Subgroups — #16017627

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125376	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123710	C₂	1	1	yes	maximal, characteristic
3	#13117003	C₂	1	2	yes	maximal, complemented
4	#13116992	C₂	1	2	yes	maximal, complemented
5	#13116987	C₂	1	2	yes	maximal, complemented
6	#13116972	C₂	1	2	yes	maximal, complemented
7	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116596	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116549	C₂	1	1	yes	maximal, characteristic
10	#13116541	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88158	C₂²	1	1	yes	characteristic
13	#11719224	—	2	1	no	—
14	#11719209	—	2	1	no	—
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
18	C₂×#88058	C₂²	1	1	yes	characteristic
19	#11718762	C₂²	1	1	yes	characteristic
20	C₂²×#6090	C₂²	1	1	yes	characteristic
21	#11718656	—	2	1	no	—
22	#11718627	—	2	1	no	—
23	#11718550	—	2	2	no	—
24	#11718539	C₂²	1	2	yes	—
25	#11718535	C₂²	1	2	yes	—
26	#11718534	C₂²	1	2	yes	—
27	#11718533	C₂²	1	2	yes	—
28	#11718517	—	2	2	no	—
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#366	C₂²	1	2	yes	—
31	#11718101	C₂²	1	2	yes	—
32	#11718095	C₂²	1	2	yes	—
33	#11718093	C₂²	1	2	yes	—
34	#11718093	C₂²	1	2	yes	—
35	#11718084	C₂²	1	1	yes	characteristic
36	#11718084	C₂²	1	1	yes	characteristic
37	#11718083	C₂²	1	1	yes	characteristic
38	#11718083	C₂²	1	1	yes	characteristic
39	#11718081	C₂²	1	2	yes	—
40	#11718081	C₂²	1	2	yes	—
41	C₂×#6127	—	2	1	no	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#117	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₄×C₂²⋊D₁₃	—	4	2	no	—
46	#88180	D₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₁₃	—	2	1	no	—
48	C₂×#6099	D₄	1	1	yes	characteristic
49	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
50	#88158	—	2	2	no	—
51	#88158	Q₈	1	4	yes	complemented
52	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
53	C₂×#6095	—	2	1	no	—
54	#88147	—	2	2	no	—
55	#88146	—	4	1	no	—
56	C₄⋊C₄×D₁₃	—	4	1	no	—
57	#88144	—	2	2	no	—
58	#88142	—	2	2	no	—
59	#88141	—	2	2	no	—
60	#88132	—	4	2	no	—
61	#88124	—	4	1	no	—
62	#88123	—	4	1	no	—
63	C₁₃×M₃₂	—	2	2	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#106	—	2	1	no	—
70	#88058	—	2	1	no	—
71	#88058	C₂³	1	2	yes	—
72	#88055	D₄	1	2	yes	—
73	C₈×Dic₁₃	C₂³	1	2	yes	—
74	C₂×#6090	—	2	2	no	—
75	C₂×#6090	C₂³	1	1	yes	characteristic
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	#88050	—	2	2	no	—
78	#88049	—	2	2	no	—
79	#88048	—	2	2	no	—
80	#88047	—	2	2	no	—
81	#33633	—	13	1	no	maximal, Sylow
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₁₃	—	4	1	no	—
85	C₂²×Dic₁₃	—	4	1	no	—
86	#6127	—	2	4	no	—
87	#6127	—	4	2	no	—
88	C₂×D₅₂	—	2	1	no	—
89	C₂×C₄×D₁₃	—	4	1	no	—
90	C₂×C₄×D₁₃	—	4	2	no	—
91	C₂×Dic₂₆	—	2	1	no	—
92	C₂×Dic₂₆	—	4	2	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₁₃	#41	1	2	yes	—
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₂²⋊C₄	—	4	1	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₂²⋊Dic₁₃	D₈	1	2	yes	—
104	#6103	—	4	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	4	1	no	—
107	C₄⋊Dic₁₃	—	4	2	no	—
108	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
109	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
110	#6101	—	4	2	no	—
111	#6101	—	4	2	no	—
112	#6101	—	4	2	no	—
113	C₄×Dic₁₃	—	4	2	no	—
114	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
115	#6099	—	2	2	no	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	Dic₅₂	—	2	1	no	—
118	Dic₅₂	C₂×Q₈	1	2	yes	—
119	D₁₀₄	—	2	1	no	—
120	D₁₀₄	C₂×Q₈	1	2	yes	—
121	#6095	—	2	4	no	—
122	C₁₃×C₁₆	—	2	2	no	—
123	#6090	—	2	1	no	—
124	#6090	—	2	1	no	—
125	#3164	—	13	1	no	—
126	#2951	—	13	1	no	—
127	#2229	—	13	2	no	—
128	#2228	—	13	2	no	—
129	#2227	—	13	2	no	—
130	#2224	—	13	2	no	—
131	#2159	D₁₃	1	1	yes	characteristic, complemented
132	C₂×#366	—	13	1	no	—
133	#2151	—	13	1	no	—
134	#2139	—	13	1	no	—
135	#2138	—	13	1	no	—
136	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂²×D₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	16	1	no	—
142	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#137	1	2	yes	—
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₁₃	—	4	4	no	—
147	C₂×Dic₁₃	—	4	1	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	16	1	no	—
150	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
151	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
152	D₅₂	—	2	2	no	—
153	D₅₂	—	4	1	no	—
154	C₄×D₁₃	—	4	4	no	—
155	C₄×D₁₃	—	16	2	no	—
156	Dic₂₆	—	2	2	no	—
157	Dic₂₆	—	4	1	no	—
158	Dic₂₆	—	16	2	no	—
159	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
160	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
161	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
162	#1055	—	2	2	no	—
163	C₂×#135	—	13	1	no	—
164	#544	—	26	1	no	—
165	#520	—	26	1	no	—
166	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
167	C₂²×C₁₆	—	13	1	no	—
168	#501	—	26	2	no	—
169	#500	—	13	2	no	—
170	#499	—	13	2	no	—
171	#440	—	26	2	no	—
172	C₄×Q₁₆	—	13	2	no	—
173	C₄×D₈	—	13	2	no	—
174	#428	—	26	1	no	—
175	#427	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#107	—	13	1	no	—
177	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
178	#417	—	26	1	no	—
179	#405	—	13	1	no	—
180	#367	C₂×D₁₃	1	2	yes	—
181	#366	—	13	2	no	—
182	#366	—	13	2	no	—
183	#366	C₂×D₁₃	1	2	yes	—
184	C₄⋊C₁₆	—	13	2	no	—
185	C₄⋊C₁₆	—	13	2	no	—
186	#358	—	13	2	no	—
187	#358	—	13	1	no	—
188	#358	—	13	1	no	—
189	#357	—	13	2	no	—
190	#357	—	13	1	no	—
191	#357	—	13	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	#474	1	2	yes	—
195	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₁₃	—	4	2	no	—
197	C₂×D₁₃	—	8	1	no	—
198	C₂×D₁₃	—	8	1	no	—
199	C₄×C₁₃	—	8	1	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	#474	1	2	yes	—
203	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
204	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
205	Dic₁₃	—	2	2	no	—
206	Dic₁₃	—	2	2	no	—
207	Dic₁₃	—	4	2	no	—
208	Dic₁₃	—	8	2	no	—
209	C₂×#41	—	26	1	no	—
210	#135	—	13	4	no	—
211	#135	—	26	2	no	—
212	C₂×Q₁₆	—	13	1	no	—
213	C₂×SD₁₆	—	26	1	no	—
214	C₂×D₈	—	13	1	no	—
215	C₂×M₁₆	—	13	1	no	—
216	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	26	2	no	—
218	C₄⋊Q₈	—	52	1	no	—
219	#125	—	26	2	no	—
220	#125	—	52	1	no	—
221	C₂²⋊Q₈	—	52	2	no	—
222	C₄×Q₈	—	26	2	no	—
223	C₄×D₄	—	26	2	no	—
224	C₄×D₄	—	52	2	no	—
225	#117	—	2	1	no	—
226	#117	—	13	1	no	—
227	#117	—	52	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	26	1	no	—
230	C₂×C₄⋊C₄	—	52	1	no	—
231	M₃₂	—	2	2	no	—
232	C₂×C₁₆	—	13	1	no	—
233	C₂×C₁₆	—	13	1	no	—
234	C₂×C₁₆	—	26	2	no	—
235	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
236	#107	—	2	1	no	—
237	#107	—	13	2	no	—
238	#107	—	26	2	no	—
239	#107	—	26	1	no	—
240	#107	C₂²×D₁₃	1	1	yes	characteristic
241	#107	C₂²×D₁₃	1	1	yes	characteristic
242	#107	C₂²×D₁₃	1	2	yes	—
243	#106	—	2	1	no	—
244	#106	—	26	2	no	—
245	#103	—	26	2	no	—
246	#102	—	26	2	no	—
247	#97	—	13	2	no	—
248	C₄×C₈	—	13	2	no	—
249	C₂×C₁₃	—	2	2	no	—
250	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
251	C₂×C₁₃	#3415	1	1	yes	characteristic
252	C₂×C₁₃	#3417	1	1	yes	characteristic
253	D₁₃	—	8	2	no	—
254	#41	—	26	4	no	—
255	#41	—	52	2	no	—
256	C₂×Q₈	—	26	1	no	—
257	C₂×Q₈	—	52	2	no	—
258	C₂×D₄	—	26	1	no	—
259	C₂×D₄	—	52	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	52	2	no	—
262	C₂²×C₄	—	52	1	no	—
263	C₂²×C₄	—	52	1	no	—
264	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
265	Q₁₆	—	13	2	no	—
266	Q₁₆	—	26	1	no	—
267	SD₁₆	—	26	4	no	—
268	D₈	—	13	2	no	—
269	D₈	—	26	1	no	—
270	M₁₆	—	26	2	no	—
271	C₂×C₈	—	13	2	no	—
272	C₂×C₈	#6127	1	2	yes	—
273	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₁₃	1	2	yes	—
275	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	13	1	no	—
281	C₄⋊C₄	—	13	1	no	—
282	C₄⋊C₄	—	26	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₄⋊C₄	—	52	2	no	—
285	C₄⋊C₄	—	52	2	no	—
286	C₄⋊C₄	—	52	2	no	—
287	C₄⋊C₄	—	52	1	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	13	2	no	—
290	C₂²⋊C₄	—	52	2	no	—
291	C₄²	—	4	1	no	—
292	C₄²	—	13	2	no	—
293	C₄²	—	52	2	no	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	26	1	no	—
296	C₁₆	—	26	1	no	—
297	C₁₃	#33633	1	1	yes	characteristic, complemented, Sylow
298	C₂³	—	52	1	no	—
299	C₂³	D₈×D₁₃	1	1	yes	characteristic
300	Q₈	—	26	2	no	—
301	Q₈	—	52	1	no	—
302	Q₈	—	208	2	no	—
303	D₄	—	26	2	no	—
304	D₄	—	52	4	no	—
305	D₄	—	52	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	13	2	no	—
311	C₂×C₄	—	16	1	no	—
312	C₂×C₄	—	52	1	no	—
313	C₂×C₄	—	52	4	no	—
314	C₂×C₄	—	52	2	no	—
315	C₂×C₄	—	208	2	no	—
316	C₂×C₄	—	208	1	no	—
317	C₂×C₄	#88150	1	2	yes	—
318	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
319	C₂×C₄	#88172	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
321	C₈	—	26	2	no	—
322	C₈	#88150	1	2	yes	—
323	C₈	C₂×#6127	1	1	yes	characteristic
324	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	52	2	no	—
328	C₂²	—	104	1	no	—
329	C₂²	—	104	1	no	—
330	C₂²	#11718551	1	2	yes	—
331	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	26	2	no	—
336	C₄	—	26	2	no	—
337	C₄	—	52	2	no	—
338	C₄	—	104	2	no	—
339	C₄	#11718551	1	2	yes	—
340	C₄	C₂×#88150	1	1	yes	characteristic
341	C₄	C₂×#88172	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	104	2	no	—
344	C₂	C₂×#11718551	1	1	yes	characteristic
345	C₂	#13127976	1	1	yes	characteristic
346	C₂	#13127988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2159	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
hypercenter	#2159	D₁₃
hypocenter	C₁₃	#33633
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
1. #13125376	C₂	semidirect
3. #13117003	C₂	semidirect
4. #13116992	C₂	semidirect
5. #13116987	C₂	semidirect
6. #13116972	C₂	semidirect
7. C₂×#11718081	C₂	semidirect
8. #13116596	C₂	semidirect
11. C₁₃×#2159	C₂	semidirect
51. #88158	Q₈	semidirect
131. #2159	D₁₃	semidirect
297. C₁₃	#33633	semidirect

Maximal subgroups

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

subgroup	index
#13125376	2
#13123710	2
#13117003	2
#13116992	2
#13116987	2
#13116972	2
C₂×#11718081	2
#13116596	2
#13116549	2
#13116541	2
C₁₃×#2159	2
#33633	13