Subgroups — #15768789

Order 3328.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
2	#13117243	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718148	C₂	1	4	yes	maximal
4	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88180	C₂²	1	2	yes	—
8	C₂²×#6099	C₂²	1	2	yes	—
9	C₁₃×#423	—	2	1	no	—
10	C₁₃×#423	—	2	1	no	—
11	C₁₃×#423	—	2	2	no	—
12	C₂×C₁₃×C₄⋊C₈	C₄	1	4	yes	complemented
13	#11718873	C₄	1	4	yes	—
14	#11718597	—	2	1	no	—
15	#11718596	—	2	1	no	—
16	C₄×C₄⋊Dic₁₃	—	2	2	no	—
17	C₄×#6099	—	2	1	no	—
18	C₄×#6099	—	2	1	no	—
19	C₄×#6099	C₄	1	2	yes	complemented
20	C₄×#6099	C₄	1	2	yes	complemented
21	C₂×#88043	—	2	1	no	—
22	C₂×C₄×#1055	—	2	1	no	—
23	#11718148	—	2	8	no	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	4	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
40	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
41	#88180	—	2	4	no	—
42	#88180	—	2	4	no	—
43	C₂×C₄⋊Dic₁₃	—	2	4	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	C₂×C₄	1	4	yes	—
50	C₂×#6099	D₄	1	4	yes	—
51	C₂×#6099	Q₈	1	4	yes	—
52	C₂²×#1055	—	2	2	no	—
53	C₁₃×C₄⋊C₈	—	2	4	no	—
54	C₁₃×C₄⋊C₈	—	2	4	no	—
55	#88071	—	2	4	no	—
56	#88043	—	2	2	no	—
57	#88043	—	2	2	no	—
58	C₄×#1055	—	2	2	no	—
59	C₄×#1055	—	2	2	no	—
60	#10855	—	13	1	no	maximal, Sylow
61	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	4	4	no	—
63	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	#41	1	2	yes	—
78	C₂²×C₄×C₁₃	#41	1	2	yes	—
79	C₂²×Dic₁₃	—	2	4	no	—
80	C₁₃×M₁₆	—	4	4	no	—
81	C₁₃×M₁₆	—	4	2	no	—
82	C₁₃×M₁₆	—	4	2	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	8	no	—
85	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
94	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
95	C₂²⋊Dic₁₃	—	2	8	no	—
96	C₄⋊Dic₁₃	—	4	8	no	—
97	C₄⋊Dic₁₃	—	4	4	no	—
98	C₄×Dic₁₃	—	4	2	no	—
99	C₄×Dic₁₃	—	4	2	no	—
100	#6099	—	2	4	no	—
101	#6099	—	2	4	no	—
102	#6099	—	2	4	no	—
103	#6099	C₄⋊C₄	1	8	yes	complemented
104	C₂×#1055	—	2	4	no	—
105	C₂×#1055	—	2	4	no	—
106	C₂×#1055	—	2	4	no	—
107	C₂×#423	D₁₃	1	1	yes	characteristic
108	C₂×C₄×M₁₆	—	13	1	no	—
109	#2285	—	13	1	no	—
110	C₂×#344	—	13	4	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂³×C₁₃	C₄×D₄	1	2	yes	—
116	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	#95	1	2	yes	—
124	C₂×C₄×C₁₃	#95	1	2	yes	—
125	C₂×C₄×C₁₃	#95	1	2	yes	—
126	C₂×C₄×C₁₃	#95	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	#121	1	2	yes	—
136	C₂×C₄×C₁₃	#121	1	2	yes	—
137	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#125	1	2	yes	—
142	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
145	C₂×Dic₁₃	—	4	8	no	—
146	C₂×Dic₁₃	—	4	4	no	—
147	C₈×C₁₃	—	4	4	no	—
148	#1055	—	2	8	no	—
149	C₂²×M₁₆	—	13	2	no	—
150	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
151	C₂×#117	—	13	2	no	—
152	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
153	#423	—	2	2	no	—
154	#423	—	2	1	no	—
155	#423	—	2	1	no	—
156	C₂×C₄⋊C₈	Dic₁₃	1	4	yes	complemented
157	C₄×M₁₆	—	13	2	no	—
158	C₄×M₁₆	—	13	2	no	—
159	C₄×M₁₆	—	26	1	no	—
160	C₄×M₁₆	—	26	1	no	—
161	C₂×#97	—	26	1	no	—
162	C₂×C₄×C₈	—	26	1	no	—
163	#384	—	26	1	no	—
164	#382	—	26	1	no	—
165	#380	—	13	4	no	—
166	C₄×C₄⋊C₄	—	26	2	no	—
167	#344	—	26	8	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	2	2	no	—
171	C₂²×C₁₃	—	4	2	no	—
172	C₂²×C₁₃	—	4	2	no	—
173	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
174	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
175	C₂²×C₁₃	#389	1	2	yes	—
176	C₂²×C₁₃	#390	1	1	yes	characteristic
177	C₂²×C₁₃	#391	1	1	yes	characteristic
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	#344	1	4	yes	complemented
182	C₄×C₁₃	#380	1	2	yes	—
183	C₄×C₁₃	#382	1	1	yes	characteristic
184	C₄×C₁₃	#384	1	1	yes	characteristic
185	C₄×C₁₃	#389	1	2	yes	—
186	C₄×C₁₃	#390	1	1	yes	characteristic
187	C₄×C₁₃	#391	1	1	yes	characteristic
188	Dic₁₃	—	8	4	no	—
189	C₂³×C₄	C₄×D₁₃	1	2	yes	—
190	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	13	4	no	—
196	C₂×M₁₆	—	13	4	no	—
197	C₂×M₁₆	—	13	4	no	—
198	C₂×M₁₆	—	26	2	no	—
199	C₂×M₁₆	—	26	2	no	—
200	C₂×M₁₆	—	26	2	no	—
201	C₂²×C₈	—	26	2	no	—
202	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
203	#117	—	26	4	no	—
204	#117	—	26	4	no	—
205	C₂×C₄⋊C₄	—	26	4	no	—
206	C₂×C₄⋊C₄	—	26	2	no	—
207	C₂×C₂²⋊C₄	—	13	4	no	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	Dic₂₆	1	2	yes	—
210	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄²	D₅₂	1	1	yes	characteristic
213	C₂×C₄²	D₅₂	1	1	yes	characteristic
214	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
215	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	2	4	no	—
220	#97	—	26	2	no	—
221	#97	—	26	2	no	—
222	C₄×C₈	—	26	2	no	—
223	C₄×C₈	—	26	2	no	—
224	#95	—	26	4	no	—
225	C₂×C₁₃	—	2	1	no	—
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	—	2	2	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#344	1	2	yes	—
231	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	26	4	no	—
234	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
235	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
236	C₂²×C₄	#6101	1	2	yes	—
237	C₂²×C₄	#6101	1	2	yes	—
238	C₂²×C₄	#6103	1	2	yes	—
239	C₂²×C₄	#6103	1	2	yes	—
240	C₂²×C₄	C₂×Dic₂₆	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	C₂²×Dic₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	2	no	—
253	M₁₆	—	13	8	no	—
254	M₁₆	—	26	4	no	—
255	M₁₆	—	26	4	no	—
256	M₁₆	—	26	4	no	—
257	C₂×C₈	—	2	8	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	26	4	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	—	26	4	no	—
262	C₄⋊C₄	—	52	8	no	—
263	C₄⋊C₄	—	52	4	no	—
264	C₂²⋊C₄	—	26	8	no	—
265	C₄²	—	52	2	no	—
266	C₄²	—	52	2	no	—
267	C₄²	#6101	1	2	yes	—
268	C₄²	#6101	1	2	yes	—
269	C₄²	C₄⋊Dic₁₃	1	2	yes	—
270	C₄²	C₄⋊Dic₁₃	1	2	yes	—
271	C₄²	#6103	1	1	yes	characteristic
272	C₄²	#6103	1	1	yes	characteristic
273	C₄²	#6103	1	1	yes	characteristic
274	C₄²	#6103	1	1	yes	characteristic
275	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
276	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
277	C₁₃	#10855	1	1	yes	characteristic, complemented, Sylow
278	C₂³	—	4	2	no	—
279	C₂³	#88141	1	2	yes	—
280	C₂³	#88147	1	2	yes	—
281	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
282	C₂³	C₂×#6101	1	1	yes	characteristic
283	C₂³	C₂×#6103	1	1	yes	characteristic
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	52	8	no	—
291	C₂×C₄	—	52	4	no	—
292	C₂×C₄	#88071	1	2	yes	—
293	C₂×C₄	#88071	1	2	yes	—
294	C₂×C₄	#88071	1	2	yes	—
295	C₂×C₄	#88071	1	2	yes	—
296	C₂×C₄	#88142	1	2	yes	—
297	C₂×C₄	#88144	1	2	yes	—
298	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
299	C₂×C₄	#88146	1	2	yes	—
300	C₂×C₄	#88149	1	2	yes	—
301	C₂×C₄	#88151	1	2	yes	—
302	C₂×C₄	C₂×#6101	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂×#6103	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
306	C₂×C₄	#88190	1	1	yes	characteristic
307	C₂×C₄	#88192	1	2	yes	—
308	C₂×C₄	#88194	1	1	yes	characteristic
309	C₂×C₄	#88198	1	1	yes	characteristic
310	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
311	C₂×C₄	#88200	1	2	yes	—
312	C₂×C₄	#88201	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
314	C₈	—	4	4	no	—
315	C₈	—	26	8	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	#11718634	1	2	yes	—
322	C₂²	#11718635	1	1	yes	characteristic
323	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
324	C₂²	#11718649	1	1	yes	characteristic
325	C₂²	C₂×#88071	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	104	4	no	—
330	C₄	#11718148	1	4	yes	complemented
331	C₄	#11718637	1	1	yes	characteristic
332	C₄	#11718638	1	1	yes	characteristic
333	C₄	#11718646	1	2	yes	—
334	C₄	#11718648	1	2	yes	—
335	C₄	#11718882	1	1	yes	characteristic
336	C₄	#11718891	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	#13113065	1	1	yes	characteristic
342	C₂	C₂×#11718148	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₃
commutator	C₂³×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#423	C₂
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#423	D₁₃
hypocenter	C₁₃	#10855
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₃×C₄⋊C₈	C₄	semidirect
19. C₄×#6099	C₄	semidirect
20. C₄×#6099	C₄	semidirect
103. #6099	C₄⋊C₄	semidirect
156. C₂×C₄⋊C₈	Dic₁₃	semidirect
181. C₄×C₁₃	#344	semidirect
277. C₁₃	#10855	semidirect
330. C₄	#11718148	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#6099	2
#13117243	2
C₂×#11718148	2
C₂×C₁₃×#423	2
#10855	13