Subgroups — C₈×#88129 #15870784

Order 3328.

974 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₄×#88129	C₂	1	1	yes	maximal, characteristic
4	#13115772	C₂	1	1	yes	maximal, characteristic, complemented
5	#13115630	C₂	1	1	yes	maximal, characteristic
6	#13115442	C₂	1	4	yes	maximal, complemented
7	#13114953	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#6103	C₂	1	4	yes	maximal, complemented
9	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
11	C₂×#88129	C₄	1	1	yes	characteristic
12	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₄³×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88060	C₂²	1	2	yes	—
20	C₂×#88060	C₂²	1	4	yes	—
21	C₂×#88054	C₂²	1	2	yes	—
22	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
23	#11718601	C₄	1	4	yes	—
24	#11718600	C₄	1	1	yes	characteristic
25	C₄×#6103	C₂²	1	4	yes	—
26	#11718597	C₄	1	1	yes	characteristic
27	#11718590	C₄	1	1	yes	characteristic
28	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
30	C₈×D₅₂	—	2	4	no	—
31	C₈×Dic₂₆	—	2	4	no	—
32	#11718142	C₂²	1	4	yes	—
33	#11718116	C₂²	1	4	yes	—
34	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₁₃	C₂³	1	2	yes	—
36	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×#6103	C₂×C₄	1	2	yes	—
46	C₂×#6103	C₂×C₄	1	4	yes	—
47	C₂×#6103	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×#6101	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
52	C₂²×#1055	C₂³	1	2	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₂×C₈×D₁₃	—	2	4	no	—
55	C₂×C₈×D₁₃	—	2	8	no	—
56	#88129	C₈	1	8	yes	direct product factor
57	C₄×D₅₂	—	2	2	no	—
58	C₄×D₅₂	—	2	2	no	—
59	C₄×Dic₂₆	—	2	2	no	—
60	C₄×Dic₂₆	—	2	2	no	—
61	C₄×C₈×C₁₃	—	2	4	no	—
62	C₄×C₈×C₁₃	—	2	4	no	—
63	C₄×C₈×C₁₃	—	2	2	no	—
64	C₄×C₈×C₁₃	D₄	1	4	yes	—
65	#88071	C₂×C₄	1	4	yes	—
66	#88071	C₂×C₄	1	4	yes	—
67	#88060	—	2	4	no	—
68	#88060	—	2	8	no	—
69	#88054	—	2	4	no	—
70	C₈×Dic₁₃	—	2	4	no	—
71	C₄⋊#1055	—	2	2	no	—
72	C₄⋊#1055	—	2	2	no	—
73	C₈×#124	—	13	1	no	maximal, Sylow
74	C₂³×D₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₂×D₅₂	—	2	2	no	—
84	C₂×D₅₂	—	2	2	no	—
85	C₂×D₅₂	C₂×C₈	1	4	yes	—
86	C₂×C₄×D₁₃	—	2	2	no	—
87	C₂×C₄×D₁₃	—	2	2	no	—
88	C₂×C₄×D₁₃	—	2	4	no	—
89	C₂×C₄×D₁₃	—	2	4	no	—
90	C₂×Dic₂₆	—	2	2	no	—
91	C₂×Dic₂₆	—	2	2	no	—
92	C₂×Dic₂₆	C₂×C₈	1	4	yes	complemented
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃	#41	1	4	yes	—
99	C₂×C₈×C₁₃	#41	1	4	yes	—
100	C₄²×C₁₃	—	2	4	no	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	C₂×C₈	1	4	yes	complemented
105	C₄²×C₁₃	C₂×D₄	1	2	yes	—
106	C₄²×C₁₃	#41	1	2	yes	—
107	#6103	—	2	4	no	—
108	#6103	—	2	8	no	—
109	#6103	C₂×C₈	1	16	yes	complemented
110	C₄⋊Dic₁₃	—	2	2	no	—
111	C₄⋊Dic₁₃	—	2	2	no	—
112	#6101	—	2	4	no	—
113	C₄×Dic₁₃	—	2	2	no	—
114	C₄×Dic₁₃	—	2	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	4	no	—
118	C₂×#1055	—	2	4	no	—
119	C₈×D₁₃	—	4	8	no	—
120	C₂×C₈×Q₈	—	13	1	no	—
121	C₂×C₈×D₄	—	13	1	no	—
122	C₄×#124	—	13	1	no	—
123	#1984	—	13	1	no	—
124	#1954	—	13	1	no	—
125	#1923	—	13	4	no	—
126	#1831	—	13	1	no	—
127	C₈×C₂²⋊C₄	—	13	4	no	—
128	C₄²×C₈	D₁₃	1	1	yes	characteristic, complemented
129	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂²×D₁₃	—	2	2	no	—
131	C₂²×D₁₃	—	2	4	no	—
132	C₂²×D₁₃	—	2	4	no	—
133	C₂²×D₁₃	—	2	2	no	—
134	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
143	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
144	C₂×C₄×C₁₃	#131	1	4	yes	—
145	C₂×C₄×C₁₃	#131	1	2	yes	—
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	—	2	2	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	2	2	no	—
154	C₂×Dic₁₃	—	2	4	no	—
155	C₂×Dic₁₃	—	2	4	no	—
156	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
157	D₅₂	—	2	8	no	—
158	C₄×D₁₃	—	4	2	no	—
159	C₄×D₁₃	—	4	4	no	—
160	C₄×D₁₃	—	4	2	no	—
161	Dic₂₆	—	2	8	no	—
162	C₈×C₁₃	—	2	4	no	—
163	C₈×C₁₃	—	2	2	no	—
164	C₈×C₁₃	#124	1	4	yes	complemented
165	#1055	—	4	2	no	—
166	#1055	—	4	2	no	—
167	C₂³×C₈	—	13	2	no	—
168	C₂×#124	—	13	1	no	—
169	C₂×C₄×Q₈	—	13	1	no	—
170	C₂×C₄×D₄	—	13	1	no	—
171	C₈×Q₈	—	26	4	no	—
172	C₈×D₄	—	26	4	no	—
173	C₂×C₄⋊C₈	—	13	2	no	—
174	C₂×C₄⋊C₈	—	13	1	no	—
175	C₂×C₄⋊C₈	—	13	1	no	—
176	C₂×C₂²⋊C₈	—	13	2	no	—
177	C₂×C₂²⋊C₈	—	13	4	no	—
178	C₂×C₄×C₈	—	13	2	no	—
179	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
180	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
182	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
183	#391	—	13	1	no	—
184	#390	—	13	1	no	—
185	#388	—	13	4	no	—
186	#382	—	13	1	no	—
187	C₄×C₂²⋊C₄	—	13	4	no	—
188	C₄³	C₂×D₁₃	1	1	yes	characteristic
189	#336	—	13	4	no	—
190	#336	—	13	4	no	—
191	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₁₃	C₄×#41	1	2	yes	—
193	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
194	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×#131	1	2	yes	—
196	C₂×D₁₃	—	2	4	no	—
197	C₂×D₁₃	—	2	8	no	—
198	C₂×D₁₃	—	4	2	no	—
199	C₂×D₁₃	—	4	4	no	—
200	C₂×D₁₃	—	4	2	no	—
201	C₄×C₁₃	—	2	2	no	—
202	C₄×C₁₃	—	2	4	no	—
203	C₄×C₁₃	—	2	4	no	—
204	C₄×C₁₃	C₈×D₄	1	4	yes	—
205	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
206	C₄×C₁₃	#529	1	2	yes	—
207	C₄×C₁₃	#532	1	1	yes	characteristic
208	Dic₁₃	—	2	4	no	—
209	Dic₁₃	—	4	2	no	—
210	Dic₁₃	—	4	2	no	—
211	C₂²×Q₈	—	13	1	no	—
212	C₂²×D₄	—	13	1	no	—
213	C₂³×C₄	—	13	2	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	26	4	no	—
217	C₂²×C₈	—	26	8	no	—
218	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
220	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
221	#124	—	13	8	no	—
222	C₄×Q₈	—	26	2	no	—
223	C₄×Q₈	—	26	2	no	—
224	C₄×D₄	—	26	2	no	—
225	C₄×D₄	—	26	2	no	—
226	C₂×C₄⋊C₄	—	13	1	no	—
227	C₂×C₄⋊C₄	—	13	2	no	—
228	C₂×C₄⋊C₄	—	13	1	no	—
229	C₂×C₂²⋊C₄	—	13	4	no	—
230	C₂×C₂²⋊C₄	—	13	2	no	—
231	C₂×C₂²⋊C₄	—	13	4	no	—
232	C₂×C₄²	—	13	2	no	—
233	C₂×C₄²	C₄×D₁₃	1	2	yes	—
234	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
237	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
238	C₄⋊C₈	—	26	4	no	—
239	C₄⋊C₈	—	26	2	no	—
240	C₄⋊C₈	—	26	2	no	—
241	C₂²⋊C₈	—	26	4	no	—
242	C₂²⋊C₈	—	26	8	no	—
243	C₄×C₈	—	2	4	no	—
244	C₄×C₈	—	2	4	no	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	—	26	4	no	—
247	C₄×C₈	D₅₂	1	4	yes	—
248	#95	—	13	4	no	—
249	#95	—	13	4	no	—
250	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
251	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
252	C₂×C₁₃	#2931	1	1	yes	characteristic
253	C₂×C₁₃	C₈×#41	1	2	yes	—
254	C₂×C₁₃	#2966	1	2	yes	—
255	D₁₃	—	4	4	no	—
256	C₂⁴	—	13	2	no	—
257	C₂×Q₈	—	13	4	no	—
258	C₂×Q₈	—	26	2	no	—
259	C₂×Q₈	—	26	2	no	—
260	C₂×D₄	—	13	4	no	—
261	C₂×D₄	—	26	2	no	—
262	C₂×D₄	—	26	2	no	—
263	C₂²×C₄	—	13	2	no	—
264	C₂²×C₄	—	13	2	no	—
265	C₂²×C₄	—	13	2	no	—
266	C₂²×C₄	—	26	2	no	—
267	C₂²×C₄	—	26	4	no	—
268	C₂²×C₄	—	26	2	no	—
269	C₂²×C₄	—	26	4	no	—
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	2	yes	—
273	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	26	2	no	—
279	C₂×C₈	—	26	2	no	—
280	C₂×C₈	—	26	4	no	—
281	C₂×C₈	—	26	4	no	—
282	C₂×C₈	—	52	8	no	—
283	C₂×C₈	C₂×D₅₂	1	2	yes	—
284	C₂×C₈	C₂×D₅₂	1	2	yes	—
285	C₂×C₈	#6127	1	4	yes	—
286	C₂×C₈	#6127	1	4	yes	—
287	C₄⋊C₄	—	26	2	no	—
288	C₄⋊C₄	—	26	4	no	—
289	C₄⋊C₄	—	26	2	no	—
290	C₂²⋊C₄	—	13	16	no	—
291	C₂²⋊C₄	—	26	4	no	—
292	C₂²⋊C₄	—	26	8	no	—
293	C₄²	—	2	4	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	26	2	no	—
298	C₄²	—	26	2	no	—
299	C₄²	C₈×D₁₃	1	4	yes	complemented
300	C₄²	C₂×D₅₂	1	2	yes	—
301	C₄²	#6127	1	2	yes	—
302	C₁₃	C₈×#124	1	1	yes	characteristic, complemented, Sylow
303	C₂³	—	13	4	no	—
304	C₂³	—	26	2	no	—
305	C₂³	—	26	4	no	—
306	C₂³	—	26	4	no	—
307	C₂³	—	26	2	no	—
308	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
309	Q₈	—	26	8	no	—
310	D₄	—	26	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	13	4	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	26	2	no	—
319	C₂×C₄	—	26	2	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	26	4	no	—
322	C₂×C₄	—	26	4	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	—	52	2	no	—
325	C₂×C₄	—	52	4	no	—
326	C₂×C₄	C₄×D₅₂	1	2	yes	—
327	C₂×C₄	C₄×D₅₂	1	2	yes	—
328	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
329	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
330	C₂×C₄	#88155	1	4	yes	—
331	C₂×C₄	#88155	1	2	yes	—
332	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
333	C₂×C₄	C₂×#6127	1	2	yes	—
334	C₂×C₄	C₂×#6127	1	2	yes	—
335	C₂×C₄	C₂×#6127	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	4	no	—
338	C₈	—	52	2	no	—
339	C₈	—	52	2	no	—
340	C₈	#88129	1	4	yes	direct product factor
341	C₂²	—	26	4	no	—
342	C₂²	—	26	8	no	—
343	C₂²	—	52	2	no	—
344	C₂²	—	52	2	no	—
345	C₂²	—	52	4	no	—
346	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
347	C₂²	C₄×#6127	1	2	yes	—
348	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
349	C₂²	C₂×#88155	1	1	yes	characteristic
350	C₂²	C₂×#88155	1	2	yes	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	4	no	—
354	C₄	—	26	4	no	—
355	C₄	—	52	2	no	—
356	C₄	—	52	2	no	—
357	C₄	C₈×D₅₂	1	4	yes	—
358	C₄	C₂×#88129	1	1	yes	characteristic
359	C₄	#11719182	1	1	yes	characteristic
360	C₄	#11719184	1	2	yes	—
361	C₂	—	52	4	no	—
362	C₂	C₄×#88129	1	1	yes	characteristic
363	C₂	C₂×C₈×D₅₂	1	1	yes	characteristic
364	C₂	#13123549	1	1	yes	characteristic
365	C₂	C₈×#6127	1	2	yes	—
366	C₂	#13123811	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	C₈×#124
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Dic₂₆	C₂	semidirect
4. #13115772	C₂	semidirect
6. #13115442	C₂	semidirect
7. #13114953	C₂	semidirect
8. C₈×#6103	C₂	semidirect
9. C₄²×C₈×C₁₃	C₂	semidirect
56. #88129	C₈	direct (N × H)
92. C₂×Dic₂₆	C₂×C₈	semidirect
104. C₄²×C₁₃	C₂×C₈	semidirect
109. #6103	C₂×C₈	semidirect
128. C₄²×C₈	D₁₃	semidirect
164. C₈×C₁₃	#124	semidirect
299. C₄²	C₈×D₁₃	semidirect
302. C₁₃	C₈×#124	semidirect
340. C₈	#88129	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₆	2
C₂×C₈×D₅₂	2
C₄×#88129	2
#13115772	2
#13115630	2
#13115442	2
#13114953	2
C₈×#6103	2
C₄²×C₈×C₁₃	2
C₈×#124	13

Subgroups — C8×#88129 #15870784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#88129 #15870784