	yes	maximal
3	#13116708	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116707	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116706	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#88069	C₂	1	2	yes	maximal, complemented
7	#13116692	C₂	1	2	yes	maximal, complemented
8	#13116619	C₂	1	1	yes	maximal, characteristic
9	#13116554	C₂	1	2	yes	maximal
10	#13116434	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₂²	1	2	yes	—
13	#11719277	—	2	2	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
17	C₂×#88069	C₂²	1	2	yes	—
18	#11718691	—	2	2	no	—
19	D₁₃×Q₃₂	C₄	1	8	yes	complemented
20	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
21	C₂×#88038	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	#11718541	C₂²	1	2	yes	—
24	#11718534	C₂²	1	2	yes	—
25	#11718463	—	2	2	no	—
26	#11718450	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	2	yes	—
28	#11718350	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₁₃×#346	C₂²	1	1	yes	characteristic
32	#11718156	C₂²	1	2	yes	—
33	#11718097	C₂²	1	1	yes	characteristic
34	#11718095	C₂²	1	2	yes	—
35	#11718094	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	#11718084	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	C₄×#6090	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₁₃	—	2	2	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	Q₈×Dic₁₃	—	2	2	no	—
44	C₂×#6107	—	2	2	no	—
45	Q₁₆×D₁₃	—	2	4	no	—
46	Q₁₆×D₁₃	C₂×C₄	1	8	yes	—
47	C₂×Dic₅₂	C₂³	1	2	yes	—
48	C₂×#6094	D₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
50	#88146	—	2	2	no	—
51	#88146	—	4	2	no	—
52	#88141	—	2	2	no	—
53	#88141	—	4	2	no	—
54	#88126	—	4	2	no	—
55	#88126	D₄	1	1	yes	characteristic
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	—	4	2	no	—
58	C₁₃×Q₃₂	C₂×C₄	1	4	yes	—
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#103	—	2	2	no	—
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88075	—	2	2	no	—
65	#88069	C₂×C₄	1	8	yes	—
66	#88058	C₂³	1	2	yes	—
67	#88056	—	2	2	no	—
68	#88055	D₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88050	—	2	2	no	—
73	#88048	—	2	2	no	—
74	#88043	D₄	1	1	yes	characteristic
75	Dic₁₀₄	C₂×C₄	1	4	yes	—
76	#88038	—	2	2	no	—
77	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
78	#33458	—	13	1	no	maximal, Sylow
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	Q₈×D₁₃	—	2	8	no	—
81	Q₈×D₁₃	—	4	4	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₂₆	—	2	2	no	—
85	C₂×Dic₂₆	—	4	2	no	—
86	C₁₃×Q₁₆	—	2	2	no	—
87	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₄²×C₁₃	—	4	2	no	—
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	#6107	—	2	8	no	—
95	#6103	—	4	2	no	—
96	#6103	D₈	1	2	yes	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	#6101	—	4	2	no	—
100	#6101	—	4	2	no	—
101	#6101	D₈	1	2	yes	—
102	C₄×Dic₁₃	—	4	2	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	C₄×Dic₁₃	—	4	2	no	—
105	C₄×Dic₁₃	—	4	2	no	—
106	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×#1055	C₂×D₄	1	1	yes	characteristic
108	C₂×#1055	C₂×D₄	1	1	yes	characteristic
109	Dic₅₂	—	2	2	no	—
110	Dic₅₂	C₂²×C₄	1	4	yes	—
111	#6094	—	2	2	no	—
112	C₈×D₁₃	C₂²×C₄	1	2	yes	—
113	C₈×D₁₃	#41	1	2	yes	—
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
116	#6090	—	2	1	no	—
117	#6090	C₂²×C₄	1	2	yes	—
118	C₂²×Q₃₂	—	13	1	no	—
119	#2950	—	13	2	no	—
120	#2176	—	13	1	no	—
121	#2176	—	13	1	no	—
122	#2176	—	13	1	no	—
123	#2176	D₁₃	1	1	yes	characteristic, complemented
124	C₄×Q₃₂	—	13	2	no	—
125	C₄×Q₃₂	—	13	2	no	—
126	#2158	—	13	1	no	—
127	#2141	—	13	2	no	—
128	#2108	—	13	1	no	—
129	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
130	Q₈×C₁₃	—	2	4	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
139	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
140	C₄×D₁₃	—	4	8	no	—
141	C₄×D₁₃	C₄×D₄	1	2	yes	—
142	C₄×D₁₃	#135	1	2	yes	—
143	Dic₂₆	—	2	4	no	—
144	Dic₂₆	—	4	2	no	—
145	Dic₂₆	—	4	8	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	#1055	—	2	1	no	—
150	#1055	C₄×D₄	1	2	yes	—
151	C₂²×Q₁₆	—	13	2	no	—
152	#519	—	26	2	no	—
153	C₂×Q₃₂	—	13	2	no	—
154	C₂×Q₃₂	—	13	8	no	—
155	C₂×Q₃₂	—	13	1	no	—
156	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
157	C₂×M₃₂	—	13	1	no	—
158	C₂²×C₁₆	—	13	1	no	—
159	#441	—	26	2	no	—
160	#441	—	26	2	no	—
161	C₄×Q₁₆	—	13	2	no	—
162	C₄×Q₁₆	—	13	2	no	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
165	#427	—	13	2	no	—
166	#419	—	26	2	no	—
167	#405	—	13	1	no	—
168	#367	—	13	1	no	—
169	#367	C₂×D₁₃	1	1	yes	characteristic
170	#366	—	13	2	no	—
171	#358	—	13	2	no	—
172	#358	—	13	2	no	—
173	#358	—	13	2	no	—
174	#358	C₂×D₁₃	1	2	yes	—
175	#346	—	13	1	no	—
176	#346	C₂×D₁₃	1	1	yes	characteristic
177	C₄×C₁₆	—	13	1	no	—
178	C₄×C₁₆	—	13	1	no	—
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	C₄×D₈	1	2	yes	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
187	Dic₁₃	—	2	2	no	—
188	Dic₁₃	—	4	4	no	—
189	Dic₁₃	—	8	2	no	—
190	Dic₁₃	C₄×D₈	1	2	yes	—
191	C₂²×Q₈	—	26	2	no	—
192	C₂×Q₁₆	—	13	8	no	—
193	C₂×Q₁₆	—	13	2	no	—
194	C₂×Q₁₆	—	26	4	no	—
195	C₂×Q₁₆	—	26	2	no	—
196	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
197	C₂×M₁₆	—	13	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	26	2	no	—
201	C₄×Q₈	—	26	2	no	—
202	C₄×Q₈	—	26	2	no	—
203	C₄×Q₈	—	52	2	no	—
204	C₄×Q₈	—	52	2	no	—
205	#117	—	13	1	no	—
206	#117	—	26	2	no	—
207	#117	—	52	2	no	—
208	#117	—	52	2	no	—
209	Q₃₂	—	13	8	no	—
210	Q₃₂	—	13	4	no	—
211	Q₃₂	C₄×D₁₃	1	4	yes	—
212	M₃₂	—	26	2	no	—
213	C₂×C₁₆	—	13	1	no	—
214	C₂×C₁₆	—	13	4	no	—
215	C₂×C₁₆	—	13	1	no	—
216	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
217	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
218	#107	—	13	2	no	—
219	#107	C₂²×D₁₃	1	2	yes	—
220	#106	—	26	2	no	—
221	#103	—	2	2	no	—
222	#103	—	26	2	no	—
223	#103	—	26	2	no	—
224	#103	—	26	2	no	—
225	#97	—	13	1	no	—
226	#97	—	13	1	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
230	C₂×C₁₃	#3417	1	2	yes	—
231	D₁₃	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	26	8	no	—
234	C₂×Q₈	—	26	2	no	—
235	C₂×Q₈	—	52	2	no	—
236	C₂×Q₈	—	52	4	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	52	2	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	13	4	no	—
241	Q₁₆	—	26	8	no	—
242	Q₁₆	—	26	2	no	—
243	Q₁₆	C₂×C₄×D₁₃	1	4	yes	—
244	M₁₆	—	26	2	no	—
245	C₂×C₈	—	13	1	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	13	2	no	—
254	C₄⋊C₄	—	26	2	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₂²⋊C₄	—	13	2	no	—
259	C₂²⋊C₄	—	52	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	13	1	no	—
262	C₄²	—	52	2	no	—
263	C₄²	—	52	2	no	—
264	C₄²	—	52	2	no	—
265	C₄²	—	52	2	no	—
266	C₄²	D₄×D₁₃	1	1	yes	characteristic
267	C₁₆	—	2	1	no	—
268	C₁₆	—	13	2	no	—
269	C₁₆	—	26	1	no	—
270	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
271	C₁₃	#33458	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	13	1	no	—
273	Q₈	—	2	4	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	26	4	no	—
276	Q₈	—	52	8	no	—
277	Q₈	—	52	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	52	8	no	—
285	C₂×C₄	—	52	2	no	—
286	C₂×C₄	—	52	2	no	—
287	C₂×C₄	D₈×D₁₃	1	2	yes	—
288	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	13	2	no	—
291	C₈	—	26	1	no	—
292	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
293	C₈	#41×D₁₃	1	1	yes	characteristic
294	C₂²	—	13	2	no	—
295	C₂²	—	26	2	no	—
296	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	13	2	no	—
301	C₄	—	26	2	no	—
302	C₄	—	52	4	no	—
303	C₄	—	104	2	no	—
304	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
305	C₄	D₁₃×#135	1	1	yes	characteristic
306	C₂	—	26	2	no	—
307	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic
308	C₂	#13127988	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2176	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2176	D₁₃
hypocenter	C₁₃	#33458
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
3. #13116708	C₂	semidirect
4. #13116707	C₂	semidirect
5. #13116706	C₂	semidirect
6. C₄×#88069	C₂	semidirect
7. #13116692	C₂	semidirect
11. C₁₃×#2176	C₂	semidirect
19. D₁₃×Q₃₂	C₄	semidirect
123. #2176	D₁₃	semidirect
271. C₁₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃×Q₃₂	2
#13123699	2
#13116708	2
#13116707	2
#13116706	2
C₄×#88069	2
#13116692	2
#13116619	2
#13116554	2
#13116434	2
C₁₃×#2176	2
#33458	13