Subgroups — #15861566

Order 3328.

938 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718492	C₂	1	1	yes	maximal, characteristic
3	C₄×#88141	C₂	1	1	yes	maximal, characteristic
4	#13114710	C₂	1	2	yes	maximal
5	#13114707	C₂	1	2	yes	maximal
6	C₄⋊C₄×#1055	C₂	1	1	yes	maximal, characteristic
7	#13114691	C₂	1	1	yes	maximal, characteristic
8	#13114688	C₂	1	2	yes	maximal
9	C₄×#88055	C₂	1	2	yes	maximal
10	C₄×#88055	C₂	1	1	yes	maximal, characteristic
11	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×#88141	C₄	1	2	yes	—
13	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×#88055	C₂²	1	1	yes	characteristic
19	C₂×#88055	C₂²	1	2	yes	—
20	C₂×#88055	C₂²	1	1	yes	characteristic
21	C₂×#88055	C₂²	1	1	yes	characteristic
22	C₂×#88055	C₂²	1	2	yes	—
23	C₂×#88055	C₂²	1	2	yes	—
24	C₂×#88055	C₂²	1	1	yes	characteristic
25	C₂×#88054	C₂²	1	2	yes	—
26	C₂×#88054	C₂²	1	2	yes	—
27	C₄⋊C₄×Dic₁₃	C₄	1	2	yes	—
28	C₄×#6101	C₂²	1	2	yes	—
29	C₄²×Dic₁₃	C₂²	1	2	yes	—
30	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
32	C₂×C₄×#1055	C₂²	1	2	yes	—
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	#11718492	C₄	1	8	yes	complemented
35	#11718340	C₄	1	8	yes	complemented
36	#11718310	C₄	1	4	yes	—
37	C₁₃×#336	C₂²	1	2	yes	—
38	#11718142	C₂²	1	2	yes	—
39	#11718116	C₂²	1	2	yes	—
40	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#6101	C₂×C₄	1	4	yes	—
49	C₂×#6101	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
54	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
56	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₂²×#1055	C₂³	1	1	yes	characteristic
58	C₂²×#1055	C₂³	1	2	yes	—
59	C₂²×#1055	C₂³	1	1	yes	characteristic
60	#88141	—	2	8	no	—
61	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
62	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
63	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₁₃	—	2	4	no	—
65	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
66	C₁₃×#95	C₂×C₄	1	2	yes	—
67	#88071	C₂×C₄	1	4	yes	—
68	#88055	—	2	2	no	—
69	#88055	—	2	2	no	—
70	#88055	—	2	4	no	—
71	#88055	—	2	2	no	—
72	#88055	C₂×C₄	1	8	yes	—
73	#88055	C₂×C₄	1	8	yes	—
74	#88055	Q₈	1	4	yes	complemented
75	#88054	C₂×C₄	1	8	yes	—
76	#88054	C₂×C₄	1	8	yes	—
77	C₄⋊#1055	C₂×C₄	1	4	yes	—
78	C₄×#1055	—	2	4	no	—
79	C₄×#1055	C₂×C₄	1	4	yes	—
80	#20520	—	13	1	no	maximal, Sylow
81	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₂₆	—	2	4	no	—
90	C₂×Dic₂₆	C₄²	1	4	yes	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	1	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₁₃	#41	1	2	yes	—
100	C₁₃×C₄⋊C₄	M₁₆	1	8	yes	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
105	C₄⋊Dic₁₃	C₄²	1	4	yes	—
106	#6101	—	2	8	no	—
107	#6101	C₄²	1	8	yes	—
108	C₄×Dic₁₃	—	2	8	no	—
109	C₄×Dic₁₃	—	2	4	no	—
110	C₄×Dic₁₃	—	2	1	no	—
111	C₄×Dic₁₃	—	2	1	no	—
112	C₄×Dic₁₃	—	4	4	no	—
113	C₄×Dic₁₃	—	4	2	no	—
114	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
115	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
116	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
117	C₄×Dic₁₃	#41	1	2	yes	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	2	1	no	—
120	C₂×#1055	—	2	1	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	C₂²×C₄	1	2	yes	—
123	C₂×#1055	C₂²×C₄	1	4	yes	—
124	C₂×#1055	C₂²×C₄	1	2	yes	—
125	C₂×#1055	C₂×Q₈	1	2	yes	—
126	C₂×#1055	#41	1	2	yes	—
127	C₂×#446	—	13	1	no	—
128	C₂×#446	—	13	1	no	—
129	C₄²×Q₈	—	13	1	no	—
130	#1771	—	13	2	no	—
131	#1771	—	13	2	no	—
132	C₈×C₄⋊C₄	—	13	1	no	—
133	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
134	#1767	—	13	1	no	—
135	#1767	—	13	2	no	—
136	C₄×#97	—	13	1	no	—
137	C₄×#97	—	13	2	no	—
138	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	4	no	—
142	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
143	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
145	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
146	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	2	4	no	—
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	2	4	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
160	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
161	C₂×Dic₁₃	C₄×Q₈	1	4	yes	—
162	Dic₂₆	—	2	8	no	—
163	C₈×C₁₃	—	2	2	no	—
164	C₈×C₁₃	—	2	4	no	—
165	C₈×C₁₃	C₄×Q₈	1	4	yes	complemented
166	#1055	—	2	4	no	—
167	#1055	—	2	2	no	—
168	#1055	C₄×Q₈	1	4	yes	complemented
169	C₂×C₄×Q₈	—	13	2	no	—
170	C₂×C₄×Q₈	—	13	1	no	—
171	#446	—	13	8	no	—
172	#446	—	13	8	no	—
173	C₂×C₄⋊C₈	—	13	2	no	—
174	C₂×C₄⋊C₈	—	13	2	no	—
175	C₂×C₄⋊C₈	—	13	1	no	—
176	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
177	C₂×#97	—	13	2	no	—
178	C₂×#97	—	13	2	no	—
179	C₂×#97	—	13	1	no	—
180	C₂×#97	—	13	2	no	—
181	C₂×#97	—	13	1	no	—
182	C₂×#97	—	13	1	no	—
183	C₂×#97	—	13	1	no	—
184	C₂×C₄×C₈	—	13	1	no	—
185	C₂×C₄×C₈	—	13	2	no	—
186	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
187	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
188	C₄×C₄⋊C₄	—	13	2	no	—
189	C₄×C₄⋊C₄	—	13	2	no	—
190	C₄×C₄⋊C₄	—	13	4	no	—
191	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
192	C₄³	—	13	2	no	—
193	C₄³	—	13	1	no	—
194	#336	—	13	2	no	—
195	#336	—	13	2	no	—
196	#336	C₂×D₁₃	1	2	yes	—
197	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
198	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
199	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
200	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
201	C₂²×C₁₃	#568	1	2	yes	—
202	C₄×C₁₃	—	2	2	no	—
203	C₄×C₁₃	—	2	8	no	—
204	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
205	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
206	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
207	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
208	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
209	Dic₁₃	—	2	4	no	—
210	Dic₁₃	—	2	4	no	—
211	Dic₁₃	—	2	4	no	—
212	Dic₁₃	—	4	2	no	—
213	C₂²×Q₈	—	13	1	no	—
214	C₂²×C₈	—	13	1	no	—
215	C₂²×C₈	—	13	1	no	—
216	C₂²×C₈	—	13	2	no	—
217	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
218	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
220	C₄×Q₈	—	13	4	no	—
221	C₄×Q₈	—	13	4	no	—
222	C₄×Q₈	—	26	8	no	—
223	C₂×C₄⋊C₄	—	13	2	no	—
224	C₂×C₄⋊C₄	—	13	1	no	—
225	C₂×C₄⋊C₄	—	13	4	no	—
226	C₂×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
227	C₂×C₄²	—	13	2	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	4	no	—
230	C₂×C₄²	—	13	2	no	—
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	4	no	—
233	C₂×C₄²	—	13	2	no	—
234	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₄⋊C₈	—	13	8	no	—
237	C₄⋊C₈	—	13	8	no	—
238	C₄⋊C₈	—	13	4	no	—
239	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
240	#97	—	13	8	no	—
241	#97	—	13	8	no	—
242	#97	—	13	4	no	—
243	#97	—	26	2	no	—
244	#97	—	26	4	no	—
245	#97	—	26	2	no	—
246	#97	—	26	2	no	—
247	C₄×C₈	—	2	4	no	—
248	C₄×C₈	—	13	4	no	—
249	C₄×C₈	—	26	4	no	—
250	C₄×C₈	C₄×D₁₃	1	4	yes	—
251	#95	—	13	4	no	—
252	#95	C₄×D₁₃	1	2	yes	—
253	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
254	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
255	C₂×C₁₃	#2875	1	1	yes	characteristic
256	C₂×C₁₃	Q₈×M₁₆	1	2	yes	—
257	C₂×C₁₃	#2970	1	2	yes	—
258	C₂×Q₈	—	13	4	no	—
259	C₂×Q₈	—	26	4	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	2	no	—
262	C₂²×C₄	—	13	2	no	—
263	C₂²×C₄	—	13	2	no	—
264	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
266	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	13	2	no	—
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	—	13	2	no	—
275	C₂×C₈	—	13	2	no	—
276	C₂×C₈	—	13	4	no	—
277	C₂×C₈	—	26	1	no	—
278	C₂×C₈	—	26	2	no	—
279	C₂×C₈	—	26	2	no	—
280	C₂×C₈	—	26	1	no	—
281	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
282	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
283	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
284	C₂×C₈	#6129	1	2	yes	—
285	C₂×C₈	Q₈×D₁₃	1	2	yes	—
286	C₄⋊C₄	—	13	8	no	—
287	C₄⋊C₄	—	13	4	no	—
288	C₄⋊C₄	—	26	8	no	—
289	C₄⋊C₄	#6094	1	8	yes	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	13	4	no	—
293	C₄²	—	13	4	no	—
294	C₄²	—	13	2	no	—
295	C₄²	—	13	2	no	—
296	C₄²	—	26	8	no	—
297	C₄²	—	26	4	no	—
298	C₄²	—	26	1	no	—
299	C₄²	—	26	1	no	—
300	C₄²	—	52	4	no	—
301	C₄²	—	52	2	no	—
302	C₄²	C₂×C₄×D₁₃	1	2	yes	—
303	C₄²	C₂×C₄×D₁₃	1	2	yes	—
304	C₁₃	#20520	1	1	yes	characteristic, complemented, Sylow
305	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
306	Q₈	—	26	8	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	13	4	no	—
311	C₂×C₄	—	13	4	no	—
312	C₂×C₄	—	13	4	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	—	26	4	no	—
315	C₂×C₄	—	26	2	no	—
316	C₂×C₄	—	26	4	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	26	2	no	—
319	C₂×C₄	—	26	2	no	—
320	C₂×C₄	C₄²×D₁₃	1	2	yes	—
321	C₂×C₄	C₄²×D₁₃	1	2	yes	—
322	C₂×C₄	C₂×#6094	1	8	yes	—
323	C₂×C₄	C₂×#6094	1	4	yes	—
324	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
326	C₂×C₄	C₂×#6129	1	1	yes	characteristic
327	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
328	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
329	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	4	no	—
332	C₈	—	13	4	no	—
333	C₈	—	26	4	no	—
334	C₈	—	26	2	no	—
335	C₈	#88141	1	4	yes	complemented
336	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
337	C₂²	C₄×#6129	1	1	yes	characteristic
338	C₂²	C₄×Q₈×D₁₃	1	1	yes	characteristic
339	C₂²	C₂²×#6094	1	2	yes	—
340	C₂²	#11719455	1	2	yes	—
341	C₄	—	2	8	no	—
342	C₄	—	2	2	no	—
343	C₄	—	26	4	no	—
344	C₄	—	26	4	no	—
345	C₄	—	26	4	no	—
346	C₄	—	52	2	no	—
347	C₄	C₄×#6094	1	4	yes	—
348	C₄	C₂×#88141	1	1	yes	characteristic
349	C₄	#11719227	1	1	yes	characteristic
350	C₄	C₄×#6129	1	1	yes	characteristic
351	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
352	C₂	C₄×#88141	1	1	yes	characteristic
353	C₂	C₂×C₄×#6094	1	1	yes	characteristic
354	C₂	#13123263	1	1	yes	characteristic
355	C₂	Q₈×#6094	1	2	yes	—
356	C₂	#13123844	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#20520
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
34. #11718492	C₄	semidirect
35. #11718340	C₄	semidirect
74. #88055	Q₈	semidirect
165. C₈×C₁₃	C₄×Q₈	semidirect
168. #1055	C₄×Q₈	semidirect
304. C₁₃	#20520	semidirect
335. C₈	#88141	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718340	2
C₂×#11718492	2
C₄×#88141	2
#13114710	2
#13114707	2
C₄⋊C₄×#1055	2
#13114691	2
#13114688	2
C₄×#88055	2
C₈×C₁₃×C₄⋊C₄	2
#20520	13