Subgroups — #16018460

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125419	C₂	1	2	yes	maximal
2	#13117110	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117087	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117062	C₂	1	2	yes	maximal, complemented
5	#13116928	C₂	1	1	yes	maximal, characteristic
6	#13116926	C₂	1	2	yes	maximal
7	#13116924	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#88069	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
11	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719283	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	#11718849	C₂²	1	2	yes	—
16	#11718842	C₂²	1	2	yes	—
17	C₂×#88069	C₂²	1	2	yes	—
18	C₄×#6107	—	2	2	no	—
19	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	#11718548	C₂²	1	2	yes	—
23	#11718535	C₂²	1	2	yes	—
24	#11718474	—	2	2	no	—
25	#11718453	—	2	2	no	—
26	C₄×Dic₅₂	C₂²	1	2	yes	—
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#358	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	#11718156	C₂²	1	2	yes	—
32	#11718096	C₂²	1	1	yes	characteristic
33	#11718095	C₂²	1	2	yes	—
34	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	#11718084	C₂²	1	2	yes	—
37	#11718081	C₂²	1	2	yes	—
38	C₄×#6090	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₄×Q₈×C₁₃	—	2	2	no	—
41	#88200	—	2	2	no	—
42	#88198	—	2	2	no	—
43	C₂×#6107	—	2	2	no	—
44	C₂×Dic₅₂	C₂³	1	2	yes	—
45	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	#88151	—	2	2	no	—
48	#88150	—	4	2	no	—
49	#88143	—	4	2	no	—
50	#88142	—	2	2	no	—
51	#88126	—	4	2	no	—
52	C₄²×D₁₃	D₄	1	1	yes	characteristic
53	C₄×Dic₂₆	—	2	2	no	—
54	C₄×Dic₂₆	—	4	2	no	—
55	C₁₃×Q₃₂	—	2	2	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#103	—	2	2	no	—
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	#88075	—	2	2	no	—
62	#88069	—	2	4	no	—
63	#88058	C₂³	1	2	yes	—
64	#88056	—	2	2	no	—
65	C₈×Dic₁₃	D₄	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
67	C₂×#6090	C₂³	1	1	yes	characteristic
68	C₂×#6090	C₂³	1	1	yes	characteristic
69	#88050	—	2	2	no	—
70	#88047	—	2	2	no	—
71	C₄×#1055	D₄	1	1	yes	characteristic
72	Dic₁₀₄	—	2	2	no	—
73	C₁₆×D₁₃	—	2	2	no	—
74	C₁₆×D₁₃	—	2	2	no	—
75	#33734	—	13	1	no	maximal, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
78	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₂₆	—	2	2	no	—
81	C₂×Dic₂₆	—	4	2	no	—
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₄²×C₁₃	—	4	2	no	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	#6107	—	4	4	no	—
91	#6103	—	4	2	no	—
92	#6103	—	4	2	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	4	2	no	—
95	#6101	—	4	2	no	—
96	#6101	—	4	2	no	—
97	#6101	—	4	2	no	—
98	#6101	—	4	2	no	—
99	#6101	—	4	2	no	—
100	C₄×Dic₁₃	—	4	2	no	—
101	C₄×Dic₁₃	D₈	1	1	yes	characteristic
102	C₄×Dic₁₃	D₈	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×#1055	C₂×D₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	1	yes	characteristic
106	Dic₅₂	—	2	2	no	—
107	Dic₅₂	—	2	2	no	—
108	C₈×D₁₃	—	2	1	no	—
109	C₈×D₁₃	—	2	1	no	—
110	C₈×D₁₃	—	2	2	no	—
111	C₁₃×C₁₆	—	2	1	no	—
112	C₁₃×C₁₆	#41	1	2	yes	—
113	#6090	—	2	1	no	—
114	#6090	#41	1	2	yes	—
115	#3170	—	13	2	no	—
116	#2252	—	13	1	no	—
117	#2247	—	13	1	no	—
118	#2241	—	13	2	no	—
119	#2216	—	13	1	no	—
120	#2216	—	13	1	no	—
121	#2216	—	13	2	no	—
122	C₄×Q₃₂	—	13	2	no	—
123	C₄×Q₃₂	—	13	1	no	—
124	C₄×Q₃₂	D₁₃	1	1	yes	characteristic, complemented
125	C₂×C₄×C₁₆	—	13	1	no	—
126	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	—	2	1	no	—
141	C₄×D₁₃	—	2	2	no	—
142	C₄×D₁₃	Q₃₂	1	4	yes	complemented
143	Dic₂₆	—	4	2	no	—
144	Dic₂₆	—	4	2	no	—
145	Dic₂₆	—	8	4	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	#1055	—	2	1	no	—
150	#1055	#135	1	2	yes	—
151	#533	—	26	2	no	—
152	C₂×Q₃₂	—	13	2	no	—
153	C₂×Q₃₂	—	13	1	no	—
154	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
155	C₂²×C₁₆	—	13	1	no	—
156	C₂²×C₁₆	—	13	1	no	—
157	#500	—	13	2	no	—
158	#494	—	13	2	no	—
159	#487	—	26	2	no	—
160	#467	—	13	2	no	—
161	#467	—	13	2	no	—
162	#467	—	26	2	no	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	—	26	2	no	—
165	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
166	C₂×C₄×C₈	—	13	1	no	—
167	#366	—	13	1	no	—
168	#366	—	13	2	no	—
169	#366	C₂×D₁₃	1	1	yes	characteristic
170	#358	—	13	2	no	—
171	#358	—	13	2	no	—
172	#358	—	13	2	no	—
173	#358	C₂×D₁₃	1	2	yes	—
174	C₄×C₁₆	—	13	1	no	—
175	C₄×C₁₆	—	13	1	no	—
176	C₄×C₁₆	—	13	1	no	—
177	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
178	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
179	C₂×D₁₃	—	2	1	no	—
180	C₂×D₁₃	—	2	1	no	—
181	C₂×D₁₃	C₂×Q₃₂	1	2	yes	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
186	C₄×C₁₃	#532	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
188	Dic₁₃	—	2	1	no	—
189	Dic₁₃	—	8	2	no	—
190	Dic₁₃	—	8	2	no	—
191	Dic₁₃	C₂×Q₃₂	1	2	yes	—
192	Dic₁₃	#508	1	2	yes	—
193	C₂×Q₁₆	—	13	2	no	—
194	C₂×Q₁₆	—	26	2	no	—
195	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	—	13	1	no	—
198	C₄⋊Q₈	—	26	2	no	—
199	C₄⋊Q₈	—	26	2	no	—
200	#125	—	52	2	no	—
201	C₂²⋊Q₈	—	26	2	no	—
202	C₂²⋊Q₈	—	26	2	no	—
203	C₂²⋊Q₈	—	52	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×Q₈	—	52	2	no	—
207	#117	—	52	2	no	—
208	C₂×C₄²	—	13	1	no	—
209	Q₃₂	—	2	2	no	—
210	Q₃₂	—	26	4	no	—
211	Q₃₂	—	26	2	no	—
212	C₂×C₁₆	—	13	1	no	—
213	C₂×C₁₆	—	13	1	no	—
214	C₂×C₁₆	—	26	2	no	—
215	C₂×C₁₆	—	26	2	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	—	26	2	no	—
220	#107	C₂²×D₁₃	1	2	yes	—
221	#103	—	2	2	no	—
222	#103	—	26	2	no	—
223	#103	—	26	2	no	—
224	#103	—	26	2	no	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	—	13	1	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₁₃	#3169	1	1	yes	characteristic
230	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
232	D₁₃	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	26	2	no	—
235	C₂×Q₈	—	52	2	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	26	2	no	—
242	Q₁₆	—	26	2	no	—
243	Q₁₆	—	52	4	no	—
244	C₂×C₈	—	13	1	no	—
245	C₂×C₈	—	13	1	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	—	26	1	no	—
248	C₂×C₈	—	26	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	—	26	2	no	—
254	C₄⋊C₄	—	52	2	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₂²⋊C₄	—	52	2	no	—
261	C₂²⋊C₄	—	52	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	13	1	no	—
264	C₄²	—	13	1	no	—
265	C₄²	—	13	1	no	—
266	C₄²	—	52	2	no	—
267	C₄²	D₄×D₁₃	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	13	2	no	—
270	C₁₆	—	26	1	no	—
271	C₁₆	#6127	1	2	yes	—
272	C₁₃	#33734	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	13	1	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	52	2	no	—
277	Q₈	—	52	2	no	—
278	Q₈	—	104	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	13	1	no	—
282	C₂×C₄	—	13	1	no	—
283	C₂×C₄	—	13	4	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	26	2	no	—
286	C₂×C₄	—	26	1	no	—
287	C₂×C₄	—	26	1	no	—
288	C₂×C₄	—	52	2	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
291	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	13	2	no	—
295	C₈	—	26	1	no	—
296	C₈	C₂×#6127	1	1	yes	characteristic
297	C₈	#41×D₁₃	1	1	yes	characteristic
298	C₂²	—	13	2	no	—
299	C₂²	—	26	1	no	—
300	C₂²	—	26	1	no	—
301	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	13	2	no	—
306	C₄	—	13	2	no	—
307	C₄	—	26	1	no	—
308	C₄	—	104	2	no	—
309	C₄	—	104	2	no	—
310	C₄	D₁₃×Q₃₂	1	2	yes	—
311	C₄	#11719253	1	1	yes	characteristic
312	C₄	D₁₃×#135	1	1	yes	characteristic
313	C₂	—	26	2	no	—
314	C₂	#13125412	1	1	yes	characteristic
315	C₂	C₂×D₁₃×Q₃₂	1	1	yes	characteristic
316	C₂	D₁₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
hypercenter	C₄×Q₃₂	D₁₃
hypocenter	C₁₃	#33734
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #13117110	C₂	semidirect
3. #13117087	C₂	semidirect
4. #13117062	C₂	semidirect
8. C₄×Dic₁₀₄	C₂	semidirect
9. C₄×#88069	C₂	semidirect
11. C₄×C₁₃×Q₃₂	C₂	semidirect
124. C₄×Q₃₂	D₁₃	semidirect
142. C₄×D₁₃	Q₃₂	semidirect
272. C₁₃	#33734	semidirect

Maximal subgroups

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

subgroup	index
#13125419	2
#13117110	2
#13117087	2
#13117062	2
#13116928	2
#13116926	2
#13116924	2
C₄×Dic₁₀₄	2
C₄×#88069	2
C₄×C₁₆×D₁₃	2
C₄×C₁₃×Q₃₂	2
#33734	13