Subgroups — #15828617

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123803	C₂	1	1	yes	maximal, characteristic
2	#13123480	C₂	1	1	yes	maximal, characteristic
3	#13113136	C₂	1	2	yes	maximal, complemented
4	#13113131	C₂	1	2	yes	maximal, complemented
5	#13113128	C₂	1	2	yes	maximal, complemented
6	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113120	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×#6099	C₂	1	2	yes	maximal, complemented
10	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
12	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₁₃	C₄	1	2	yes	—
14	C₈×D₄×C₁₃	C₄	1	2	yes	—
15	C₈×D₄×C₁₃	C₄	1	4	yes	—
16	C₁₃×#431	C₂²	1	1	yes	characteristic
17	C₁₃×#431	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
20	#11718895	C₄	1	2	yes	—
21	#11718685	C₄	1	2	yes	—
22	#11718673	C₄	1	2	yes	—
23	#11718672	C₄	1	4	yes	—
24	#11718661	C₄	1	4	yes	—
25	#11718660	C₄	1	2	yes	—
26	#11718586	C₂²	1	2	yes	—
27	#11718586	C₂²	1	1	yes	characteristic
28	#11718586	C₂²	1	2	yes	—
29	#11718586	C₂²	1	2	yes	—
30	C₄×#6099	C₂²	1	2	yes	—
31	C₄×#6099	C₂²	1	1	yes	characteristic
32	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	#11718047	C₂²	1	2	yes	—
35	#11718046	C₂²	1	1	yes	characteristic
36	#11718046	C₂²	1	1	yes	characteristic
37	#11718046	C₂²	1	1	yes	characteristic
38	#11718046	C₂²	1	1	yes	characteristic
39	#11718046	C₂²	1	2	yes	—
40	#11718046	C₂²	1	2	yes	—
41	#11718046	C₂²	1	2	yes	—
42	C₈×#1055	C₂²	1	2	yes	—
43	C₈×#1055	C₂²	1	2	yes	—
44	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
54	C₁₃×#117	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
57	C₂×#6099	C₂×C₄	1	2	yes	—
58	C₂×#6099	C₂×C₄	1	4	yes	—
59	C₂²×#1055	C₂×C₄	1	2	yes	—
60	C₂²×#1055	C₂×C₄	1	2	yes	—
61	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₂²⋊#1055	C₂×C₄	1	4	yes	—
71	C₂²⋊#1055	C₂×C₄	1	4	yes	—
72	C₂²⋊#1055	C₂×C₄	1	2	yes	—
73	C₂²⋊#1055	C₂×C₄	1	4	yes	—
74	C₄⋊#1055	C₂×C₄	1	4	yes	—
75	C₄⋊#1055	C₂×C₄	1	4	yes	—
76	C₄⋊#1055	C₂×C₄	1	4	yes	—
77	C₄⋊#1055	C₂×C₄	1	2	yes	—
78	#88043	C₂×C₄	1	2	yes	—
79	#88043	C₂×C₄	1	4	yes	—
80	C₄×#1055	C₂×C₄	1	2	yes	—
81	C₄×#1055	C₂×C₄	1	2	yes	—
82	C₄×#1055	C₂³	1	1	yes	characteristic
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	2	yes	—
85	C₄×#1055	C₂³	1	2	yes	—
86	C₄×#1055	C₂³	1	1	yes	characteristic
87	C₄×#1055	C₂³	1	1	yes	characteristic
88	#17331	—	13	1	no	maximal, Sylow
89	C₁₃×#41	—	2	4	no	—
90	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
91	C₂×D₄×C₁₃	C₄²	1	2	yes	—
92	C₂×D₄×C₁₃	C₄²	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	—	4	2	no	—
100	C₂×C₈×C₁₃	—	4	4	no	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
104	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
105	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
107	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
108	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
109	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
110	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
111	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
115	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
116	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
118	#6099	—	2	4	no	—
119	#6099	—	2	8	no	—
120	C₂×#1055	—	2	4	no	—
121	C₂×#1055	—	2	4	no	—
122	C₂×#1055	C₂²×C₄	1	2	yes	—
123	C₂×#1055	C₂²×C₄	1	2	yes	—
124	C₂×#1055	C₂²×C₄	1	2	yes	—
125	C₂×#1055	C₂²×C₄	1	4	yes	—
126	C₂×#1055	C₂²×C₄	1	4	yes	—
127	C₂×#1055	C₂²×C₄	1	2	yes	—
128	#2965	—	13	1	no	—
129	C₈×#41	D₁₃	1	1	yes	characteristic
130	#2922	—	13	1	no	—
131	#1454	—	13	2	no	—
132	#1453	—	13	2	no	—
133	#1452	—	13	2	no	—
134	#1451	—	13	1	no	—
135	#1450	—	13	1	no	—
136	C₈×M₁₆	—	13	1	no	—
137	C₈×M₁₆	—	13	2	no	—
138	C₂×#322	—	13	1	no	—
139	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
140	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
141	Q₈×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	2	4	no	—
143	D₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	4	no	—
146	C₂×C₄×C₁₃	—	2	4	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	4	no	—
149	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₁₃	#131	1	4	yes	—
156	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	—	2	4	no	—
161	C₈×C₁₃	—	2	2	no	—
162	#1055	—	2	2	no	—
163	#1055	—	2	2	no	—
164	#1055	—	2	2	no	—
165	#1055	—	2	4	no	—
166	#1055	—	2	4	no	—
167	#1055	#131	1	4	yes	—
168	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
169	#446	—	13	2	no	—
170	C₈×Q₈	Dic₁₃	1	2	yes	—
171	#436	—	13	2	no	—
172	#435	—	13	4	no	—
173	C₈×D₄	Dic₁₃	1	2	yes	—
174	C₈×D₄	Dic₁₃	1	4	yes	—
175	#433	—	13	4	no	—
176	#431	—	13	2	no	—
177	#431	—	13	2	no	—
178	#431	—	13	1	no	—
179	#431	—	13	2	no	—
180	#431	C₂×D₁₃	1	1	yes	characteristic
181	#431	C₂×D₁₃	1	2	yes	—
182	#424	—	13	2	no	—
183	#408	—	13	2	no	—
184	C₄×M₁₆	—	13	1	no	—
185	C₄×M₁₆	—	13	2	no	—
186	C₂×C₄×C₈	—	13	1	no	—
187	C₂×C₄×C₈	—	13	1	no	—
188	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
190	#322	—	13	1	no	—
191	#322	—	13	1	no	—
192	#322	—	13	1	no	—
193	#322	—	13	1	no	—
194	#322	—	13	2	no	—
195	#322	—	13	2	no	—
196	#322	—	13	2	no	—
197	#322	—	13	2	no	—
198	C₈²	—	13	2	no	—
199	C₈²	—	13	2	no	—
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	—	2	2	no	—
203	C₂²×C₁₃	—	2	2	no	—
204	C₂²×C₁₃	—	2	2	no	—
205	C₂²×C₁₃	#405	1	2	yes	—
206	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
207	C₄×C₁₃	—	2	2	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	2	2	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	2	no	—
212	C₄×C₁₃	#405	1	2	yes	—
213	C₄×C₁₃	C₂×#131	1	2	yes	—
214	C₄×C₁₃	C₂×#131	1	2	yes	—
215	C₄×C₁₃	#568	1	2	yes	—
216	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
217	C₂×M₁₆	—	13	2	no	—
218	C₂×M₁₆	—	13	4	no	—
219	C₂²×C₈	—	13	2	no	—
220	C₂²×C₈	—	13	2	no	—
221	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
222	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
223	C₄×Q₈	C₄×D₁₃	1	1	yes	characteristic
224	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
225	C₄×D₄	C₄×D₁₃	1	2	yes	—
226	C₄×D₄	C₄×D₁₃	1	1	yes	characteristic
227	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
228	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
229	#117	C₄×D₁₃	1	1	yes	characteristic
230	#117	C₄×D₁₃	1	2	yes	—
231	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
233	C₄⋊C₈	—	13	4	no	—
234	C₄⋊C₈	—	13	4	no	—
235	C₄⋊C₈	—	13	4	no	—
236	C₄⋊C₈	—	13	2	no	—
237	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
238	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
239	C₂²⋊C₈	—	13	4	no	—
240	C₂²⋊C₈	—	13	4	no	—
241	C₂²⋊C₈	—	13	2	no	—
242	C₂²⋊C₈	—	13	4	no	—
243	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
244	C₂²⋊C₈	C₂×Dic₁₃	1	4	yes	—
245	#97	—	13	2	no	—
246	#97	—	13	4	no	—
247	C₄×C₈	—	13	2	no	—
248	C₄×C₈	—	13	2	no	—
249	C₄×C₈	—	13	1	no	—
250	C₄×C₈	—	13	1	no	—
251	C₄×C₈	—	13	2	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	13	1	no	—
254	C₄×C₈	—	13	1	no	—
255	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
256	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
257	C₄×C₈	C₂²×D₁₃	1	2	yes	—
258	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
260	C₂×C₁₃	—	2	2	no	—
261	C₂×C₁₃	—	4	2	no	—
262	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
263	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
264	C₂×C₁₃	#2875	1	1	yes	characteristic
265	#41	—	2	4	no	—
266	C₂×Q₈	C₄×Dic₁₃	1	1	yes	characteristic
267	C₂×D₄	C₄×Dic₁₃	1	2	yes	—
268	C₂×D₄	C₄×Dic₁₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
270	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
272	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
274	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
275	M₁₆	—	26	8	no	—
276	M₁₆	—	26	4	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	4	4	no	—
279	C₂×C₈	—	13	2	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	13	2	no	—
283	C₂×C₈	—	13	4	no	—
284	C₂×C₈	—	13	4	no	—
285	C₂×C₈	—	26	4	no	—
286	C₂×C₈	—	26	4	no	—
287	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
288	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
289	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
290	C₄⋊C₄	C₄×Dic₁₃	1	2	yes	—
291	C₄⋊C₄	C₄×Dic₁₃	1	1	yes	characteristic
292	C₄⋊C₄	C₄×Dic₁₃	1	2	yes	—
293	C₄⋊C₄	C₄×Dic₁₃	1	1	yes	characteristic
294	C₂²⋊C₄	C₄×Dic₁₃	1	2	yes	—
295	C₂²⋊C₄	C₄×Dic₁₃	1	2	yes	—
296	C₂²⋊C₄	C₄×Dic₁₃	1	2	yes	—
297	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
298	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₁₃	1	2	yes	—
300	C₄²	C₂×C₄×D₁₃	1	2	yes	—
301	C₄²	C₂²×Dic₁₃	1	2	yes	—
302	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
303	C₄²	C₂³×D₁₃	1	1	yes	characteristic
304	C₁₃	#17331	1	1	yes	characteristic, complemented, Sylow
305	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
306	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	#88155	1	4	yes	—
316	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
318	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
320	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
321	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	4	no	—
327	C₈	—	2	2	no	—
328	C₈	—	13	4	no	—
329	C₈	—	26	2	no	—
330	C₈	—	26	2	no	—
331	C₈	—	26	4	no	—
332	C₈	—	26	2	no	—
333	C₈	—	26	4	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	#11718762	1	2	yes	—
340	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	#11718762	1	2	yes	—
347	C₄	C₂×#88155	1	2	yes	—
348	C₄	D₁₃×#131	1	2	yes	—
349	C₄	#11719524	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×#11718762	1	1	yes	characteristic
353	C₂	#131×Dic₁₃	1	1	yes	characteristic
354	C₂	#13123264	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₈×C₁₃×#41	C₂
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×#41	D₁₃
hypocenter	C₁₃	#17331
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
3. #13113136	C₂	semidirect
4. #13113131	C₂	semidirect
5. #13113128	C₂	semidirect
6. #13113125	C₂	semidirect
7. #13113120	C₂	semidirect
8. #13113117	C₂	semidirect
9. C₈×#6099	C₂	semidirect
10. C₂×#11718046	C₂	semidirect
304. C₁₃	#17331	semidirect

Maximal subgroups

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

subgroup	index
#13123803	2
#13123480	2
#13113136	2
#13113131	2
#13113128	2
#13113125	2
#13113120	2
#13113117	2
C₈×#6099	2
C₂×#11718046	2
C₈×C₁₃×#41	2
#17331	13