Subgroups — #15768713

Order 3328.

1062 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718116	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	2	yes	—
9	C₂³×#1055	C₂²	1	2	yes	—
10	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
11	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
12	C₂×C₁₃×#95	C₄	1	2	yes	—
13	C₂×C₄⋊#1055	—	2	2	no	—
14	C₂×C₄⋊#1055	—	2	2	no	—
15	C₂×C₄⋊#1055	C₄	1	4	yes	complemented
16	C₂×C₄×#1055	—	2	2	no	—
17	C₂×C₄×#1055	—	2	2	no	—
18	C₂×C₄×#1055	C₄	1	4	yes	complemented
19	#11718116	—	2	8	no	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₂×C₁₃×C₄⋊C₄	C₈	1	8	yes	complemented
27	C₂×C₄²×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	—	2	2	no	—
29	C₂×C₄²×C₁₃	—	2	4	no	—
30	C₂×C₄²×C₁₃	—	2	4	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
37	C₂²×#1055	—	2	2	no	—
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	4	no	—
44	C₂²×#1055	—	2	4	no	—
45	C₂²×#1055	C₂×C₄	1	4	yes	—
46	C₂²×#1055	C₂×C₄	1	4	yes	—
47	C₂²×#1055	D₄	1	4	yes	—
48	C₂²×#1055	Q₈	1	4	yes	—
49	C₁₃×#95	—	2	4	no	—
50	C₄⋊#1055	—	2	8	no	—
51	C₄×#1055	—	2	8	no	—
52	#10834	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	—	2	8	no	—
59	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₃	#41	1	2	yes	—
77	C₂²×C₄×C₁₃	#41	1	2	yes	—
78	C₁₃×C₄⋊C₄	—	2	16	no	—
79	C₄²×C₁₃	—	4	4	no	—
80	C₄²×C₁₃	—	4	2	no	—
81	C₄²×C₁₃	—	4	2	no	—
82	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
83	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
84	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
85	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	8	no	—
94	C₂×#1055	—	4	4	no	—
95	C₂×#1055	—	4	2	no	—
96	C₂×#1055	—	4	2	no	—
97	C₂×#1055	C₄⋊C₄	1	8	yes	complemented
98	C₂²×C₄⋊C₈	—	13	1	no	—
99	C₂²×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
101	C₂×#336	—	13	4	no	—
102	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₁₃	C₄×D₄	1	2	yes	—
106	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
107	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
108	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂³×C₁₃	#131	1	4	yes	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	8	no	—
114	C₂×C₄×C₁₃	—	2	16	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	8	no	—
118	C₂×C₄×C₁₃	#95	1	4	yes	—
119	C₂×C₄×C₁₃	#95	1	4	yes	—
120	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
121	C₂×C₄×C₁₃	#97	1	4	yes	—
122	C₂×C₄×C₁₃	C₂²⋊C₈	1	8	yes	—
123	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#117	1	2	yes	—
130	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	#121	1	2	yes	—
133	C₂×C₄×C₁₃	#121	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#125	1	2	yes	—
139	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
142	#1055	—	2	8	no	—
143	#1055	—	4	4	no	—
144	C₂³×C₈	—	13	2	no	—
145	C₂³×C₈	—	13	2	no	—
146	C₂²×C₄⋊C₄	Dic₁₃	1	2	yes	—
147	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
148	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₈	—	13	4	no	—
150	C₂×C₄⋊C₈	—	26	2	no	—
151	C₂×C₄⋊C₈	—	26	2	no	—
152	C₂×C₄×C₈	—	13	4	no	—
153	C₂×C₄×C₈	—	26	2	no	—
154	C₂×C₄×C₈	—	26	2	no	—
155	C₄×C₄⋊C₄	—	2	2	no	—
156	C₄×C₄⋊C₄	—	2	2	no	—
157	C₂×#95	Dic₁₃	1	2	yes	—
158	#336	—	26	8	no	—
159	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
160	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
161	C₂²×C₁₃	#389	1	2	yes	—
162	C₂²×C₁₃	#390	1	1	yes	characteristic
163	C₂²×C₁₃	#391	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
166	C₂²×C₁₃	#405	1	2	yes	—
167	C₂²×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
168	C₂²×C₁₃	#408	1	2	yes	—
169	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
170	C₂²×C₁₃	#424	1	2	yes	—
171	C₂²×C₁₃	C₈×D₄	1	4	yes	—
172	C₂²×C₁₃	#436	1	4	yes	—
173	C₂²×C₁₃	C₈×Q₈	1	4	yes	—
174	C₂²×C₁₃	#446	1	4	yes	—
175	C₄×C₁₃	—	2	4	no	—
176	C₄×C₁₃	—	4	8	no	—
177	C₄×C₁₃	#336	1	8	yes	complemented
178	C₄×C₁₃	#380	1	2	yes	—
179	C₄×C₁₃	#382	1	1	yes	characteristic
180	C₄×C₁₃	#384	1	1	yes	characteristic
181	C₄×C₁₃	#389	1	2	yes	—
182	C₄×C₁₃	#390	1	1	yes	characteristic
183	C₄×C₁₃	#391	1	1	yes	characteristic
184	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
185	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
188	C₂²×C₈	—	13	4	no	—
189	C₂²×C₈	—	13	4	no	—
190	C₂²×C₈	—	13	4	no	—
191	C₂²×C₈	—	13	4	no	—
192	C₂²×C₈	—	26	2	no	—
193	C₂²×C₈	—	26	2	no	—
194	C₂²×C₈	—	26	2	no	—
195	C₂²×C₈	—	26	2	no	—
196	C₂²×C₈	—	26	2	no	—
197	C₂²×C₈	—	26	4	no	—
198	C₂²×C₈	—	26	4	no	—
199	C₂²×C₈	—	26	2	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	#1055	1	8	yes	complemented
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	Dic₂₆	1	2	yes	—
208	C₂×C₄²	C₄×D₁₃	1	2	yes	—
209	C₂×C₄²	C₄×D₁₃	1	2	yes	—
210	C₂×C₄²	D₅₂	1	2	yes	—
211	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
212	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
213	C₄⋊C₈	—	26	8	no	—
214	C₄×C₈	—	26	8	no	—
215	#95	—	2	4	no	—
216	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
218	C₂×C₁₃	#1729	1	1	yes	characteristic
219	C₂×C₁₃	C₈×C₄⋊C₄	1	2	yes	—
220	C₂×C₁₃	#1770	1	2	yes	—
221	C₂×C₁₃	#1939	1	4	yes	—
222	C₂×C₁₃	#1954	1	2	yes	—
223	C₂×C₁₃	#1984	1	2	yes	—
224	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂×#1055	1	8	yes	—
231	C₂²×C₄	C₂×#1055	1	4	yes	—
232	C₂²×C₄	#6099	1	4	yes	—
233	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
234	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
235	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
237	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
238	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
239	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₁₃	1	2	yes	—
244	C₂²×C₄	#6129	1	2	yes	—
245	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
246	C₂²×C₄	#6131	1	2	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	—	13	8	no	—
250	C₂×C₈	—	26	4	no	—
251	C₂×C₈	—	26	4	no	—
252	C₂×C₈	—	26	4	no	—
253	C₂×C₈	—	26	4	no	—
254	C₂×C₈	—	26	4	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	—	26	8	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	—	52	4	no	—
259	C₂×C₈	—	52	2	no	—
260	C₂×C₈	—	52	2	no	—
261	C₄⋊C₄	—	2	16	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	4	no	—
264	C₄²	—	4	2	no	—
265	C₄²	#6101	1	4	yes	—
266	C₄²	#6101	1	4	yes	—
267	C₄²	#6103	1	4	yes	—
268	C₄²	#6103	1	4	yes	—
269	C₁₃	#10834	1	1	yes	characteristic, complemented, Sylow
270	C₂³	C₂²×#1055	1	2	yes	—
271	C₂³	C₂×#6099	1	2	yes	—
272	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
273	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
274	C₂³	D₄×Dic₁₃	1	2	yes	—
275	C₂³	Q₈×Dic₁₃	1	2	yes	—
276	C₂³	#88202	1	4	yes	—
277	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	16	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₄×#1055	1	4	yes	—
287	C₂×C₄	#88043	1	4	yes	—
288	C₂×C₄	C₄⋊#1055	1	8	yes	—
289	C₂×C₄	C₂²⋊#1055	1	8	yes	—
290	C₂×C₄	#88071	1	4	yes	—
291	C₂×C₄	#88071	1	4	yes	—
292	C₂×C₄	#88141	1	2	yes	—
293	C₂×C₄	#88142	1	2	yes	—
294	C₂×C₄	#88144	1	2	yes	—
295	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
296	C₂×C₄	#88146	1	2	yes	—
297	C₂×C₄	#88147	1	2	yes	—
298	C₂×C₄	#88149	1	2	yes	—
299	C₂×C₄	#88151	1	2	yes	—
300	C₂×C₄	C₂×#6101	1	1	yes	characteristic
301	C₂×C₄	C₂×#6101	1	1	yes	characteristic
302	C₂×C₄	C₂×#6103	1	1	yes	characteristic
303	C₂×C₄	C₂×#6103	1	1	yes	characteristic
304	C₂×C₄	#88190	1	1	yes	characteristic
305	C₂×C₄	#88192	1	2	yes	—
306	C₂×C₄	#88194	1	1	yes	characteristic
307	C₂×C₄	#88198	1	1	yes	characteristic
308	C₂×C₄	#88200	1	2	yes	—
309	C₂×C₄	#88201	1	1	yes	characteristic
310	C₈	—	26	8	no	—
311	C₈	—	52	4	no	—
312	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
313	C₂²	C₂×#88043	1	1	yes	characteristic
314	C₂²	C₂×C₄⋊#1055	1	2	yes	—
315	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
316	C₂²	#11718646	1	2	yes	—
317	C₂²	#11718658	1	2	yes	—
318	C₂²	#11718660	1	2	yes	—
319	C₂²	D₄×#1055	1	4	yes	—
320	C₂²	#11718673	1	4	yes	—
321	C₂²	Q₈×#1055	1	4	yes	—
322	C₂²	#11718685	1	4	yes	—
323	C₂²	C₂×C₂²⋊#1055	1	2	yes	—
324	C₂²	C₂×#88071	1	1	yes	characteristic
325	C₂²	#11718882	1	1	yes	characteristic
326	C₂²	#11718891	1	1	yes	characteristic
327	C₂²	#11718895	1	2	yes	—
328	C₄	—	2	4	no	—
329	C₄	—	4	8	no	—
330	C₄	#11718116	1	8	yes	complemented
331	C₄	#11718634	1	2	yes	—
332	C₄	#11718635	1	1	yes	characteristic
333	C₄	#11718637	1	1	yes	characteristic
334	C₄	#11718638	1	1	yes	characteristic
335	C₄	#11718648	1	2	yes	—
336	C₄	#11718649	1	1	yes	characteristic
337	C₂	#13113065	1	1	yes	characteristic
338	C₂	C₂×#11718116	1	1	yes	characteristic
339	C₂	#13114533	1	1	yes	characteristic
340	C₂	C₄⋊C₄×#1055	1	2	yes	—
341	C₂	#13114705	1	2	yes	—
342	C₂	#13115555	1	4	yes	—
343	C₂	#13115631	1	2	yes	—
344	C₂	#13115773	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#10834
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
15. C₂×C₄⋊#1055	C₄	semidirect
18. C₂×C₄×#1055	C₄	semidirect
26. C₂×C₁₃×C₄⋊C₄	C₈	semidirect
97. C₂×#1055	C₄⋊C₄	semidirect
177. C₄×C₁₃	#336	semidirect
202. C₂×C₄⋊C₄	#1055	semidirect
269. C₁₃	#10834	semidirect
330. C₄	#11718116	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#1055	2
C₂²×C₄×#1055	2
C₂×#11718116	2
C₂×C₄×C₁₃×C₄⋊C₄	2
#10834	13