Subgroups — #15867653

Order 3328.

746 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	#13115772	C₂	1	2	yes	maximal
3	#13115554	C₂	1	2	yes	maximal
4	#13115400	C₂	1	4	yes	maximal
5	#13114690	C₂	1	2	yes	maximal
6	#13114689	C₂	1	1	yes	maximal, characteristic
7	#13114584	C₂	1	2	yes	maximal
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88054	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
19	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
20	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	C₄×#6101	C₄	1	2	yes	—
22	C₄×#6101	C₄	1	2	yes	—
23	C₄×#6101	C₂²	1	2	yes	—
24	C₄²×Dic₁₃	C₂²	1	2	yes	—
25	C₂×C₄⋊#1055	C₂²	1	2	yes	—
26	C₂×#88043	C₂²	1	2	yes	—
27	#11718142	C₂²	1	4	yes	—
28	#11718142	C₂²	1	4	yes	—
29	#11718142	C₂²	1	4	yes	—
30	#11718116	C₂²	1	2	yes	—
31	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×#6101	C₂×C₄	1	2	yes	—
45	C₂×#6101	C₂×C₄	1	2	yes	—
46	C₂×#6101	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂²×#1055	C₂³	1	4	yes	—
54	C₄×Dic₂₆	—	2	4	no	—
55	C₄×Dic₂₆	—	2	4	no	—
56	C₄×Dic₂₆	—	2	4	no	—
57	C₁₃×#97	—	2	4	no	—
58	C₁₃×#97	—	2	2	no	—
59	C₁₃×#97	Q₈	1	4	yes	complemented
60	C₄×C₈×C₁₃	—	2	4	no	—
61	#88071	C₂×C₄	1	2	yes	—
62	#88071	C₂×C₄	1	2	yes	—
63	#88071	C₂×C₄	1	2	yes	—
64	#88054	—	2	8	no	—
65	C₄⋊#1055	—	2	4	no	—
66	#88043	—	2	4	no	—
67	#21345	—	13	1	no	maximal, Sylow
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₆	—	2	2	no	—
80	C₂×Dic₂₆	—	2	2	no	—
81	C₂×Dic₂₆	M₁₆	1	4	yes	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
87	C₂×C₈×C₁₃	#41	1	4	yes	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	4	1	no	—
92	C₄²×C₁₃	—	4	1	no	—
93	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
94	C₄²×C₁₃	#41	1	4	yes	—
95	C₄²×C₁₃	#41	1	2	yes	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	M₁₆	1	4	yes	—
99	#6101	—	2	4	no	—
100	#6101	—	2	4	no	—
101	#6101	—	2	4	no	—
102	C₄×Dic₁₃	—	2	4	no	—
103	C₄×Dic₁₃	—	2	4	no	—
104	C₄×Dic₁₃	—	2	4	no	—
105	C₄×Dic₁₃	—	2	4	no	—
106	C₄×Dic₁₃	—	4	2	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₂×#1055	—	2	4	no	—
109	C₂×#1055	—	2	4	no	—
110	C₂×#1055	—	2	4	no	—
111	C₄²×Q₈	—	13	1	no	—
112	#1984	—	13	2	no	—
113	#1939	—	13	2	no	—
114	#1917	—	13	4	no	—
115	#1767	—	13	2	no	—
116	#1767	—	13	1	no	—
117	#1745	—	13	2	no	—
118	C₄×#97	D₁₃	1	1	yes	characteristic
119	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	#117	1	2	yes	—
133	C₂×C₄×C₁₃	#117	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	2	2	no	—
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	—	2	2	no	—
152	C₂×Dic₁₃	C₂×M₁₆	1	4	yes	—
153	C₂×Dic₁₃	C₂×M₁₆	1	4	yes	—
154	Dic₂₆	—	2	8	no	—
155	C₈×C₁₃	—	2	4	no	—
156	C₈×C₁₃	—	2	4	no	—
157	#1055	—	4	4	no	—
158	C₂×C₄×Q₈	—	13	1	no	—
159	C₂×C₄×Q₈	—	13	1	no	—
160	C₂×C₄×Q₈	—	13	1	no	—
161	C₂×C₄⋊C₈	—	13	4	no	—
162	C₂×C₄⋊C₈	—	13	2	no	—
163	C₂×#97	—	13	2	no	—
164	C₂×#97	C₂×D₁₃	1	2	yes	—
165	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
166	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
168	C₄×C₄⋊C₄	—	13	2	no	—
169	C₄×C₄⋊C₄	—	13	2	no	—
170	C₄×C₄⋊C₄	—	13	1	no	—
171	C₄×C₄⋊C₄	—	13	1	no	—
172	C₄×C₄⋊C₄	—	13	2	no	—
173	C₄×C₄⋊C₄	—	13	1	no	—
174	C₄³	—	13	2	no	—
175	C₄³	C₂×D₁₃	1	1	yes	characteristic
176	#336	—	13	4	no	—
177	#336	—	13	4	no	—
178	#336	—	13	4	no	—
179	#336	—	13	2	no	—
180	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
181	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
182	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
183	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
184	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
185	C₂²×C₁₃	#568	1	1	yes	characteristic
186	C₂²×C₁₃	#568	1	1	yes	characteristic
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	4	1	no	—
191	C₄×C₁₃	—	4	1	no	—
192	C₄×C₁₃	#432	1	4	yes	—
193	C₄×C₁₃	#529	1	2	yes	—
194	C₄×C₁₃	#533	1	2	yes	—
195	Dic₁₃	—	2	4	no	—
196	Dic₁₃	—	2	4	no	—
197	Dic₁₃	—	4	2	no	—
198	Dic₁₃	—	4	2	no	—
199	C₂²×Q₈	—	13	1	no	—
200	C₂²×C₈	—	13	4	no	—
201	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
202	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
203	C₄×Q₈	—	26	4	no	—
204	C₄×Q₈	—	26	4	no	—
205	C₄×Q₈	—	26	4	no	—
206	C₂×C₄⋊C₄	—	13	1	no	—
207	C₂×C₄⋊C₄	—	13	2	no	—
208	C₂×C₄⋊C₄	—	13	2	no	—
209	C₂×C₄⋊C₄	—	13	2	no	—
210	C₂×C₄⋊C₄	—	13	1	no	—
211	C₂×C₄⋊C₄	—	13	1	no	—
212	C₂×C₄²	—	13	2	no	—
213	C₂×C₄²	—	13	2	no	—
214	C₂×C₄²	—	13	2	no	—
215	C₂×C₄²	—	13	2	no	—
216	C₂×C₄²	—	13	2	no	—
217	C₂×C₄²	—	13	2	no	—
218	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
220	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
221	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
225	C₄⋊C₈	—	26	8	no	—
226	C₄⋊C₈	—	26	4	no	—
227	#97	—	2	4	no	—
228	#97	—	2	2	no	—
229	#97	—	26	4	no	—
230	#97	Dic₂₆	1	4	yes	complemented
231	C₄×C₈	—	2	4	no	—
232	#95	—	13	2	no	—
233	#95	—	13	2	no	—
234	#95	—	13	2	no	—
235	C₂×C₁₃	#2295	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
237	C₂×C₁₃	#2923	1	1	yes	characteristic
238	C₂×C₁₃	Q₈×M₁₆	1	1	yes	characteristic
239	C₂×C₁₃	Q₈×M₁₆	1	1	yes	characteristic
240	C₂×C₁₃	#2970	1	1	yes	characteristic
241	C₂×C₁₃	#2971	1	1	yes	characteristic
242	C₂×Q₈	—	13	4	no	—
243	C₂×Q₈	—	26	2	no	—
244	C₂×Q₈	—	26	2	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	—	26	4	no	—
262	C₂×C₈	—	26	4	no	—
263	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
264	C₂×C₈	#6127	1	4	yes	—
265	C₄⋊C₄	—	13	4	no	—
266	C₄⋊C₄	—	26	4	no	—
267	C₄⋊C₄	—	26	4	no	—
268	C₄⋊C₄	—	26	4	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	26	4	no	—
277	C₄²	—	26	4	no	—
278	C₄²	—	26	4	no	—
279	C₄²	—	26	4	no	—
280	C₄²	—	52	2	no	—
281	C₄²	—	52	2	no	—
282	C₄²	C₂×Dic₂₆	1	2	yes	—
283	C₄²	#6127	1	2	yes	—
284	C₄²	#6127	1	4	yes	—
285	C₁₃	#21345	1	1	yes	characteristic, complemented, Sylow
286	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
287	Q₈	—	26	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	13	4	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	26	2	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	26	2	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	2	no	—
312	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
313	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
314	C₂×C₄	#88126	1	2	yes	—
315	C₂×C₄	#88126	1	2	yes	—
316	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
317	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
318	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
319	C₂×C₄	C₂×#6127	1	2	yes	—
320	C₂×C₄	C₂×#6127	1	2	yes	—
321	C₂×C₄	C₂×#6127	1	1	yes	characteristic
322	C₈	—	2	4	no	—
323	C₈	—	2	4	no	—
324	C₈	—	52	4	no	—
325	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
326	C₂²	C₂×#88126	1	1	yes	characteristic
327	C₂²	C₄×#6127	1	1	yes	characteristic
328	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
329	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
330	C₂²	#11719450	1	1	yes	characteristic
331	C₂²	#11719450	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	26	4	no	—
338	C₄	—	26	4	no	—
339	C₄	—	52	2	no	—
340	C₄	—	52	2	no	—
341	C₄	#11718366	1	4	yes	—
342	C₄	#11719175	1	2	yes	—
343	C₄	#11719184	1	2	yes	—
344	C₂	#13117313	1	1	yes	characteristic
345	C₂	C₂×#11718366	1	1	yes	characteristic
346	C₂	#13123485	1	1	yes	characteristic
347	C₂	M₁₆×Dic₂₆	1	1	yes	characteristic
348	C₂	M₁₆×Dic₂₆	1	1	yes	characteristic
349	C₂	#13123851	1	1	yes	characteristic
350	C₂	#13123854	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×#97	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#21345
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
59. C₁₃×#97	Q₈	semidirect
230. #97	Dic₂₆	semidirect
285. C₁₃	#21345	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₂₆	2
#13115772	2
#13115554	2
#13115400	2
#13114690	2
#13114689	2
#13114584	2
C₄×C₁₃×#97	2
#21345	13