Subgroups — #15755614

Order 3328.

568 conjugacy classes of subgroups.

The subgroup lattice has 380 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	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718076	C₂	1	2	yes	maximal
4	#13112675	C₂	1	2	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
14	C₄×#6101	—	2	1	no	—
15	C₄×#6101	—	2	1	no	—
16	C₄×#6101	—	2	1	no	—
17	C₄²×Dic₁₃	—	2	1	no	—
18	C₂×C₄⋊#1055	—	2	1	no	—
19	C₂×C₄×#1055	C₂²	1	2	yes	—
20	C₁₃×#336	—	2	1	no	—
21	C₁₃×#336	—	2	1	no	—
22	#11718116	—	2	1	no	—
23	#11718076	—	2	4	no	—
24	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#6101	—	2	1	no	—
37	C₂×#6101	—	2	1	no	—
38	C₂×#6101	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂×C₄×Dic₁₃	—	2	1	no	—
41	C₂×C₄×Dic₁₃	—	2	1	no	—
42	C₂×C₄×Dic₁₃	—	2	1	no	—
43	C₂×C₄×Dic₁₃	—	2	1	no	—
44	C₂×C₄×Dic₁₃	—	2	1	no	—
45	C₂²×#1055	—	2	2	no	—
46	C₄×Dic₂₆	—	2	2	no	—
47	C₄×Dic₂₆	—	2	2	no	—
48	C₄×Dic₂₆	—	2	1	no	—
49	C₄×Dic₂₆	—	2	1	no	—
50	C₄×Dic₂₆	—	2	2	no	—
51	C₄×Dic₂₆	C₈	1	4	yes	complemented
52	C₄×Dic₂₆	D₄	1	2	yes	—
53	C₄×Dic₂₆	Q₈	1	2	yes	—
54	C₁₃×C₄⋊C₈	—	2	2	no	—
55	C₁₃×C₄⋊C₈	—	2	2	no	—
56	#88071	—	2	1	no	—
57	#88071	—	2	1	no	—
58	#88071	—	2	1	no	—
59	C₄⋊#1055	—	2	4	no	—
60	C₄×#1055	—	2	2	no	—
61	C₄×#1055	—	2	2	no	—
62	#8566	—	13	1	no	maximal, Sylow
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	1	no	—
71	C₂²×Dic₁₃	—	2	1	no	—
72	C₂²×Dic₁₃	—	2	1	no	—
73	C₂²×Dic₁₃	—	2	1	no	—
74	C₂×Dic₂₆	—	2	1	no	—
75	C₂×Dic₂₆	—	2	1	no	—
76	C₂×Dic₂₆	—	2	1	no	—
77	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
78	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
79	C₂×Dic₂₆	M₁₆	1	2	yes	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	4	1	no	—
89	C₄²×C₁₃	C₂×C₈	1	2	yes	—
90	C₄²×C₁₃	M₁₆	1	2	yes	—
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	#41	1	1	yes	characteristic
95	C₄⋊Dic₁₃	—	2	1	no	—
96	C₄⋊Dic₁₃	—	2	1	no	—
97	C₄⋊Dic₁₃	—	2	1	no	—
98	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
99	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₁₃	M₁₆	1	2	yes	—
101	#6101	—	2	4	no	—
102	#6101	—	2	4	no	—
103	#6101	—	2	2	no	—
104	#6101	—	2	2	no	—
105	C₄×Dic₁₃	—	2	4	no	—
106	C₄×Dic₁₃	—	2	4	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	4	2	no	—
110	C₄×Dic₁₃	—	4	2	no	—
111	C₄×Dic₁₃	—	4	2	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂×#1055	—	4	2	no	—
114	C₂×#1055	—	4	2	no	—
115	C₄²×Q₈	—	13	1	no	—
116	#1831	D₁₃	1	1	yes	characteristic
117	C₄×C₄⋊C₈	—	13	1	no	—
118	C₂×Q₈⋊C₈	—	13	2	no	—
119	#1279	—	13	2	no	—
120	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
134	C₂×C₄×C₁₃	#104	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#135	1	2	yes	—
146	C₂×C₄×C₁₃	#137	1	2	yes	—
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	—	2	2	no	—
153	C₂×Dic₁₃	—	4	1	no	—
154	C₂×Dic₁₃	—	4	1	no	—
155	C₂×Dic₁₃	—	4	1	no	—
156	C₂×Dic₁₃	—	4	1	no	—
157	C₂×Dic₁₃	—	4	1	no	—
158	C₂×Dic₁₃	—	4	1	no	—
159	Dic₂₆	—	2	2	no	—
160	Dic₂₆	—	2	2	no	—
161	Dic₂₆	—	2	2	no	—
162	Dic₂₆	C₄⋊C₈	1	4	yes	complemented
163	C₈×C₁₃	—	8	2	no	—
164	#1055	—	4	4	no	—
165	C₂×C₄×Q₈	—	13	1	no	—
166	C₂×C₄×Q₈	—	13	1	no	—
167	C₂×C₄×Q₈	—	13	1	no	—
168	C₂×C₄⋊C₈	—	26	1	no	—
169	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
170	C₂×C₄×C₈	—	13	2	no	—
171	C₄×C₄⋊C₄	—	13	1	no	—
172	C₄×C₄⋊C₄	—	13	1	no	—
173	C₄×C₄⋊C₄	—	13	1	no	—
174	C₄×C₄⋊C₄	—	26	1	no	—
175	C₄×C₄⋊C₄	—	26	1	no	—
176	C₄×C₄⋊C₄	—	26	1	no	—
177	C₄³	—	26	1	no	—
178	C₄³	C₂×D₁₃	1	1	yes	characteristic
179	#336	—	2	1	no	—
180	#336	—	2	1	no	—
181	#336	—	26	1	no	—
182	Q₈⋊C₈	—	26	4	no	—
183	C₂²×C₁₃	#380	1	1	yes	characteristic
184	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
185	C₂²×C₁₃	#407	1	1	yes	characteristic
186	C₂²×C₁₃	#416	1	1	yes	characteristic
187	C₂²×C₁₃	#419	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
189	C₂²×C₁₃	#421	1	1	yes	characteristic
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	2	2	no	—
195	C₄×C₁₃	—	4	1	no	—
196	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
197	C₄×C₁₃	#423	1	1	yes	characteristic
198	C₄×C₁₃	#431	1	1	yes	characteristic
199	C₄×C₁₃	#432	1	1	yes	characteristic
200	C₄×C₁₃	#464	1	2	yes	—
201	C₄×C₁₃	#479	1	2	yes	—
202	Dic₁₃	—	2	4	no	—
203	Dic₁₃	—	4	2	no	—
204	Dic₁₃	—	4	2	no	—
205	Dic₁₃	—	4	2	no	—
206	C₂²×Q₈	—	13	1	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	26	2	no	—
209	C₄×Q₈	—	13	4	no	—
210	C₄×Q₈	—	13	2	no	—
211	C₄×Q₈	—	13	2	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	—	26	2	no	—
214	C₄×Q₈	—	26	2	no	—
215	C₄×Q₈	—	26	1	no	—
216	C₄×Q₈	—	26	1	no	—
217	C₂×C₄⋊C₄	—	13	1	no	—
218	C₂×C₄⋊C₄	—	13	1	no	—
219	C₂×C₄⋊C₄	—	13	1	no	—
220	C₂×C₄⋊C₄	—	26	1	no	—
221	C₂×C₄⋊C₄	—	26	1	no	—
222	C₂×C₄⋊C₄	—	26	1	no	—
223	C₂×C₄²	—	26	1	no	—
224	C₂×C₄²	—	26	1	no	—
225	C₂×C₄²	—	26	1	no	—
226	C₂×C₄²	—	26	1	no	—
227	C₂×C₄²	—	26	1	no	—
228	C₂×C₄²	—	26	1	no	—
229	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
230	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
231	C₂×C₄²	D₅₂	1	1	yes	characteristic
232	C₂×C₄²	D₅₂	1	1	yes	characteristic
233	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	26	4	no	—
239	C₄×C₈	—	26	2	no	—
240	C₄×C₈	—	26	2	no	—
241	#95	—	26	1	no	—
242	#95	—	26	1	no	—
243	#95	—	26	1	no	—
244	C₂×C₁₃	#1476	1	1	yes	characteristic
245	C₂×C₁₃	#1479	1	1	yes	characteristic
246	C₂×C₁₃	#1488	1	1	yes	characteristic
247	C₂×C₁₃	#1491	1	1	yes	characteristic
248	C₂×C₁₃	#1797	1	1	yes	characteristic
249	C₂×C₁₃	#1805	1	1	yes	characteristic
250	C₂×C₁₃	#1806	1	1	yes	characteristic
251	C₂×Q₈	—	13	2	no	—
252	C₂×Q₈	—	13	2	no	—
253	C₂×Q₈	—	13	2	no	—
254	C₂×Q₈	—	26	1	no	—
255	C₂×Q₈	—	26	1	no	—
256	C₂×Q₈	—	26	1	no	—
257	C₂²×C₄	—	26	1	no	—
258	C₂²×C₄	—	26	1	no	—
259	C₂²×C₄	—	26	1	no	—
260	C₂²×C₄	—	26	1	no	—
261	C₂²×C₄	#6103	1	1	yes	characteristic
262	C₂²×C₄	#6103	1	1	yes	characteristic
263	C₂²×C₄	#6103	1	1	yes	characteristic
264	C₂²×C₄	#6103	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	26	4	no	—
272	C₂×C₈	—	52	2	no	—
273	C₂×C₈	—	52	2	no	—
274	C₄⋊C₄	—	13	2	no	—
275	C₄⋊C₄	—	13	2	no	—
276	C₄⋊C₄	—	13	2	no	—
277	C₄⋊C₄	—	26	4	no	—
278	C₄⋊C₄	—	26	1	no	—
279	C₄⋊C₄	—	26	2	no	—
280	C₄⋊C₄	—	26	2	no	—
281	C₄⋊C₄	—	26	1	no	—
282	C₄⋊C₄	—	26	1	no	—
283	C₄⋊C₄	—	26	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	26	4	no	—
291	C₄²	—	26	2	no	—
292	C₄²	—	26	2	no	—
293	C₄²	—	26	4	no	—
294	C₄²	—	52	2	no	—
295	C₄²	—	52	2	no	—
296	C₄²	—	52	2	no	—
297	C₄²	C₈×D₁₃	1	2	yes	—
298	C₄²	#6094	1	2	yes	—
299	C₄²	D₄×D₁₃	1	1	yes	characteristic
300	C₄²	#6129	1	1	yes	characteristic
301	C₄²	Q₈×D₁₃	1	1	yes	characteristic
302	C₄²	#6131	1	1	yes	characteristic
303	C₁₃	#8566	1	1	yes	characteristic, complemented, Sylow
304	C₂³	C₂×#6103	1	1	yes	characteristic
305	Q₈	—	13	4	no	—
306	Q₈	—	26	2	no	—
307	Q₈	—	26	2	no	—
308	Q₈	—	26	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	26	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	26	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	52	1	no	—
327	C₂×C₄	—	52	1	no	—
328	C₂×C₄	—	52	1	no	—
329	C₂×C₄	—	52	1	no	—
330	C₂×C₄	—	52	1	no	—
331	C₂×C₄	—	52	1	no	—
332	C₂×C₄	#88060	1	2	yes	—
333	C₂×C₄	#88060	1	2	yes	—
334	C₂×C₄	#88065	1	2	yes	—
335	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
336	C₂×C₄	#88146	1	1	yes	characteristic
337	C₂×C₄	#88149	1	1	yes	characteristic
338	C₂×C₄	#88151	1	1	yes	characteristic
339	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
340	C₂×C₄	C₂×#6094	1	1	yes	characteristic
341	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
342	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
343	C₂×C₄	#88192	1	1	yes	characteristic
344	C₂×C₄	#88200	1	1	yes	characteristic
345	C₂×C₄	#88204	1	2	yes	—
346	C₂×C₄	#88205	1	2	yes	—
347	C₈	—	8	2	no	—
348	C₈	—	52	4	no	—
349	C₂²	#11718648	1	1	yes	characteristic
350	C₂²	#11718663	1	1	yes	characteristic
351	C₂²	#11718665	1	1	yes	characteristic
352	C₂²	C₂×#88060	1	1	yes	characteristic
353	C₂²	#11718787	1	1	yes	characteristic
354	C₂²	C₂×#88065	1	1	yes	characteristic
355	C₂²	#11718800	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	26	4	no	—
363	C₄	—	52	2	no	—
364	C₄	—	52	2	no	—
365	C₄	—	52	2	no	—
366	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
367	C₄	#11718494	1	1	yes	characteristic
368	C₄	#11718496	1	1	yes	characteristic
369	C₄	#11718498	1	1	yes	characteristic
370	C₄	#11718715	1	2	yes	—
371	C₄	#11718724	1	2	yes	—
372	C₂	#13113216	1	1	yes	characteristic
373	C₂	#13113230	1	1	yes	characteristic
374	C₂	#13113268	1	1	yes	characteristic
375	C₂	#13113282	1	1	yes	characteristic
376	C₂	#13114813	1	1	yes	characteristic
377	C₂	#13114848	1	1	yes	characteristic
378	C₂	#13114853	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8566
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
51. C₄×Dic₂₆	C₈	semidirect
162. Dic₂₆	C₄⋊C₈	semidirect
303. C₁₃	#8566	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
C₄×C₄⋊#1055	2
C₂×#11718076	2
#13112675	2
C₁₃×#1831	2
#8566	13