Subgroups — #15866445

Order 3328.

906 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718785	C₂	1	2	yes	maximal
2	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
4	#13115611	C₂	1	8	yes	maximal, complemented
5	#13114562	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₄×C₁₃×M₁₆	—	2	2	no	—
13	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	#11718785	—	2	8	no	—
17	C₂×#88062	C₂²	1	8	yes	—
18	#11718585	—	2	1	no	—
19	#11718585	—	2	1	no	—
20	#11718585	C₄	1	2	yes	—
21	#11718585	C₄	1	4	yes	—
22	#11718585	C₄	1	2	yes	—
23	#11718585	C₂²	1	4	yes	—
24	C₂×C₄⋊#1055	—	2	2	no	—
25	C₂×C₄⋊#1055	C₂²	1	4	yes	—
26	#11718150	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
34	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
35	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
36	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	—	2	1	no	—
38	C₂²×C₈×C₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₂×#6099	—	2	4	no	—
49	C₂×#6099	—	2	4	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	—	2	4	no	—
52	C₂×#6099	—	2	4	no	—
53	C₂×#6099	C₂×C₄	1	4	yes	—
54	C₂×#6099	C₂×C₄	1	4	yes	—
55	C₂×#6099	C₂³	1	4	yes	—
56	C₂²×#1055	—	2	2	no	—
57	C₂²×#1055	C₂³	1	4	yes	—
58	C₁₃×#97	—	2	4	no	—
59	C₄×C₈×C₁₃	—	2	4	no	—
60	#88062	—	2	16	no	—
61	C₄⋊#1055	—	2	4	no	—
62	C₄⋊#1055	—	2	4	no	—
63	C₄⋊#1055	C₂×C₄	1	8	yes	—
64	C₄⋊#1055	C₂×C₄	1	8	yes	—
65	#21177	—	13	1	no	maximal, Sylow
66	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	—	4	4	no	—
68	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₁₃	#41	1	2	yes	—
82	C₂²×C₄×C₁₃	#41	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×M₁₆	—	2	4	no	—
85	C₁₃×M₁₆	—	2	4	no	—
86	C₁₃×M₁₆	—	2	2	no	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₁₃×M₁₆	C₂×D₄	1	4	yes	—
89	C₁₃×M₁₆	C₂×Q₈	1	4	yes	—
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₂×C₈×C₁₃	—	2	4	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
98	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
99	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	C₂×D₄	1	2	yes	—
103	C₄²×C₁₃	C₂×D₄	1	2	yes	—
104	#6099	—	2	8	no	—
105	#6099	—	2	8	no	—
106	#6099	—	2	4	no	—
107	#6099	—	2	4	no	—
108	#6099	—	4	4	no	—
109	C₂×#1055	—	2	4	no	—
110	C₂×#1055	—	2	4	no	—
111	C₂×#1055	—	2	4	no	—
112	C₂×#1055	—	4	4	no	—
113	C₂×#1055	C₂²×C₄	1	8	yes	—
114	C₂×#430	—	13	2	no	—
115	C₂×#423	—	13	1	no	—
116	C₂×#423	—	13	1	no	—
117	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
118	#1950	—	13	8	no	—
119	#1738	—	13	2	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂³×C₁₃	#125	1	2	yes	—
124	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
125	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₁₃	#121	1	2	yes	—
143	C₂×C₄×C₁₃	#121	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₄×C₁₃	#125	1	2	yes	—
147	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₈×C₁₃	—	2	4	no	—
155	C₈×C₁₃	—	2	4	no	—
156	#1055	—	2	8	no	—
157	#1055	—	4	4	no	—
158	C₂²×M₁₆	—	13	2	no	—
159	C₂²×M₁₆	—	13	2	no	—
160	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
161	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
163	#430	—	26	8	no	—
164	C₂×#108	—	13	8	no	—
165	#423	—	13	2	no	—
166	#423	—	13	4	no	—
167	#423	—	13	2	no	—
168	#423	—	13	4	no	—
169	#423	—	26	1	no	—
170	#423	—	26	1	no	—
171	C₂×C₄⋊C₈	—	13	4	no	—
172	C₂×C₄⋊C₈	—	26	2	no	—
173	C₄×M₁₆	—	2	2	no	—
174	C₄×M₁₆	C₂×D₁₃	1	4	yes	—
175	C₂×#97	—	2	1	no	—
176	C₂×C₄×C₈	—	2	1	no	—
177	#341	—	13	8	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	4	2	no	—
183	C₂²×C₁₃	#389	1	2	yes	—
184	C₂²×C₁₃	#389	1	2	yes	—
185	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	#389	1	2	yes	—
192	C₄×C₁₃	#389	1	2	yes	—
193	C₄×C₁₃	#430	1	4	yes	complemented
194	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
197	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
198	C₂³×C₄	Dic₂₆	1	2	yes	—
199	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	13	4	no	—
205	C₂×M₁₆	—	13	4	no	—
206	C₂×M₁₆	—	13	4	no	—
207	C₂×M₁₆	—	26	4	no	—
208	C₂×M₁₆	—	26	4	no	—
209	C₂×M₁₆	—	26	2	no	—
210	C₂×M₁₆	—	26	4	no	—
211	C₂×M₁₆	—	26	4	no	—
212	C₂×M₁₆	Dic₂₆	1	4	yes	—
213	C₂×M₁₆	D₅₂	1	4	yes	—
214	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
215	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	13	4	no	—
219	C₂²×C₈	—	26	2	no	—
220	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
221	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
222	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₁₃	1	2	yes	—
224	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
228	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
229	#108	—	26	16	no	—
230	C₄⋊C₈	—	13	8	no	—
231	C₄⋊C₈	—	13	8	no	—
232	C₄⋊C₈	—	26	4	no	—
233	C₄⋊C₈	—	26	4	no	—
234	#97	—	2	4	no	—
235	C₄×C₈	—	2	4	no	—
236	C₂×C₁₃	—	2	1	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	2	2	no	—
239	C₂×C₁₃	—	2	2	no	—
240	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#430	1	2	yes	—
242	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	#6101	1	2	yes	—
245	C₂²×C₄	#6101	1	2	yes	—
246	C₂²×C₄	#6101	1	2	yes	—
247	C₂²×C₄	#6101	1	2	yes	—
248	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
249	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
250	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
251	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₅₂	1	2	yes	—
255	C₂²×C₄	#6127	1	2	yes	—
256	C₂²×C₄	#6127	1	2	yes	—
257	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	2	2	no	—
264	M₁₆	—	26	8	no	—
265	M₁₆	—	26	8	no	—
266	M₁₆	—	26	4	no	—
267	M₁₆	—	26	4	no	—
268	M₁₆	—	52	4	no	—
269	M₁₆	C₂×Dic₂₆	1	4	yes	—
270	M₁₆	C₂×D₅₂	1	4	yes	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	13	8	no	—
276	C₂×C₈	—	26	4	no	—
277	C₂×C₈	—	26	4	no	—
278	C₂×C₈	—	26	4	no	—
279	C₂×C₈	—	52	4	no	—
280	C₄²	#6101	1	1	yes	characteristic
281	C₄²	#6101	1	1	yes	characteristic
282	C₄²	#6101	1	1	yes	characteristic
283	C₄²	#6101	1	2	yes	—
284	C₄²	#6101	1	2	yes	—
285	C₄²	#6101	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₁₃	1	2	yes	—
287	C₄²	C₂×C₄×D₁₃	1	2	yes	—
288	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
289	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
290	C₁₃	#21177	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	4	2	no	—
292	C₂³	#88123	1	2	yes	—
293	C₂³	#88124	1	2	yes	—
294	C₂³	C₂×#6101	1	1	yes	characteristic
295	C₂³	C₂×#6101	1	1	yes	characteristic
296	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
304	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
305	C₂×C₄	#88123	1	2	yes	—
306	C₂×C₄	#88124	1	2	yes	—
307	C₂×C₄	C₄×D₅₂	1	2	yes	—
308	C₂×C₄	C₄×D₅₂	1	2	yes	—
309	C₂×C₄	C₂×#6101	1	2	yes	—
310	C₂×C₄	C₂×#6101	1	2	yes	—
311	C₂×C₄	C₂×#6101	1	2	yes	—
312	C₂×C₄	C₂×#6101	1	2	yes	—
313	C₂×C₄	C₂×#6101	1	1	yes	characteristic
314	C₂×C₄	C₂×#6101	1	1	yes	characteristic
315	C₂×C₄	#88178	1	2	yes	—
316	C₂×C₄	#88178	1	2	yes	—
317	C₂×C₄	#88184	1	2	yes	—
318	C₂×C₄	#88184	1	2	yes	—
319	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
322	C₂×C₄	C₂×#6127	1	1	yes	characteristic
323	C₂×C₄	C₂×#6127	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	2	4	no	—
327	C₈	—	26	8	no	—
328	C₈	—	52	4	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	#11718589	1	2	yes	—
335	C₂²	#11718589	1	2	yes	—
336	C₂²	C₂×#88123	1	1	yes	characteristic
337	C₂²	C₂×#88124	1	1	yes	characteristic
338	C₂²	C₂²×#6101	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	#11718589	1	2	yes	—
343	C₄	#11718589	1	2	yes	—
344	C₄	#11718785	1	4	yes	complemented
345	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
346	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
347	C₄	C₂×#88178	1	1	yes	characteristic
348	C₄	C₂×#88184	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	2	no	—
352	C₂	—	2	2	no	—
353	C₂	C₂×#11718589	1	1	yes	characteristic
354	C₂	C₂×#11718785	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#21177
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #13115611	C₂	semidirect
193. C₄×C₁₃	#430	semidirect
290. C₁₃	#21177	semidirect
344. C₄	#11718785	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718785	2
C₂×#11718585	2
#13115611	2
#13114562	2
C₂×C₄×C₁₃×M₁₆	2
#21177	13