Subgroups — #16019859

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127260	C₂	1	1	yes	maximal, characteristic, complemented
2	#13126557	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117064	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117046	C₂	1	2	yes	maximal, complemented
5	#13116983	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116931	C₂	1	2	yes	maximal, complemented
7	#13116881	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116756	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116540	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88166	C₂²	1	2	yes	—
13	#11719268	—	2	1	no	—
14	#11719260	—	2	1	no	—
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
18	#11718818	C₂²	1	2	yes	—
19	C₂×#88067	C₂²	1	2	yes	—
20	#11718680	—	2	1	no	—
21	#11718669	—	2	1	no	—
22	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
23	#11718548	C₂²	1	2	yes	—
24	#11718541	C₂²	1	2	yes	—
25	#11718445	—	2	2	no	—
26	#11718444	—	2	2	no	—
27	#11718360	C₂²	1	1	yes	characteristic
28	C₈×D₅₂	C₂²	1	1	yes	characteristic
29	#11718343	C₂²	1	1	yes	characteristic
30	C₁₃×#366	C₂²	1	1	yes	characteristic
31	C₁₃×#357	C₂²	1	1	yes	characteristic
32	C₁₃×#357	C₂²	1	1	yes	characteristic
33	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
34	#11718154	C₂²	1	2	yes	—
35	#11718099	C₂²	1	2	yes	—
36	#11718095	C₂²	1	2	yes	—
37	#11718095	C₂²	1	2	yes	—
38	#11718084	C₂²	1	2	yes	—
39	#11718082	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₂×#6129	—	2	2	no	—
42	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₃	—	2	1	no	—
45	C₄×D₄×C₁₃	—	2	1	no	—
46	#88192	—	2	2	no	—
47	C₂×#6105	—	2	2	no	—
48	#88184	—	4	1	no	—
49	#88184	—	4	1	no	—
50	#88180	—	4	1	no	—
51	C₂×C₄⋊Dic₁₃	—	4	1	no	—
52	#88166	—	2	4	no	—
53	#88166	—	4	2	no	—
54	C₂×Dic₅₂	—	2	1	no	—
55	C₂×Dic₅₂	C₂³	1	2	yes	—
56	C₂×C₈×D₁₃	C₂³	1	2	yes	—
57	#88151	—	2	2	no	—
58	#88146	—	2	2	no	—
59	#88140	—	4	2	no	—
60	#88139	—	4	2	no	—
61	C₄×D₅₂	D₄	1	1	yes	characteristic
62	#88123	—	2	1	no	—
63	#88123	—	2	1	no	—
64	C₁₃×D₁₆	—	2	2	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#102	—	2	1	no	—
70	C₁₃×#102	—	2	1	no	—
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#88072	—	2	2	no	—
73	#88067	—	4	2	no	—
74	#88060	D₄	1	2	yes	—
75	#88058	—	2	1	no	—
76	#88058	C₂³	1	2	yes	—
77	C₂×#6090	C₂³	1	2	yes	—
78	#88050	—	2	2	no	—
79	#88048	—	2	2	no	—
80	C₄⋊#1055	D₄	1	1	yes	characteristic
81	C₁₆×D₁₃	—	4	2	no	—
82	#33870	—	13	1	no	maximal, Sylow
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂×C₂²⋊D₁₃	—	4	2	no	—
88	C₂×C₂²⋊D₁₃	—	4	2	no	—
89	C₂²×Dic₁₃	—	4	2	no	—
90	#6129	—	4	4	no	—
91	#6129	—	8	2	no	—
92	C₂×D₅₂	D₈	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
94	C₂×Dic₂₆	—	2	2	no	—
95	C₂×Dic₂₆	—	4	1	no	—
96	C₁₃×D₈	—	2	1	no	—
97	C₁₃×D₈	—	2	1	no	—
98	C₁₃×D₈	—	2	1	no	—
99	C₁₃×D₈	#41	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
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×C₂²⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	1	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂²⋊Dic₁₃	—	4	2	no	—
108	#6105	—	4	4	no	—
109	#6103	—	4	2	no	—
110	#6103	—	4	2	no	—
111	#6103	D₈	1	2	yes	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	4	1	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	C₄⋊Dic₁₃	—	4	1	no	—
116	C₄⋊Dic₁₃	—	4	2	no	—
117	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
118	C₄×Dic₁₃	—	4	2	no	—
119	C₂×#1055	C₂×D₄	1	2	yes	—
120	Dic₅₂	—	2	2	no	—
121	Dic₅₂	—	2	2	no	—
122	Dic₅₂	—	8	1	no	—
123	C₈×D₁₃	—	2	2	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₁₃×C₁₆	—	2	1	no	—
126	C₁₃×C₁₆	—	2	1	no	—
127	#6090	—	2	2	no	—
128	#3341	—	13	1	no	—
129	#3283	—	13	1	no	—
130	#2241	—	13	1	no	—
131	#2237	—	13	2	no	—
132	#2226	—	13	1	no	—
133	#2217	—	13	2	no	—
134	#2208	—	13	1	no	—
135	#2186	—	13	2	no	—
136	C₄×D₁₆	D₁₃	1	1	yes	characteristic, complemented
137	D₄×C₁₆	—	13	1	no	—
138	#2138	—	13	2	no	—
139	C₂³×C₁₃	—	4	1	no	—
140	C₂³×C₁₃	—	4	1	no	—
141	C₂²×D₁₃	C₂×D₈	1	2	yes	—
142	D₄×C₁₃	—	2	2	no	—
143	D₄×C₁₃	—	4	1	no	—
144	D₄×C₁₃	—	4	1	no	—
145	D₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	8	2	no	—
149	C₂×C₄×C₁₃	—	8	2	no	—
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₁₃	—	8	4	no	—
154	C₂²⋊D₁₃	—	16	2	no	—
155	C₂×Dic₁₃	—	4	2	no	—
156	C₂×Dic₁₃	—	4	2	no	—
157	C₂×Dic₁₃	—	8	4	no	—
158	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
159	D₅₂	—	2	2	no	—
160	C₄×D₁₃	—	2	2	no	—
161	C₄×D₁₃	—	2	2	no	—
162	Dic₂₆	—	4	2	no	—
163	Dic₂₆	—	4	2	no	—
164	Dic₂₆	—	16	1	no	—
165	C₈×C₁₃	—	2	1	no	—
166	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
167	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
168	#1055	—	2	2	no	—
169	C₂×#135	—	13	2	no	—
170	#555	—	26	1	no	—
171	#547	—	26	1	no	—
172	C₂×SD₃₂	—	13	2	no	—
173	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
174	C₂²×C₁₆	—	13	2	no	—
175	#500	—	13	1	no	—
176	#494	—	13	1	no	—
177	#482	—	26	2	no	—
178	#476	—	26	1	no	—
179	#467	—	13	2	no	—
180	#466	—	13	2	no	—
181	#461	—	26	1	no	—
182	#452	—	26	2	no	—
183	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
184	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
185	C₈×D₄	—	13	1	no	—
186	#427	—	13	2	no	—
187	#367	—	13	2	no	—
188	#366	C₂×D₁₃	1	1	yes	characteristic
189	C₄⋊C₁₆	—	13	1	no	—
190	#358	—	13	2	no	—
191	#358	—	13	2	no	—
192	#358	—	13	2	no	—
193	#357	—	13	2	no	—
194	#357	C₂×D₁₃	1	1	yes	characteristic
195	#357	C₂×D₁₃	1	1	yes	characteristic
196	C₂²⋊C₁₆	—	13	2	no	—
197	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
198	C₂²×C₁₃	—	4	2	no	—
199	C₂²×C₁₃	—	8	1	no	—
200	C₂²×C₁₃	—	8	1	no	—
201	C₂²×C₁₃	—	8	1	no	—
202	C₂²×C₁₃	—	8	1	no	—
203	C₂²×C₁₃	—	8	1	no	—
204	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
205	C₂×D₁₃	—	2	2	no	—
206	C₂×D₁₃	—	2	2	no	—
207	C₂×D₁₃	—	2	2	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	—	8	1	no	—
211	C₄×C₁₃	#508	1	2	yes	—
212	C₄×C₁₃	#546	1	1	yes	characteristic
213	C₄×C₁₃	#577	1	1	yes	characteristic
214	Dic₁₃	—	2	2	no	—
215	Dic₁₃	—	8	2	no	—
216	Dic₁₃	—	8	2	no	—
217	C₂×#41	—	26	2	no	—
218	#135	—	26	4	no	—
219	#135	—	52	2	no	—
220	C₂×Q₁₆	—	13	2	no	—
221	C₂×Q₁₆	—	26	1	no	—
222	C₂×SD₁₆	—	26	2	no	—
223	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₂²×C₈	—	13	2	no	—
226	C₄⋊Q₈	—	26	1	no	—
227	C₄⋊Q₈	—	26	1	no	—
228	#124	—	52	2	no	—
229	#123	—	52	2	no	—
230	C₂²⋊Q₈	—	26	2	no	—
231	#121	—	26	2	no	—
232	#121	—	52	1	no	—
233	#121	—	52	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	13	1	no	—
237	#117	—	26	2	no	—
238	#117	—	52	1	no	—
239	C₂×C₄⋊C₄	—	52	1	no	—
240	SD₃₂	—	52	2	no	—
241	D₁₆	—	2	2	no	—
242	C₂×C₁₆	—	13	2	no	—
243	C₂×C₁₆	—	52	2	no	—
244	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
245	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
246	#107	—	13	2	no	—
247	#107	—	26	1	no	—
248	#107	C₂²×D₁₃	1	1	yes	characteristic
249	#107	C₂²×D₁₃	1	1	yes	characteristic
250	#106	—	26	2	no	—
251	C₄⋊C₈	—	13	1	no	—
252	#103	—	26	2	no	—
253	#102	—	2	1	no	—
254	#102	—	2	1	no	—
255	#102	—	26	2	no	—
256	C₂²⋊C₈	—	13	2	no	—
257	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
258	C₂×C₁₃	—	8	2	no	—
259	C₂×C₁₃	—	16	1	no	—
260	C₂×C₁₃	#3294	1	1	yes	characteristic
261	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
262	C₂×C₁₃	#3416	1	1	yes	characteristic
263	D₁₃	—	4	2	no	—
264	#41	—	52	4	no	—
265	#41	—	104	2	no	—
266	C₂×Q₈	—	26	2	no	—
267	C₂×Q₈	—	52	1	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	13	1	no	—
271	C₂×D₄	—	52	2	no	—
272	C₂×D₄	—	52	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	13	2	no	—
276	C₂²×C₄	—	52	2	no	—
277	Q₁₆	—	26	2	no	—
278	Q₁₆	—	26	2	no	—
279	Q₁₆	—	104	1	no	—
280	SD₁₆	—	52	4	no	—
281	D₈	—	2	1	no	—
282	D₈	—	2	1	no	—
283	D₈	—	2	1	no	—
284	D₈	#6127	1	2	yes	—
285	C₂×C₈	—	13	2	no	—
286	C₂×C₈	—	26	2	no	—
287	C₂×C₈	—	26	2	no	—
288	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	13	1	no	—
293	C₄⋊C₄	—	26	2	no	—
294	C₄⋊C₄	—	52	2	no	—
295	C₄⋊C₄	—	52	1	no	—
296	C₄⋊C₄	—	52	1	no	—
297	C₄⋊C₄	—	52	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	13	2	no	—
301	C₂²⋊C₄	—	52	2	no	—
302	C₂²⋊C₄	—	52	2	no	—
303	C₂²⋊C₄	—	52	2	no	—
304	C₄²	—	52	2	no	—
305	C₄²	D₄×D₁₃	1	1	yes	characteristic
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	26	2	no	—
309	C₁₃	#33870	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	4	1	no	—
311	C₂³	—	4	1	no	—
312	C₂³	—	13	2	no	—
313	Q₈	—	52	2	no	—
314	Q₈	—	52	2	no	—
315	Q₈	—	208	1	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	26	2	no	—
321	D₄	—	104	4	no	—
322	D₄	—	208	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	13	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	—	52	2	no	—
332	C₂×C₄	—	104	4	no	—
333	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
334	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	26	2	no	—
338	C₈	C₂×#6127	1	1	yes	characteristic
339	C₈	#88256	1	1	yes	characteristic
340	C₂²	—	4	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	1	no	—
343	C₂²	—	8	1	no	—
344	C₂²	—	8	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	26	2	no	—
347	C₂²	—	26	2	no	—
348	C₂²	—	26	2	no	—
349	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	8	1	no	—
352	C₄	—	8	1	no	—
353	C₄	—	26	2	no	—
354	C₄	—	104	2	no	—
355	C₄	—	104	2	no	—
356	C₄	#11718573	1	2	yes	—
357	C₄	#11719255	1	1	yes	characteristic
358	C₄	#11719475	1	1	yes	characteristic
359	C₂	—	8	2	no	—
360	C₂	—	16	1	no	—
361	C₂	—	52	2	no	—
362	C₂	#13126651	1	1	yes	characteristic
363	C₂	C₂×#11718573	1	1	yes	characteristic
364	C₂	#13127979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×D₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₄×D₁₆	D₁₃
hypocenter	C₁₃	#33870
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #13127260	C₂	semidirect
2. #13126557	C₂	semidirect
3. #13117064	C₂	semidirect
4. #13117046	C₂	semidirect
5. #13116983	C₂	semidirect
6. #13116931	C₂	semidirect
7. #13116881	C₂	semidirect
8. #13116756	C₂	semidirect
9. C₁₆×D₅₂	C₂	semidirect
10. #13116540	C₂	semidirect
11. C₄×C₁₃×D₁₆	C₂	semidirect
136. C₄×D₁₆	D₁₃	semidirect
309. C₁₃	#33870	semidirect

Maximal subgroups

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

subgroup	index
#13127260	2
#13126557	2
#13117064	2
#13117046	2
#13116983	2
#13116931	2
#13116881	2
#13116756	2
C₁₆×D₅₂	2
#13116540	2
C₄×C₁₃×D₁₆	2
#33870	13