Subgroups — #15881796

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117793	C₂	1	2	yes	maximal
2	#13115606	C₂	1	2	yes	maximal
3	#13115606	C₂	1	2	yes	maximal
4	#13115600	C₂	1	2	yes	maximal
5	#13115004	C₂	1	2	yes	maximal
6	#13115000	C₂	1	1	yes	maximal, characteristic
7	#13114733	C₂	1	2	yes	maximal
8	C₄×#88057	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718891	—	2	2	no	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88058	C₂²	1	2	yes	—
19	C₂×#88057	C₂²	1	1	yes	characteristic
20	C₂×#88057	C₂²	1	1	yes	characteristic
21	C₂×#88057	C₂²	1	1	yes	characteristic
22	C₂×#88057	C₂²	1	2	yes	—
23	C₂×#88057	C₂²	1	1	yes	characteristic
24	#11718642	—	2	2	no	—
25	#11718642	—	2	2	no	—
26	C₄⋊C₄×Dic₁₃	—	2	2	no	—
27	#11718597	C₂²	1	2	yes	—
28	#11718597	C₂²	1	2	yes	—
29	#11718596	C₂²	1	2	yes	—
30	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
31	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
33	#11718143	C₂²	1	2	yes	—
34	#11718143	C₂²	1	2	yes	—
35	#11718143	C₂²	1	2	yes	—
36	C₂²×Q₈×C₁₃	D₄	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	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
42	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×C₄⋊Dic₁₃	—	2	2	no	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
52	C₂×C₄×Dic₁₃	—	2	2	no	—
53	C₂×C₄×Dic₁₃	—	2	2	no	—
54	C₂×C₄×Dic₁₃	—	2	2	no	—
55	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
57	C₁₃×#97	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
59	#88071	—	2	2	no	—
60	#88071	—	2	2	no	—
61	#88071	—	2	2	no	—
62	#88071	—	2	2	no	—
63	#88071	—	2	2	no	—
64	#88071	—	2	2	no	—
65	#88071	—	2	2	no	—
66	#88058	—	2	4	no	—
67	#88058	—	4	2	no	—
68	#88057	—	2	2	no	—
69	#88057	—	4	1	no	—
70	#88057	—	4	2	no	—
71	#88057	—	4	1	no	—
72	#88057	Q₈	1	4	yes	complemented
73	#22730	—	13	1	no	maximal, Sylow
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	—	2	2	no	—
82	C₂²×Dic₁₃	—	2	2	no	—
83	C₂²×Dic₁₃	—	2	2	no	—
84	C₂²×Dic₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₁₃	#41	1	2	yes	—
94	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
95	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
99	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	2	2	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	—	2	2	no	—
107	C₄⋊Dic₁₃	—	4	4	no	—
108	C₄⋊Dic₁₃	—	8	2	no	—
109	C₄⋊Dic₁₃	C₂×Q₈	1	4	yes	—
110	C₄×Dic₁₃	—	4	4	no	—
111	C₄×Dic₁₃	—	8	2	no	—
112	C₄×Dic₁₃	—	8	2	no	—
113	C₂×#446	D₁₃	1	1	yes	characteristic
114	#2355	—	13	2	no	—
115	#1949	—	13	2	no	—
116	#1949	—	13	2	no	—
117	#1948	—	13	2	no	—
118	#1847	—	13	2	no	—
119	#1846	—	13	1	no	—
120	#1776	—	13	2	no	—
121	C₄×#106	—	13	1	no	—
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	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×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×Dic₁₃	—	2	4	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	—	4	2	no	—
147	C₂×Dic₁₃	—	4	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	4	2	no	—
151	C₂×Dic₁₃	—	4	2	no	—
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	2	4	no	—
154	C₈×C₁₃	C₄×Q₈	1	4	yes	—
155	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
156	#446	Dic₁₃	1	8	yes	complemented
157	C₂×#107	—	13	2	no	—
158	C₂×#107	—	13	2	no	—
159	C₂×#106	—	13	1	no	—
160	C₂×#106	—	13	1	no	—
161	C₂×#106	—	13	1	no	—
162	C₂×#106	—	13	1	no	—
163	C₂×#106	—	13	2	no	—
164	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₂×#97	C₂×D₁₃	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
168	#391	—	26	2	no	—
169	#387	—	26	2	no	—
170	#387	—	26	2	no	—
171	#384	—	13	2	no	—
172	#382	—	13	2	no	—
173	#382	—	13	2	no	—
174	C₄×C₄⋊C₄	—	13	2	no	—
175	C₄×C₄⋊C₄	—	13	2	no	—
176	C₄×C₄⋊C₄	—	13	2	no	—
177	C₄×C₄⋊C₄	—	26	2	no	—
178	#340	—	13	2	no	—
179	#340	—	13	2	no	—
180	#340	—	13	2	no	—
181	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
183	C₂²×C₁₃	#550	1	1	yes	characteristic
184	C₂²×C₁₃	#575	1	2	yes	—
185	C₂²×C₁₃	#577	1	2	yes	—
186	C₄×C₁₃	—	2	4	no	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	4	no	—
190	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₄×C₁₃	#520	1	1	yes	characteristic
192	C₄×C₁₃	#557	1	1	yes	characteristic
193	C₄×C₁₃	Q₈²	1	1	yes	characteristic
194	Dic₁₃	—	4	4	no	—
195	Dic₁₃	—	8	2	no	—
196	Dic₁₃	—	8	2	no	—
197	Dic₁₃	—	8	2	no	—
198	C₂²×Q₈	D₅₂	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
201	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
202	C₄×Q₈	Dic₂₆	1	4	yes	—
203	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
204	C₂×C₄⋊C₄	—	13	2	no	—
205	C₂×C₄⋊C₄	—	13	2	no	—
206	C₂×C₄⋊C₄	—	13	2	no	—
207	C₂×C₄⋊C₄	—	13	2	no	—
208	C₂×C₄⋊C₄	—	26	2	no	—
209	C₂×C₄⋊C₄	—	26	2	no	—
210	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
211	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
212	C₂×C₄²	—	26	2	no	—
213	C₂×C₄²	—	26	2	no	—
214	C₂×C₄²	—	26	2	no	—
215	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
217	#107	—	26	4	no	—
218	#107	—	52	2	no	—
219	#106	—	13	4	no	—
220	#106	—	26	2	no	—
221	#106	—	52	1	no	—
222	#106	—	52	2	no	—
223	#106	—	52	1	no	—
224	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
225	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
226	#97	C₂×Dic₁₃	1	8	yes	—
227	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
228	#95	—	26	2	no	—
229	#95	—	26	2	no	—
230	#95	—	26	2	no	—
231	#95	—	26	2	no	—
232	#95	—	26	2	no	—
233	#95	—	26	2	no	—
234	#95	—	26	2	no	—
235	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₁₃	#2911	1	1	yes	characteristic
237	C₂×C₁₃	#2912	1	1	yes	characteristic
238	C₂×C₁₃	#3388	1	2	yes	—
239	C₂×C₁₃	#3401	1	2	yes	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
242	C₂²×C₄	—	26	2	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	—	26	2	no	—
245	C₂²×C₄	—	26	2	no	—
246	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₅₂	1	2	yes	—
249	C₂²×C₄	C₂×D₅₂	1	2	yes	—
250	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	Q₈×D₁₃	1	2	yes	—
256	C₂×C₈	#6131	1	2	yes	—
257	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
258	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
259	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
260	C₄⋊C₄	—	13	4	no	—
261	C₄⋊C₄	—	26	2	no	—
262	C₄⋊C₄	—	26	2	no	—
263	C₄⋊C₄	—	26	2	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	52	4	no	—
269	C₄⋊C₄	—	104	2	no	—
270	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
271	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
272	C₄²	—	52	4	no	—
273	C₄²	—	104	2	no	—
274	C₄²	—	104	2	no	—
275	C₄²	C₂×Dic₂₆	1	2	yes	—
276	C₄²	C₂×Dic₂₆	1	4	yes	—
277	C₄²	C₂²×Dic₁₃	1	2	yes	—
278	C₄²	C₂²×Dic₁₃	1	4	yes	—
279	C₁₃	#22730	1	1	yes	characteristic, complemented, Sylow
280	C₂³	C₂²×D₅₂	1	1	yes	characteristic
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	26	4	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	52	2	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	—	52	2	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	—	52	2	no	—
298	C₂×C₄	—	52	2	no	—
299	C₂×C₄	—	52	2	no	—
300	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
302	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
304	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×#6131	1	1	yes	characteristic
307	C₂×C₄	#88256	1	1	yes	characteristic
308	C₂×C₄	#88257	1	1	yes	characteristic
309	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	Q₈×Dic₁₃	1	4	yes	—
313	C₂²	Q₈×D₅₂	1	1	yes	characteristic
314	C₂²	#11719282	1	1	yes	characteristic
315	C₂²	#11719453	1	2	yes	—
316	C₂²	#11719456	1	2	yes	—
317	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	52	4	no	—
323	C₄	—	104	2	no	—
324	C₄	—	104	2	no	—
325	C₄	—	104	2	no	—
326	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
327	C₄	#11719275	1	1	yes	characteristic
328	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
329	C₄	#11719532	1	1	yes	characteristic
330	C₂	Q₈×C₄⋊Dic₁₃	1	1	yes	characteristic
331	C₂	#13123426	1	1	yes	characteristic
332	C₂	#13123431	1	1	yes	characteristic
333	C₂	#13127783	1	2	yes	—
334	C₂	#13127904	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#22730
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
11. C₁₃×#446	C₄	semidirect
72. #88057	Q₈	semidirect
156. #446	Dic₁₃	semidirect
279. C₁₃	#22730	semidirect

Maximal subgroups

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

subgroup	index
#13117793	2
#13115606	2
#13115600	2
#13115004	2
#13115000	2
#13114733	2
C₄×#88057	2
C₂×C₁₃×#446	2
#22730	13