Subgroups — #15833471

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125395	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123549	C₂	1	1	yes	maximal, characteristic
4	#13113740	C₂	1	4	yes	maximal, complemented
5	#13113650	C₂	1	4	yes	maximal, complemented
6	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113292	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113288	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88155	—	2	1	no	—
11	#11719182	C₂²	1	1	yes	characteristic
12	#11719175	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718775	C₄	1	4	yes	—
17	#11718771	—	2	1	no	—
18	#11718763	—	2	1	no	—
19	#11718762	—	2	1	no	—
20	#11718585	C₂²	1	1	yes	characteristic
21	#11718585	C₂²	1	1	yes	characteristic
22	#11718355	C₄	1	1	yes	characteristic
23	#11718354	C₄	1	1	yes	characteristic
24	C₄×#6095	C₂²	1	4	yes	—
25	#11718349	—	2	1	no	—
26	#11718349	C₂²	1	2	yes	—
27	C₈×D₅₂	—	2	1	no	—
28	C₈×D₅₂	C₂²	1	2	yes	—
29	#11718342	C₄	1	1	yes	characteristic
30	#11718341	C₄	1	1	yes	characteristic
31	#11718340	—	2	1	no	—
32	#11718340	C₂²	1	2	yes	—
33	C₈×Dic₂₆	—	2	1	no	—
34	C₈×Dic₂₆	C₂²	1	2	yes	—
35	C₁₃×#322	C₂²	1	2	yes	—
36	C₁₃×#322	C₂²	1	2	yes	—
37	#11718051	C₂²	1	4	yes	—
38	#11718050	C₂²	1	4	yes	—
39	#11718048	C₂²	1	4	yes	—
40	C₂×#6127	—	2	1	no	—
41	C₂²×C₈×C₁₃	—	2	1	no	—
42	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	#88184	—	2	1	no	—
47	#88184	C₂×C₄	1	2	yes	—
48	#88180	—	2	1	no	—
49	#88178	—	2	1	no	—
50	#88178	C₂×C₄	1	2	yes	—
51	C₂×#6099	—	2	1	no	—
52	C₂×#6099	—	2	1	no	—
53	C₂×#6095	C₂×C₄	1	4	yes	—
54	#88155	—	4	4	no	—
55	C₂×#6094	—	2	2	no	—
56	C₂×C₈×D₁₃	—	2	2	no	—
57	#88129	—	2	1	no	—
58	#88128	C₂×C₄	1	1	yes	characteristic
59	C₄×D₅₂	—	2	1	no	—
60	C₄×D₅₂	C₂³	1	2	yes	—
61	#88124	—	2	1	no	—
62	#88123	C₂×C₄	1	1	yes	characteristic
63	#88123	C₂×C₄	1	1	yes	characteristic
64	#88123	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₆	—	2	1	no	—
66	C₄×Dic₂₆	C₂³	1	2	yes	—
67	C₄×C₈×C₁₃	—	2	1	no	—
68	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	2	yes	—
73	C₄×C₈×C₁₃	C₂³	1	2	yes	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#88061	C₂×C₄	1	4	yes	—
77	#88060	—	2	2	no	—
78	#88060	—	2	2	no	—
79	#88057	C₂×C₄	1	4	yes	—
80	#88056	C₂×C₄	1	4	yes	—
81	#88055	—	2	2	no	—
82	#88054	—	2	2	no	—
83	#88054	—	2	2	no	—
84	C₈×Dic₁₃	—	2	2	no	—
85	C₄⋊#1055	—	2	1	no	—
86	C₄⋊#1055	—	2	1	no	—
87	C₄⋊#1055	C₂³	1	2	yes	—
88	C₄⋊#1055	C₂³	1	2	yes	—
89	#17960	—	13	1	no	maximal, Sylow
90	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	—	2	2	no	—
94	#6127	—	4	2	no	—
95	#6127	—	4	2	no	—
96	C₂×D₅₂	—	2	1	no	—
97	C₂×D₅₂	—	2	1	no	—
98	C₂×D₅₂	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₁₃	—	2	2	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₂×Dic₂₆	—	2	1	no	—
102	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	4	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	—	4	2	no	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	#41	1	1	yes	characteristic
117	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊Dic₁₃	—	2	2	no	—
119	#6103	—	2	2	no	—
120	#6103	—	2	2	no	—
121	C₄⋊Dic₁₃	—	2	1	no	—
122	C₄⋊Dic₁₃	—	2	1	no	—
123	C₄⋊Dic₁₃	—	2	1	no	—
124	C₄⋊Dic₁₃	—	2	1	no	—
125	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
126	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
127	#6101	—	2	2	no	—
128	#6101	—	2	2	no	—
129	C₄×Dic₁₃	—	2	2	no	—
130	#6099	—	4	2	no	—
131	#6099	—	4	2	no	—
132	C₂×#1055	—	2	2	no	—
133	C₂×#1055	—	2	2	no	—
134	#6095	—	2	8	no	—
135	#6094	—	4	4	no	—
136	C₈×D₁₃	—	4	4	no	—
137	#3167	—	13	1	no	—
138	#2960	—	13	1	no	—
139	#2931	—	13	1	no	—
140	#1592	—	13	4	no	—
141	SD₁₆⋊C₈	—	13	4	no	—
142	#1567	—	13	1	no	—
143	#1493	—	13	1	no	—
144	#1492	—	13	1	no	—
145	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
146	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂²×D₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₁₃	#137	1	2	yes	—
155	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₂²⋊D₁₃	—	4	4	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	—	2	2	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	D₅₂	—	2	4	no	—
163	D₅₂	—	4	2	no	—
164	D₅₂	—	4	2	no	—
165	C₄×D₁₃	—	4	2	no	—
166	C₄×D₁₃	—	4	2	no	—
167	Dic₂₆	—	2	4	no	—
168	Dic₂₆	—	4	2	no	—
169	Dic₂₆	—	4	2	no	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	—	2	1	no	—
172	C₈×C₁₃	—	2	1	no	—
173	C₈×C₁₃	—	2	1	no	—
174	C₈×C₁₃	—	2	1	no	—
175	C₈×C₁₃	—	4	2	no	—
176	C₈×C₁₃	#131	1	4	yes	complemented
177	#1055	—	4	2	no	—
178	#1055	—	4	2	no	—
179	C₂×#131	—	26	1	no	—
180	#533	—	13	1	no	—
181	#532	—	13	1	no	—
182	#498	—	13	1	no	—
183	#492	—	13	1	no	—
184	#487	—	13	1	no	—
185	#486	—	13	1	no	—
186	#465	—	13	4	no	—
187	#446	—	13	2	no	—
188	#446	—	26	1	no	—
189	C₈×Q₈	—	13	2	no	—
190	C₈×Q₈	—	26	1	no	—
191	C₄×SD₁₆	—	13	4	no	—
192	#436	—	13	2	no	—
193	#436	—	26	1	no	—
194	C₈×D₄	—	13	2	no	—
195	C₈×D₄	—	26	1	no	—
196	#424	—	26	1	no	—
197	#423	—	13	1	no	—
198	#423	—	13	1	no	—
199	#408	—	26	1	no	—
200	#405	—	26	1	no	—
201	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
203	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
204	#335	—	13	4	no	—
205	Q₈⋊C₈	—	13	4	no	—
206	D₄⋊C₈	—	13	4	no	—
207	#322	C₂×D₁₃	1	2	yes	—
208	#322	C₂×D₁₃	1	2	yes	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	—	2	1	no	—
211	C₂²×C₁₃	#440	1	2	yes	—
212	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂×D₁₃	—	4	2	no	—
214	C₂×D₁₃	—	4	2	no	—
215	C₂×D₁₃	—	4	2	no	—
216	C₄×C₁₃	—	2	1	no	—
217	C₄×C₁₃	—	2	1	no	—
218	C₄×C₁₃	#440	1	2	yes	—
219	C₄×C₁₃	#444	1	2	yes	—
220	C₄×C₁₃	C₂×#131	1	2	yes	—
221	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
223	Dic₁₃	—	4	2	no	—
224	Dic₁₃	—	4	2	no	—
225	Dic₁₃	—	4	2	no	—
226	C₂×#41	—	26	1	no	—
227	C₂×SD₁₆	—	13	4	no	—
228	#131	—	52	4	no	—
229	C₂×M₁₆	—	26	1	no	—
230	C₂×M₁₆	—	26	1	no	—
231	C₂×M₁₆	—	26	2	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	26	2	no	—
234	C₂²×C₈	C₄×D₁₃	1	2	yes	—
235	C₂²×C₈	D₅₂	1	1	yes	characteristic
236	C₂²×C₈	D₅₂	1	1	yes	characteristic
237	C₄⋊Q₈	—	13	1	no	—
238	C₄⋊Q₈	—	13	1	no	—
239	C₄⋊Q₈	—	13	1	no	—
240	#127	—	13	1	no	—
241	#125	—	26	1	no	—
242	#124	—	26	1	no	—
243	C₂²⋊Q₈	—	13	2	no	—
244	C₂²⋊Q₈	—	26	1	no	—
245	#121	—	13	2	no	—
246	#121	—	26	1	no	—
247	C₄×Q₈	—	13	2	no	—
248	C₄×Q₈	—	26	1	no	—
249	C₄×D₄	—	13	2	no	—
250	C₄×D₄	—	26	1	no	—
251	#117	—	26	1	no	—
252	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
253	#106	—	13	4	no	—
254	C₄⋊C₈	—	13	2	no	—
255	C₄⋊C₈	—	13	2	no	—
256	C₄⋊C₈	—	26	2	no	—
257	C₄⋊C₈	—	26	2	no	—
258	C₄⋊C₈	—	26	1	no	—
259	C₄⋊C₈	—	26	1	no	—
260	#103	—	13	4	no	—
261	#102	—	13	4	no	—
262	C₂²⋊C₈	—	26	2	no	—
263	C₂²⋊C₈	—	26	2	no	—
264	#97	—	26	2	no	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	26	2	no	—
267	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
268	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
269	C₄×C₈	D₅₂	1	1	yes	characteristic
270	C₄×C₈	D₅₂	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₁₃	1	2	yes	—
272	C₄×C₈	C₂²×D₁₃	1	2	yes	—
273	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
275	C₂×C₁₃	—	2	2	no	—
276	C₂×C₁₃	#2931	1	1	yes	characteristic
277	C₂×C₁₃	C₂×#440	1	1	yes	characteristic
278	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
279	D₁₃	—	8	2	no	—
280	#41	—	52	2	no	—
281	#41	—	52	2	no	—
282	C₂×Q₈	—	13	2	no	—
283	C₂×Q₈	—	26	1	no	—
284	C₂×Q₈	—	26	1	no	—
285	C₂×D₄	—	13	2	no	—
286	C₂×D₄	—	26	2	no	—
287	C₂×D₄	—	26	1	no	—
288	C₂×D₄	—	26	1	no	—
289	C₂²×C₄	—	26	2	no	—
290	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
292	C₂²×C₄	#6127	1	1	yes	characteristic
293	SD₁₆	—	26	8	no	—
294	M₁₆	—	52	4	no	—
295	M₁₆	—	52	2	no	—
296	M₁₆	—	52	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	4	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	26	2	no	—
303	C₂×C₈	—	26	2	no	—
304	C₂×C₈	—	52	4	no	—
305	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
306	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
307	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
308	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
309	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
310	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
311	C₄⋊C₄	—	13	2	no	—
312	C₄⋊C₄	—	13	2	no	—
313	C₄⋊C₄	—	26	1	no	—
314	C₄⋊C₄	—	26	1	no	—
315	C₄⋊C₄	—	26	1	no	—
316	C₄⋊C₄	—	26	2	no	—
317	C₄⋊C₄	—	26	1	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₂²⋊C₄	—	26	2	no	—
320	C₂²⋊C₄	—	26	2	no	—
321	C₂²⋊C₄	—	26	2	no	—
322	C₄²	—	26	2	no	—
323	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
324	C₄²	C₂×D₅₂	1	1	yes	characteristic
325	C₄²	#6127	1	1	yes	characteristic
326	C₄²	C₂³×D₁₃	1	1	yes	characteristic
327	C₁₃	#17960	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	26	2	no	—
329	C₂³	C₄×D₅₂	1	1	yes	characteristic
330	Q₈	—	26	4	no	—
331	Q₈	—	52	2	no	—
332	Q₈	—	52	2	no	—
333	D₄	—	26	4	no	—
334	D₄	—	52	4	no	—
335	D₄	—	52	2	no	—
336	D₄	—	52	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	26	2	no	—
340	C₂×C₄	—	26	2	no	—
341	C₂×C₄	—	26	2	no	—
342	C₂×C₄	—	52	2	no	—
343	C₂×C₄	—	52	2	no	—
344	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
345	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
346	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
347	C₂×C₄	#88162	1	2	yes	—
348	C₂×C₄	#88163	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
351	C₂×C₄	C₂×#6127	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	52	2	no	—
359	C₈	—	52	2	no	—
360	C₈	#88155	1	4	yes	complemented
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	52	2	no	—
364	C₂²	—	52	2	no	—
365	C₂²	—	52	2	no	—
366	C₂²	#11718371	1	2	yes	—
367	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	52	2	no	—
371	C₄	—	52	2	no	—
372	C₄	—	52	2	no	—
373	C₄	#11718371	1	2	yes	—
374	C₄	#11718378	1	2	yes	—
375	C₄	C₂×#88155	1	2	yes	—
376	C₄	C₂×#88162	1	1	yes	characteristic
377	C₄	C₂×#88163	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	104	2	no	—
380	C₂	#13123549	1	1	yes	characteristic
381	C₂	C₂×#11718371	1	1	yes	characteristic
382	C₂	C₂×#11718378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#17960
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #13123770	C₂	semidirect
4. #13113740	C₂	semidirect
5. #13113650	C₂	semidirect
6. #13113595	C₂	semidirect
7. #13113292	C₂	semidirect
9. C₂×C₁₃×#322	C₂	semidirect
145. C₂×#322	D₁₃	semidirect
176. C₈×C₁₃	#131	semidirect
327. C₁₃	#17960	semidirect
360. C₈	#88155	semidirect

Maximal subgroups

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

subgroup	index
#13125395	2
#13123770	2
#13123549	2
#13113740	2
#13113650	2
#13113595	2
#13113292	2
#13113288	2
C₂×C₁₃×#322	2
#17960	13