Subgroups — #16016292

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#13116705	C₂	1	2	yes	maximal, complemented
4	#13116704	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116694	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116617	C₂	1	1	yes	maximal, characteristic
9	#13116554	C₂	1	2	yes	maximal
10	#13116431	C₂	1	1	yes	maximal, characteristic
11	C₄×C₁₃×Q₃₂	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×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
32	#11718156	C₂²	1	2	yes	—
33	#11718096	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	#11718082	C₂²	1	2	yes	—
39	#11718053	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	1	yes	characteristic
97	#6103	D₈	1	1	yes	characteristic
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	4	2	no	—
100	#6101	—	4	2	no	—
101	#6101	—	4	2	no	—
102	#6101	D₈	1	1	yes	characteristic
103	#6101	D₈	1	1	yes	characteristic
104	C₄×Dic₁₃	—	4	2	no	—
105	C₄×Dic₁₃	—	4	2	no	—
106	C₄×Dic₁₃	—	4	2	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×#1055	C₂×D₄	1	1	yes	characteristic
110	C₂×#1055	C₂×D₄	1	1	yes	characteristic
111	Dic₅₂	—	2	2	no	—
112	Dic₅₂	C₂²×C₄	1	4	yes	—
113	#6094	—	2	2	no	—
114	C₈×D₁₃	C₂²×C₄	1	2	yes	—
115	C₈×D₁₃	#41	1	2	yes	—
116	C₁₃×C₁₆	—	2	1	no	—
117	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
118	#6090	—	2	1	no	—
119	#6090	C₂²×C₄	1	2	yes	—
120	C₂²×Q₃₂	—	13	1	no	—
121	#2950	—	13	2	no	—
122	#2176	—	13	2	no	—
123	#2176	—	13	2	no	—
124	C₄×Q₃₂	—	13	1	no	—
125	C₄×Q₃₂	—	13	1	no	—
126	C₄×Q₃₂	—	13	1	no	—
127	C₄×Q₃₂	D₁₃	1	1	yes	characteristic, complemented
128	#2158	—	13	1	no	—
129	#2141	—	13	2	no	—
130	#2108	—	13	1	no	—
131	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
132	Q₈×C₁₃	—	2	4	no	—
133	Q₈×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
144	C₄×D₁₃	—	4	8	no	—
145	C₄×D₁₃	C₄×D₄	1	2	yes	—
146	C₄×D₁₃	#135	1	2	yes	—
147	Dic₂₆	—	2	4	no	—
148	Dic₂₆	—	4	2	no	—
149	Dic₂₆	—	4	8	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
153	#1055	—	2	1	no	—
154	#1055	C₄×D₄	1	2	yes	—
155	C₂²×Q₁₆	—	13	2	no	—
156	#519	—	26	2	no	—
157	C₂×Q₃₂	—	13	2	no	—
158	C₂×Q₃₂	—	13	8	no	—
159	C₂×Q₃₂	—	13	1	no	—
160	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
161	C₂×M₃₂	—	13	1	no	—
162	C₂²×C₁₆	—	13	1	no	—
163	#441	—	26	2	no	—
164	#441	—	26	2	no	—
165	C₄×Q₁₆	—	13	2	no	—
166	C₄×Q₁₆	—	13	2	no	—
167	C₄×Q₁₆	—	13	2	no	—
168	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
169	#427	—	13	2	no	—
170	#419	—	26	2	no	—
171	#405	—	13	1	no	—
172	#367	—	13	2	no	—
173	#366	—	13	1	no	—
174	#366	C₂×D₁₃	1	1	yes	characteristic
175	#358	—	13	2	no	—
176	#358	—	13	2	no	—
177	#358	—	13	2	no	—
178	#358	C₂×D₁₃	1	2	yes	—
179	#346	—	13	1	no	—
180	#346	—	13	1	no	—
181	C₄×C₁₆	—	13	1	no	—
182	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
183	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
184	C₂×D₁₃	—	2	1	no	—
185	C₂×D₁₃	—	2	1	no	—
186	C₂×D₁₃	C₄×D₈	1	2	yes	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	4	4	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
192	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
193	Dic₁₃	—	2	1	no	—
194	Dic₁₃	—	2	1	no	—
195	Dic₁₃	—	4	4	no	—
196	Dic₁₃	—	8	2	no	—
197	Dic₁₃	C₄×D₈	1	2	yes	—
198	C₂²×Q₈	—	26	2	no	—
199	C₂×Q₁₆	—	13	8	no	—
200	C₂×Q₁₆	—	13	2	no	—
201	C₂×Q₁₆	—	26	4	no	—
202	C₂×Q₁₆	—	26	2	no	—
203	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
204	C₂×M₁₆	—	13	1	no	—
205	C₂²×C₈	—	13	1	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	26	2	no	—
208	C₄×Q₈	—	26	2	no	—
209	C₄×Q₈	—	26	2	no	—
210	C₄×Q₈	—	52	2	no	—
211	C₄×Q₈	—	52	2	no	—
212	#117	—	13	1	no	—
213	#117	—	26	2	no	—
214	#117	—	52	2	no	—
215	#117	—	52	2	no	—
216	Q₃₂	—	13	8	no	—
217	Q₃₂	—	13	4	no	—
218	Q₃₂	C₄×D₁₃	1	4	yes	—
219	M₃₂	—	26	2	no	—
220	C₂×C₁₆	—	13	1	no	—
221	C₂×C₁₆	—	13	4	no	—
222	C₂×C₁₆	—	13	1	no	—
223	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
225	#107	—	13	2	no	—
226	#107	C₂²×D₁₃	1	2	yes	—
227	#106	—	26	2	no	—
228	#103	—	2	2	no	—
229	#103	—	26	2	no	—
230	#103	—	26	2	no	—
231	#103	—	26	2	no	—
232	#97	—	13	1	no	—
233	#97	—	13	1	no	—
234	C₄×C₈	—	13	1	no	—
235	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
237	C₂×C₁₃	#3417	1	1	yes	characteristic
238	C₂×C₁₃	#3417	1	1	yes	characteristic
239	D₁₃	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	26	8	no	—
242	C₂×Q₈	—	26	2	no	—
243	C₂×Q₈	—	52	2	no	—
244	C₂×Q₈	—	52	4	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	52	2	no	—
247	Q₁₆	—	2	2	no	—
248	Q₁₆	—	13	4	no	—
249	Q₁₆	—	26	8	no	—
250	Q₁₆	—	26	2	no	—
251	Q₁₆	C₂×C₄×D₁₃	1	4	yes	—
252	M₁₆	—	26	2	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	13	1	no	—
262	C₄⋊C₄	—	13	1	no	—
263	C₄⋊C₄	—	26	2	no	—
264	C₄⋊C₄	—	52	2	no	—
265	C₄⋊C₄	—	52	2	no	—
266	C₄⋊C₄	—	52	2	no	—
267	C₂²⋊C₄	—	13	1	no	—
268	C₂²⋊C₄	—	13	1	no	—
269	C₂²⋊C₄	—	52	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	13	1	no	—
272	C₄²	—	52	2	no	—
273	C₄²	—	52	2	no	—
274	C₄²	—	52	2	no	—
275	C₄²	—	52	2	no	—
276	C₄²	D₄×D₁₃	1	1	yes	characteristic
277	C₁₆	—	2	1	no	—
278	C₁₆	—	13	2	no	—
279	C₁₆	—	26	1	no	—
280	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
281	C₁₃	#33458	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	13	1	no	—
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	26	4	no	—
286	Q₈	—	52	8	no	—
287	Q₈	—	52	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	52	8	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	—	52	2	no	—
298	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
299	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	13	2	no	—
303	C₈	—	26	1	no	—
304	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
305	C₈	#41×D₁₃	1	1	yes	characteristic
306	C₂²	—	13	2	no	—
307	C₂²	—	26	1	no	—
308	C₂²	—	26	1	no	—
309	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	—	13	2	no	—
315	C₄	—	26	1	no	—
316	C₄	—	26	1	no	—
317	C₄	—	52	4	no	—
318	C₄	—	104	2	no	—
319	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
320	C₄	D₁₃×#135	1	1	yes	characteristic
321	C₂	—	26	2	no	—
322	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic
323	C₂	#13127985	1	1	yes	characteristic
324	C₂	#13127986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×Q₃₂	D₁₃
hypocenter	C₁₃	#33458
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #13116705	C₂	semidirect
4. #13116704	C₂	semidirect
5. C₄×Dic₁₀₄	C₂	semidirect
6. Q₃₂×Dic₁₃	C₂	semidirect
7. #13116694	C₂	semidirect
11. C₄×C₁₃×Q₃₂	C₂	semidirect
19. D₁₃×Q₃₂	C₄	semidirect
127. C₄×Q₃₂	D₁₃	semidirect
281. 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
#13116705	2
#13116704	2
C₄×Dic₁₀₄	2
Q₃₂×Dic₁₃	2
#13116694	2
#13116617	2
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