Subgroups — #15863362

Order 3328.

1070 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718782	C₂	1	2	yes	maximal
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
4	#13117459	C₂	1	2	yes	maximal
5	#13115004	C₂	1	4	yes	maximal, complemented
6	#13115000	C₂	1	2	yes	maximal, complemented
7	#13114993	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₂×#88180	C₂²	1	2	yes	—
13	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₁₃×#405	C₄	1	8	yes	complemented
15	C₂×C₁₃×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	#11718782	—	2	8	no	—
18	#11718767	—	2	4	no	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	4	yes	—
21	C₂×#88057	—	2	4	no	—
22	C₂×#88057	C₂²	1	2	yes	—
23	C₂×#88057	C₂²	1	2	yes	—
24	C₂×#88057	C₂²	1	1	yes	characteristic
25	C₂×#88057	C₂²	1	1	yes	characteristic
26	C₂×#88057	C₂²	1	4	yes	—
27	#11718646	—	2	4	no	—
28	#11718611	—	2	4	no	—
29	#11718597	C₂²	1	4	yes	—
30	#11718596	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂³×Dic₁₃	—	2	2	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
35	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	D₄	1	4	yes	—
38	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#117	C₂×C₄	1	4	yes	—
43	C₁₃×#117	Q₈	1	4	yes	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
47	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
48	#88180	—	2	4	no	—
49	#88180	—	2	4	no	—
50	C₂×C₄⋊Dic₁₃	—	2	4	no	—
51	C₂×C₄⋊Dic₁₃	—	2	4	no	—
52	C₂×C₄⋊Dic₁₃	—	2	2	no	—
53	C₂×C₄⋊Dic₁₃	—	2	8	no	—
54	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
55	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
56	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
57	C₂×C₄×Dic₁₃	—	2	2	no	—
58	C₁₃×#97	C₂×C₄	1	8	yes	complemented
59	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
60	#88071	—	2	8	no	—
61	#88058	—	2	4	no	—
62	#88058	—	2	8	no	—
63	#88057	—	2	4	no	—
64	#88057	—	2	4	no	—
65	#88057	—	2	2	no	—
66	#88057	—	2	2	no	—
67	#88057	—	2	8	no	—
68	#20802	—	13	1	no	maximal, Sylow
69	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂²×Dic₁₃	—	2	4	no	—
81	C₂²×Dic₁₃	—	2	2	no	—
82	C₂²×Dic₁₃	—	4	4	no	—
83	C₂²×Dic₁₃	—	8	4	no	—
84	C₁₃×M₁₆	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₈×C₁₃	C₄⋊C₄	1	16	yes	complemented
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
95	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
96	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
99	C₂²⋊Dic₁₃	—	4	4	no	—
100	C₄⋊Dic₁₃	—	2	4	no	—
101	C₄⋊Dic₁₃	—	2	4	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	4	4	no	—
107	C₄⋊Dic₁₃	—	8	8	no	—
108	C₄×Dic₁₃	—	4	4	no	—
109	C₂×#428	—	13	2	no	—
110	C₂×#427	—	13	1	no	—
111	C₂²×#106	—	13	1	no	—
112	C₂×#405	D₁₃	1	1	yes	characteristic
113	#2317	—	13	2	no	—
114	#1847	—	13	4	no	—
115	#1846	—	13	2	no	—
116	#1844	—	13	2	no	—
117	C₂³×C₁₃	—	2	4	no	—
118	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂³×C₁₃	#127	1	2	yes	—
120	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
121	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₃	#127	1	2	yes	—
128	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
129	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
130	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	—	4	2	no	—
139	C₂×Dic₁₃	—	4	4	no	—
140	C₂×Dic₁₃	—	4	4	no	—
141	C₂×Dic₁₃	—	4	8	no	—
142	C₂×Dic₁₃	—	4	4	no	—
143	C₂×Dic₁₃	—	8	8	no	—
144	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
145	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
146	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
147	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
148	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
149	C₂×#117	—	13	2	no	—
150	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
151	C₂²×C₄⋊C₄	—	13	2	no	—
152	#428	—	26	8	no	—
153	#427	—	26	4	no	—
154	C₂×#107	—	13	4	no	—
155	C₂×#107	—	13	2	no	—
156	C₂×#106	—	13	1	no	—
157	C₂×#106	—	13	2	no	—
158	C₂×#106	—	13	4	no	—
159	C₂×#106	—	13	1	no	—
160	C₂×#106	—	13	2	no	—
161	C₂×#106	—	26	4	no	—
162	#405	Dic₁₃	1	8	yes	complemented
163	C₂×#97	C₂×D₁₃	1	2	yes	—
164	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
165	#389	—	26	4	no	—
166	#385	—	26	4	no	—
167	#384	—	13	4	no	—
168	#382	—	13	4	no	—
169	C₂²×C₁₃	—	2	4	no	—
170	C₂²×C₁₃	—	2	4	no	—
171	C₂²×C₁₃	—	2	4	no	—
172	C₂²×C₁₃	#384	1	4	yes	—
173	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
176	C₂²×C₁₃	#577	1	4	yes	—
177	C₄×C₁₃	—	2	8	no	—
178	C₄×C₁₃	#384	1	4	yes	—
179	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
182	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
183	Dic₁₃	—	8	2	no	—
184	Dic₁₃	—	8	2	no	—
185	Dic₁₃	—	8	4	no	—
186	C₂³×C₄	—	26	2	no	—
187	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
188	C₂×M₁₆	Dic₂₆	1	4	yes	—
189	C₂×M₁₆	D₅₂	1	4	yes	—
190	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
191	C₂²×C₈	Dic₂₆	1	4	yes	—
192	C₂²×C₈	D₅₂	1	4	yes	—
193	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
197	#117	—	26	4	no	—
198	#117	—	26	4	no	—
199	#117	Dic₂₆	1	4	yes	—
200	#117	C₂×Dic₁₃	1	4	yes	—
201	C₂×C₄⋊C₄	—	13	4	no	—
202	C₂×C₄⋊C₄	—	13	2	no	—
203	C₂×C₄⋊C₄	—	13	2	no	—
204	C₂×C₄⋊C₄	—	26	4	no	—
205	C₂×C₄⋊C₄	—	26	4	no	—
206	C₂×C₄⋊C₄	—	26	2	no	—
207	C₂×C₄⋊C₄	—	26	8	no	—
208	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
209	C₂×C₂²⋊C₄	—	26	2	no	—
210	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
211	C₂×C₄²	—	26	2	no	—
212	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
213	#107	—	26	4	no	—
214	#107	—	26	8	no	—
215	#106	—	26	4	no	—
216	#106	—	26	4	no	—
217	#106	—	26	2	no	—
218	#106	—	26	2	no	—
219	#106	—	26	8	no	—
220	#97	C₂×Dic₁₃	1	8	yes	complemented
221	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
222	#95	—	26	8	no	—
223	C₂×C₁₃	—	2	4	no	—
224	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
225	C₂×C₁₃	#2912	1	2	yes	—
226	C₂×C₁₃	#3154	1	2	yes	—
227	C₂×C₁₃	#3163	1	2	yes	—
228	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
229	C₂²×C₄	—	26	2	no	—
230	C₂²×C₄	—	26	4	no	—
231	C₂²×C₄	—	26	2	no	—
232	C₂²×C₄	—	52	4	no	—
233	C₂²×C₄	—	104	4	no	—
234	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
235	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
236	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
237	C₂²×C₄	C₂×D₅₂	1	4	yes	—
238	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₅₂	1	2	yes	—
240	C₂²×C₄	C₂×D₅₂	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
242	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
243	M₁₆	C₄⋊Dic₁₃	1	16	yes	complemented
244	C₂×C₈	C₄⋊Dic₁₃	1	16	yes	complemented
245	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
246	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
247	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
248	C₂×C₈	C₂×D₅₂	1	2	yes	—
249	C₂×C₈	C₂×D₅₂	1	2	yes	—
250	C₂×C₈	C₂×D₅₂	1	4	yes	—
251	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
252	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
253	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
254	C₄⋊C₄	—	26	4	no	—
255	C₄⋊C₄	—	26	2	no	—
256	C₄⋊C₄	—	26	4	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₄⋊C₄	—	26	2	no	—
259	C₄⋊C₄	—	26	2	no	—
260	C₄⋊C₄	—	52	4	no	—
261	C₄⋊C₄	—	104	8	no	—
262	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
263	C₂²⋊C₄	—	52	4	no	—
264	C₂²⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
265	C₄²	—	52	4	no	—
266	C₄²	C₂×Dic₂₆	1	4	yes	—
267	C₄²	C₂²×Dic₁₃	1	4	yes	—
268	C₁₃	#20802	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	2	4	no	—
270	C₂³	#88123	1	2	yes	—
271	C₂³	#88128	1	2	yes	—
272	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
273	C₂³	C₂²×D₅₂	1	1	yes	characteristic
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	52	2	no	—
276	C₂×C₄	—	52	2	no	—
277	C₂×C₄	—	52	4	no	—
278	C₂×C₄	—	52	4	no	—
279	C₂×C₄	—	52	8	no	—
280	C₂×C₄	—	52	4	no	—
281	C₂×C₄	—	104	8	no	—
282	C₂×C₄	#88123	1	4	yes	—
283	C₂×C₄	#88123	1	4	yes	—
284	C₂×C₄	#88123	1	2	yes	—
285	C₂×C₄	#88128	1	2	yes	—
286	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
289	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
290	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
291	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
296	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
297	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
298	C₈	C₂×C₄⋊Dic₁₃	1	8	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#11718596	1	4	yes	—
303	C₂²	C₂×#88123	1	1	yes	characteristic
304	C₂²	C₂×#88128	1	1	yes	characteristic
305	C₂²	#11719456	1	4	yes	—
306	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
307	C₄	—	2	8	no	—
308	C₄	—	104	2	no	—
309	C₄	—	104	2	no	—
310	C₄	—	104	4	no	—
311	C₄	#11718596	1	4	yes	—
312	C₄	C₂×#88123	1	1	yes	characteristic
313	C₄	C₂×#88123	1	1	yes	characteristic
314	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
315	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×#11718596	1	1	yes	characteristic
318	C₂	#13123431	1	2	yes	—
319	C₂	#13125312	1	2	yes	—
320	C₂	#13125367	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#20802
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #13115004	C₂	semidirect
6. #13115000	C₂	semidirect
7. #13114993	C₂	semidirect
14. C₁₃×#405	C₄	semidirect
58. C₁₃×#97	C₂×C₄	semidirect
59. C₄×C₈×C₁₃	C₂×C₄	semidirect
84. C₁₃×M₁₆	C₄⋊C₄	semidirect
85. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
162. #405	Dic₁₃	semidirect
220. #97	C₂×Dic₁₃	semidirect
221. C₄×C₈	C₂×Dic₁₃	semidirect
243. M₁₆	C₄⋊Dic₁₃	semidirect
244. C₂×C₈	C₄⋊Dic₁₃	semidirect
268. C₁₃	#20802	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718782	2
C₂×#11718767	2
C₂²×#88057	2
#13117459	2
#13115004	2
#13115000	2
#13114993	2
C₂×C₁₃×#405	2
#20802	13