Subgroups — #16017777

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124492	C₂	1	1	yes	maximal, characteristic
2	#13123433	C₂	1	1	yes	maximal, characteristic
3	#13117047	C₂	1	2	yes	maximal, complemented
4	#13117046	C₂	1	2	yes	maximal, complemented
5	#13117040	C₂	1	2	yes	maximal, complemented
6	#13117035	C₂	1	2	yes	maximal, complemented
7	#13116625	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116555	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718095	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116449	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	C₂×#88155	C₂²	1	1	yes	characteristic
14	#11719231	—	2	1	no	—
15	#11719228	—	2	1	no	—
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#427	C₂²	1	1	yes	characteristic
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88052	C₂²	1	1	yes	characteristic
21	#11718665	—	2	1	no	—
22	#11718656	—	2	1	no	—
23	#11718548	C₂²	1	2	yes	—
24	#11718548	C₂²	1	2	yes	—
25	#11718541	C₂²	1	2	yes	—
26	D₁₃×#107	C₂²	1	2	yes	—
27	#11718529	—	2	2	no	—
28	#11718522	—	2	2	no	—
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#367	C₂²	1	1	yes	characteristic
31	C₁₃×#366	C₂²	1	1	yes	characteristic
32	C₁₃×#366	C₂²	1	1	yes	characteristic
33	#11718099	C₂²	1	2	yes	—
34	#11718099	C₂²	1	2	yes	—
35	#11718095	C₂²	1	2	yes	—
36	#11718095	C₂²	1	2	yes	—
37	#11718084	C₂²	1	2	yes	—
38	#11718084	C₂²	1	2	yes	—
39	#11718082	C₂²	1	2	yes	—
40	#11718081	C₂²	1	2	yes	—
41	C₂×#6127	D₄	1	1	yes	characteristic
42	C₂²×Dic₂₆	—	2	1	no	—
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#117	—	2	1	no	—
45	C₁₃×#117	—	2	1	no	—
46	C₂×C₄⋊Dic₁₃	—	2	1	no	—
47	C₂×#6099	D₄	1	1	yes	characteristic
48	C₂×Dic₅₂	—	2	2	no	—
49	C₂×Dic₅₂	—	2	2	no	—
50	C₂×Dic₅₂	—	2	1	no	—
51	C₂×Dic₅₂	C₂³	1	2	yes	—
52	#88155	—	2	2	no	—
53	#88155	—	2	2	no	—
54	C₂×#6094	D₄	1	2	yes	—
55	C₂×C₈×D₁₃	C₂³	1	2	yes	—
56	#88151	—	2	2	no	—
57	#88151	—	2	2	no	—
58	#88146	—	2	2	no	—
59	C₄⋊C₄×D₁₃	—	2	2	no	—
60	#88140	—	4	2	no	—
61	#88135	—	4	2	no	—
62	#88129	—	4	1	no	—
63	C₄×D₅₂	—	4	1	no	—
64	#88123	—	4	1	no	—
65	C₄×Dic₂₆	—	4	1	no	—
66	C₂×C₁₃×C₁₆	—	2	2	no	—
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×#107	C₂³	1	1	yes	characteristic
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#106	—	2	1	no	—
75	#88058	—	2	1	no	—
76	#88058	C₂³	1	2	yes	—
77	#88052	—	2	2	no	—
78	C₂×#6090	C₂³	1	2	yes	—
79	#88050	—	2	2	no	—
80	#88050	—	2	2	no	—
81	#88048	—	2	2	no	—
82	#88047	—	2	2	no	—
83	#33651	—	13	1	no	maximal, Sylow
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₁₃	D₈	1	2	yes	—
86	C₂²×Dic₁₃	—	4	1	no	—
87	C₂²×Dic₁₃	—	4	1	no	—
88	#6127	—	2	2	no	—
89	#6127	—	2	2	no	—
90	C₂×D₅₂	D₈	1	1	yes	characteristic
91	C₂×C₄×D₁₃	—	4	2	no	—
92	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
93	C₂×Dic₂₆	—	2	2	no	—
94	C₂×Dic₂₆	—	4	2	no	—
95	C₂×Dic₂₆	—	4	2	no	—
96	C₂×Dic₂₆	—	4	1	no	—
97	C₂×Dic₂₆	D₈	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	2	yes	—
100	C₂×C₈×C₁₃	#41	1	2	yes	—
101	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₁₃×C₄⋊C₄	—	2	1	no	—
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₄⋊C₄	—	2	1	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₂²⋊C₄	—	4	1	no	—
107	C₁₃×C₂²⋊C₄	—	4	1	no	—
108	C₄²×C₁₃	—	4	1	no	—
109	C₄²×C₁₃	—	4	1	no	—
110	#6103	—	4	2	no	—
111	#6103	—	4	2	no	—
112	#6103	—	4	2	no	—
113	C₄⋊Dic₁₃	—	2	2	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	C₄⋊Dic₁₃	—	4	2	no	—
116	C₄⋊Dic₁₃	—	4	2	no	—
117	#6101	—	4	2	no	—
118	C₄×Dic₁₃	—	4	2	no	—
119	#6099	—	2	2	no	—
120	C₂×#1055	C₂×D₄	1	2	yes	—
121	Dic₅₂	—	2	2	no	—
122	Dic₅₂	—	2	2	no	—
123	Dic₅₂	—	4	2	no	—
124	#6094	—	2	4	no	—
125	C₈×D₁₃	—	2	2	no	—
126	C₈×D₁₃	—	2	2	no	—
127	C₁₃×C₁₆	—	2	1	no	—
128	C₁₃×C₁₆	—	2	1	no	—
129	#6090	—	2	2	no	—
130	#3059	—	13	1	no	—
131	#2913	—	13	1	no	—
132	#2237	—	13	2	no	—
133	#2237	—	13	2	no	—
134	#2236	—	13	2	no	—
135	#2235	—	13	2	no	—
136	#2159	—	13	1	no	—
137	#2158	D₁₃	1	1	yes	characteristic, complemented
138	#2141	—	13	1	no	—
139	C₂×#358	—	13	1	no	—
140	#2113	—	13	1	no	—
141	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂²×D₁₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	#123	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#137	1	2	yes	—
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₁₃	—	2	4	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	—	4	2	no	—
154	C₂×Dic₁₃	—	8	2	no	—
155	C₂×Dic₁₃	—	16	1	no	—
156	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
157	D₅₂	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	16	2	no	—
161	Dic₂₆	—	2	2	no	—
162	Dic₂₆	—	4	2	no	—
163	Dic₂₆	—	4	2	no	—
164	Dic₂₆	—	8	2	no	—
165	C₈×C₁₃	#123	1	2	yes	—
166	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
167	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
168	#1055	—	2	2	no	—
169	C₂²×Q₁₆	—	13	1	no	—
170	C₂×#131	—	13	1	no	—
171	#536	—	26	1	no	—
172	#520	—	26	1	no	—
173	C₂×M₃₂	—	13	1	no	—
174	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
175	#470	—	26	2	no	—
176	#467	—	13	2	no	—
177	#467	—	13	2	no	—
178	#428	—	26	2	no	—
179	#428	—	26	1	no	—
180	#427	—	13	2	no	—
181	#427	C₂×D₁₃	1	1	yes	characteristic
182	#427	C₂×D₁₃	1	1	yes	characteristic
183	C₂×#107	—	13	1	no	—
184	C₂×#107	—	13	2	no	—
185	#419	—	26	1	no	—
186	#367	—	13	2	no	—
187	#367	C₂×D₁₃	1	1	yes	characteristic
188	#367	C₂×D₁₃	1	1	yes	characteristic
189	#366	—	13	2	no	—
190	#366	C₂×D₁₃	1	1	yes	characteristic
191	#366	C₂×D₁₃	1	1	yes	characteristic
192	#358	—	13	2	no	—
193	#358	—	13	2	no	—
194	#358	—	13	2	no	—
195	#358	—	13	2	no	—
196	C₂²⋊C₁₆	—	13	2	no	—
197	C₂²⋊C₁₆	—	13	2	no	—
198	C₂²×C₁₃	—	2	1	no	—
199	C₂²×C₁₃	—	2	1	no	—
200	C₂²×C₁₃	#480	1	2	yes	—
201	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
202	C₂×D₁₃	—	2	2	no	—
203	C₂×D₁₃	—	2	2	no	—
204	C₂×D₁₃	—	2	2	no	—
205	C₄×C₁₃	—	8	1	no	—
206	C₄×C₁₃	—	8	1	no	—
207	C₄×C₁₃	—	8	1	no	—
208	C₄×C₁₃	—	8	1	no	—
209	C₄×C₁₃	#480	1	2	yes	—
210	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
211	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
212	Dic₁₃	—	2	2	no	—
213	Dic₁₃	—	8	2	no	—
214	Dic₁₃	—	8	2	no	—
215	C₂×#41	—	13	1	no	—
216	C₂²×Q₈	—	26	1	no	—
217	C₂×Q₁₆	—	13	2	no	—
218	C₂×Q₁₆	—	26	2	no	—
219	C₂×Q₁₆	—	26	1	no	—
220	C₂×Q₁₆	—	26	2	no	—
221	#131	—	26	2	no	—
222	#131	—	26	2	no	—
223	C₂×M₁₆	—	13	1	no	—
224	C₂×M₁₆	—	13	2	no	—
225	C₂²×C₈	—	13	2	no	—
226	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₄⋊Q₈	—	52	1	no	—
228	#124	—	52	1	no	—
229	#123	—	52	2	no	—
230	C₂²⋊Q₈	—	26	2	no	—
231	C₂²⋊Q₈	—	26	2	no	—
232	C₄×Q₈	—	52	1	no	—
233	C₄×D₄	—	52	1	no	—
234	C₄×D₄	—	52	2	no	—
235	#117	—	2	1	no	—
236	#117	—	2	1	no	—
237	#117	—	26	2	no	—
238	C₂×C₄⋊C₄	—	26	2	no	—
239	C₂×C₄⋊C₄	—	26	1	no	—
240	M₃₂	—	26	2	no	—
241	C₂×C₁₆	—	2	2	no	—
242	C₂×C₁₆	—	13	2	no	—
243	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
245	#107	—	13	2	no	—
246	#107	—	26	2	no	—
247	#107	—	26	1	no	—
248	#107	C₂²×D₁₃	1	1	yes	characteristic
249	#107	C₂²×D₁₃	1	1	yes	characteristic
250	#107	C₂²×D₁₃	1	1	yes	characteristic
251	#107	C₂²×D₁₃	1	1	yes	characteristic
252	#106	—	2	1	no	—
253	#106	—	2	1	no	—
254	#106	—	26	2	no	—
255	#103	—	26	2	no	—
256	#103	—	26	2	no	—
257	C₂×C₁₃	—	2	2	no	—
258	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
259	C₂×C₁₃	#3416	1	1	yes	characteristic
260	C₂×C₁₃	#3417	1	1	yes	characteristic
261	D₁₃	—	4	2	no	—
262	#41	—	26	2	no	—
263	#41	—	26	2	no	—
264	C₂×Q₈	—	13	1	no	—
265	C₂×Q₈	—	26	2	no	—
266	C₂×Q₈	—	52	1	no	—
267	C₂×Q₈	—	52	2	no	—
268	C₂×Q₈	—	52	2	no	—
269	C₂×D₄	—	13	2	no	—
270	C₂×D₄	—	13	1	no	—
271	C₂²×C₄	—	13	2	no	—
272	C₂²×C₄	—	52	2	no	—
273	C₂²×C₄	—	52	1	no	—
274	C₂²×C₄	—	52	1	no	—
275	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
276	Q₁₆	—	26	2	no	—
277	Q₁₆	—	26	2	no	—
278	Q₁₆	—	52	2	no	—
279	M₁₆	—	26	2	no	—
280	M₁₆	—	26	4	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	26	2	no	—
283	C₂×C₈	—	26	2	no	—
284	C₂×C₈	#6127	1	2	yes	—
285	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
286	C₂×C₈	#6131	1	2	yes	—
287	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	26	2	no	—
293	C₄⋊C₄	—	52	2	no	—
294	C₄⋊C₄	—	52	2	no	—
295	C₄⋊C₄	—	52	2	no	—
296	C₄⋊C₄	—	52	1	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	52	2	no	—
300	C₂²⋊C₄	—	52	2	no	—
301	C₂²⋊C₄	—	52	2	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	52	2	no	—
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	26	2	no	—
308	C₁₃	#33651	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	13	2	no	—
310	C₂³	D₈×D₁₃	1	1	yes	characteristic
311	Q₈	—	26	2	no	—
312	Q₈	—	52	2	no	—
313	Q₈	—	52	2	no	—
314	Q₈	—	104	2	no	—
315	D₄	—	26	4	no	—
316	D₄	—	26	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	13	2	no	—
322	C₂×C₄	—	26	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	52	2	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	—	104	2	no	—
327	C₂×C₄	—	208	2	no	—
328	C₂×C₄	—	208	1	no	—
329	C₂×C₄	#88148	1	2	yes	—
330	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
331	C₂×C₄	#88172	1	2	yes	—
332	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
333	C₈	—	26	2	no	—
334	C₈	#88148	1	2	yes	—
335	C₈	C₂×#6127	1	1	yes	characteristic
336	C₈	C₂×#6131	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	26	2	no	—
340	C₂²	—	26	2	no	—
341	C₂²	—	26	2	no	—
342	C₂²	#11718543	1	2	yes	—
343	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	26	2	no	—
349	C₄	—	104	2	no	—
350	C₄	—	104	2	no	—
351	C₄	#11718543	1	2	yes	—
352	C₄	C₂×#88148	1	1	yes	characteristic
353	C₄	C₂×#88172	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	52	2	no	—
356	C₂	C₂×#11718543	1	1	yes	characteristic
357	C₂	#13127979	1	1	yes	characteristic
358	C₂	#13127986	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×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#6127	D₄
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#33651
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #13117047	C₂	semidirect
4. #13117046	C₂	semidirect
5. #13117040	C₂	semidirect
6. #13117035	C₂	semidirect
7. #13116625	C₂	semidirect
8. #13116555	C₂	semidirect
9. C₂×#11718095	C₂	semidirect
11. C₁₃×#2158	C₂	semidirect
137. #2158	D₁₃	semidirect
308. C₁₃	#33651	semidirect

Maximal subgroups

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

subgroup	index
#13124492	2
#13123433	2
#13117047	2
#13117046	2
#13117040	2
#13117035	2
#13116625	2
#13116555	2
C₂×#11718095	2
#13116449	2
C₁₃×#2158	2
#33651	13