Subgroups — #15879691

Order 3328.

970 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719275	C₂	1	1	yes	maximal, characteristic
2	Q₈×#6099	C₂	1	1	yes	maximal, characteristic
3	#13115680	C₂	1	3	yes	maximal
4	#13115679	C₂	1	3	yes	maximal
5	#13114829	C₂	1	3	yes	maximal
6	C₄×#88064	C₂	1	3	yes	maximal
7	Q₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₃	C₄	1	2	yes	—
10	#11719275	—	2	1	no	—
11	#11719275	—	2	3	no	—
12	#11719275	—	2	1	no	—
13	#11719275	—	2	3	no	—
14	C₂×#88144	C₄	1	6	yes	—
15	C₂×#88123	C₂²	1	3	yes	—
16	C₂×C₄×Dic₂₆	C₂²	1	3	yes	—
17	C₁₃×#446	—	2	3	no	—
18	C₈×Q₈×C₁₃	—	2	1	no	—
19	C₁₃×#423	C₂²	1	3	yes	—
20	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
21	C₂×#88064	C₂²	1	3	yes	—
22	C₂×#88064	C₂²	1	3	yes	—
23	C₂×#88064	C₂²	1	6	yes	—
24	#11718685	—	2	3	no	—
25	Q₈×#1055	—	2	1	no	—
26	#11718585	C₂²	1	3	yes	—
27	C₄×#6099	C₂²	1	3	yes	—
28	#11718146	C₂²	1	1	yes	characteristic
29	#11718146	C₂²	1	3	yes	—
30	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₂₆	C₂³	1	3	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	3	yes	—
33	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	D₄	1	3	yes	—
36	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	D₄	1	3	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
40	Q₈×Dic₁₃	—	2	6	no	—
41	Q₈×Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	6	yes	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	6	yes	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
45	C₂×#6101	C₂×C₄	1	6	yes	—
46	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
47	C₂×#6099	C₂³	1	3	yes	—
48	C₂×#6099	C₂³	1	1	yes	characteristic
49	#88144	—	2	12	no	—
50	#88144	—	2	6	no	—
51	#88144	—	2	6	no	—
52	#88123	—	2	3	no	—
53	#88123	—	2	3	no	—
54	#88123	—	2	3	no	—
55	#88123	—	2	3	no	—
56	C₄×Dic₂₆	—	2	3	no	—
57	C₄×Dic₂₆	—	2	3	no	—
58	C₄×Dic₂₆	—	2	3	no	—
59	C₄×Dic₂₆	—	2	3	no	—
60	C₁₃×C₄⋊C₈	—	2	3	no	—
61	C₁₃×C₄⋊C₈	—	2	3	no	—
62	C₁₃×#97	—	2	3	no	—
63	C₄×C₈×C₁₃	—	2	3	no	—
64	#88064	—	2	6	no	—
65	#88064	—	2	6	no	—
66	#88064	—	4	6	no	—
67	C₄⋊#1055	—	2	3	no	—
68	C₄⋊#1055	—	2	3	no	—
69	#88043	—	2	3	no	—
70	C₄×#1055	—	2	3	no	—
71	#22538	—	13	1	no	maximal, Sylow
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	C₂²⋊C₄	1	3	yes	—
74	C₂×Q₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
79	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₆	—	2	3	no	—
81	C₂×Dic₂₆	—	2	3	no	—
82	C₂×Dic₂₆	—	2	3	no	—
83	C₂×Dic₂₆	—	2	3	no	—
84	C₂×Dic₂₆	—	2	3	no	—
85	C₂×Dic₂₆	—	2	3	no	—
86	C₁₃×M₁₆	—	2	1	no	—
87	C₁₃×M₁₆	—	2	3	no	—
88	C₁₃×M₁₆	—	2	3	no	—
89	C₁₃×M₁₆	#41	1	2	yes	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	3	no	—
92	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
93	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	3	yes	—
94	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	3	yes	—
95	C₄²×C₁₃	C₂×D₄	1	3	yes	—
96	C₄²×C₁₃	C₂×D₄	1	3	yes	—
97	C₄²×C₁₃	C₂×D₄	1	3	yes	—
98	C₄²×C₁₃	C₂×D₄	1	3	yes	—
99	C₄⋊Dic₁₃	—	2	6	no	—
100	C₄⋊Dic₁₃	—	2	6	no	—
101	C₄⋊Dic₁₃	—	2	6	no	—
102	C₄⋊Dic₁₃	—	2	6	no	—
103	C₄⋊Dic₁₃	—	2	1	no	—
104	C₄⋊Dic₁₃	—	2	1	no	—
105	#6101	—	2	6	no	—
106	#6101	—	2	6	no	—
107	C₄×Dic₁₃	—	2	6	no	—
108	C₄×Dic₁₃	—	2	6	no	—
109	#6099	—	2	1	no	—
110	#6099	—	2	3	no	—
111	#6099	—	2	3	no	—
112	#6099	#41	1	2	yes	—
113	C₂×#1055	—	2	1	no	—
114	C₂×#1055	—	2	3	no	—
115	C₂×#557	—	13	1	no	—
116	Q₈×M₁₆	—	13	1	no	—
117	Q₈×M₁₆	D₁₃	1	1	yes	characteristic
118	#1966	—	13	3	no	—
119	#1966	—	13	3	no	—
120	#1801	—	13	3	no	—
121	C₄×#101	—	13	3	no	—
122	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	6	no	—
124	Q₈×C₁₃	#101	1	4	yes	complemented
125	C₂×C₄×C₁₃	—	2	6	no	—
126	C₂×C₄×C₁₃	—	4	3	no	—
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
129	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
130	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	6	no	—
139	C₂×Dic₁₃	—	2	6	no	—
140	C₂×Dic₁₃	—	2	6	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	Dic₂₆	—	4	3	no	—
143	Dic₂₆	—	4	3	no	—
144	Dic₂₆	—	4	3	no	—
145	Dic₂₆	—	4	3	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	4	3	no	—
148	#1055	—	2	2	no	—
149	#1055	—	4	3	no	—
150	#557	—	26	1	no	—
151	#557	—	26	3	no	—
152	#557	—	26	1	no	—
153	#557	—	26	3	no	—
154	C₂×C₄⋊Q₈	—	13	3	no	—
155	C₂×#125	—	13	6	no	—
156	C₂×C₄×Q₈	—	13	2	no	—
157	C₂×C₄×Q₈	—	13	3	no	—
158	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
159	#446	—	2	3	no	—
160	#446	—	26	3	no	—
161	C₈×Q₈	—	2	1	no	—
162	C₈×Q₈	—	26	1	no	—
163	#423	—	13	3	no	—
164	#423	C₂×D₁₃	1	3	yes	—
165	C₂×#101	—	13	3	no	—
166	C₂×#101	—	13	3	no	—
167	C₂×#101	—	13	6	no	—
168	C₄×M₁₆	—	13	3	no	—
169	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
170	#343	—	13	3	no	—
171	#343	—	13	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
176	C₂²×C₁₃	#520	1	1	yes	characteristic
177	C₄×C₁₃	—	2	3	no	—
178	C₄×C₁₃	—	2	3	no	—
179	C₄×C₁₃	—	2	3	no	—
180	C₄×C₁₃	C₂×#101	1	6	yes	—
181	C₄×C₁₃	#548	1	1	yes	characteristic
182	C₄×C₁₃	#548	1	1	yes	characteristic
183	C₄×C₁₃	#550	1	1	yes	characteristic
184	C₄×C₁₃	#550	1	1	yes	characteristic
185	Dic₁₃	—	4	6	no	—
186	Dic₁₃	—	4	2	no	—
187	C₂²×Q₈	—	13	3	no	—
188	C₂²×Q₈	C₄×D₁₃	1	1	yes	characteristic
189	C₂×M₁₆	—	13	1	no	—
190	C₂×M₁₆	—	13	3	no	—
191	C₂×M₁₆	C₂²×D₁₃	1	3	yes	—
192	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	26	3	no	—
194	C₄⋊Q₈	—	26	3	no	—
195	C₄⋊Q₈	—	26	3	no	—
196	C₄⋊Q₈	—	26	3	no	—
197	#125	—	26	12	no	—
198	#125	—	26	6	no	—
199	#125	—	26	6	no	—
200	C₄×Q₈	—	26	6	no	—
201	C₄×Q₈	—	26	2	no	—
202	C₄×Q₈	—	26	3	no	—
203	C₄×Q₈	—	26	3	no	—
204	C₄×Q₈	—	26	3	no	—
205	C₄×Q₈	—	26	3	no	—
206	C₄×Q₈	D₅₂	1	1	yes	characteristic
207	C₄×Q₈	D₅₂	1	3	yes	—
208	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
209	C₄×Q₈	C₂²⋊D₁₃	1	3	yes	—
210	C₂×C₄⋊C₄	—	13	6	no	—
211	C₂×C₄⋊C₄	—	13	6	no	—
212	C₂×C₄⋊C₄	—	13	6	no	—
213	C₂×C₄⋊C₄	—	13	1	no	—
214	C₂×C₄⋊C₄	C₄×D₁₃	1	3	yes	—
215	C₂×C₄²	—	13	6	no	—
216	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
217	C₄⋊C₈	—	2	3	no	—
218	C₄⋊C₈	—	2	3	no	—
219	C₄⋊C₈	—	26	3	no	—
220	C₄⋊C₈	—	26	3	no	—
221	#101	—	26	6	no	—
222	#101	—	26	6	no	—
223	#101	—	52	6	no	—
224	#97	—	2	3	no	—
225	#97	—	26	3	no	—
226	C₄×C₈	—	2	3	no	—
227	C₄×C₈	—	26	3	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	#2341	1	1	yes	characteristic
231	C₂×C₁₃	C₂²×#101	1	1	yes	characteristic
232	C₂×C₁₃	#2889	1	1	yes	characteristic
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	26	3	no	—
235	C₂×Q₈	—	26	3	no	—
236	C₂×Q₈	—	26	3	no	—
237	C₂×Q₈	—	26	3	no	—
238	C₂×Q₈	—	26	3	no	—
239	C₂×Q₈	—	26	3	no	—
240	C₂×Q₈	#6103	1	3	yes	—
241	C₂×Q₈	#6103	1	1	yes	characteristic
242	C₂²×C₄	—	13	6	no	—
243	C₂²×C₄	—	13	2	no	—
244	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
245	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
246	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
247	M₁₆	—	2	1	no	—
248	M₁₆	—	2	3	no	—
249	M₁₆	—	2	3	no	—
250	M₁₆	—	13	2	no	—
251	M₁₆	—	26	1	no	—
252	M₁₆	—	26	3	no	—
253	M₁₆	—	26	3	no	—
254	M₁₆	#6131	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	3	no	—
257	C₂×C₈	—	26	1	no	—
258	C₂×C₈	—	26	3	no	—
259	C₄⋊C₄	—	26	6	no	—
260	C₄⋊C₄	—	26	6	no	—
261	C₄⋊C₄	—	26	6	no	—
262	C₄⋊C₄	—	26	6	no	—
263	C₄⋊C₄	—	26	6	no	—
264	C₄⋊C₄	—	26	6	no	—
265	C₄⋊C₄	—	26	1	no	—
266	C₄⋊C₄	—	26	1	no	—
267	C₄⋊C₄	#6103	1	6	yes	—
268	C₄⋊C₄	#6103	1	3	yes	—
269	C₄⋊C₄	#6103	1	3	yes	—
270	C₄²	—	26	6	no	—
271	C₄²	—	26	6	no	—
272	C₄²	C₂×D₅₂	1	3	yes	—
273	C₄²	C₂×D₅₂	1	3	yes	—
274	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
275	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
276	C₁₃	#22538	1	1	yes	characteristic, complemented, Sylow
277	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
278	Q₈	—	2	6	no	—
279	Q₈	—	52	3	no	—
280	Q₈	—	52	3	no	—
281	Q₈	—	52	3	no	—
282	Q₈	—	52	3	no	—
283	Q₈	#88064	1	4	yes	complemented
284	C₂×C₄	—	2	6	no	—
285	C₂×C₄	—	4	3	no	—
286	C₂×C₄	—	26	2	no	—
287	C₂×C₄	—	26	6	no	—
288	C₂×C₄	—	26	6	no	—
289	C₂×C₄	—	26	6	no	—
290	C₂×C₄	—	52	2	no	—
291	C₂×C₄	C₂×#6103	1	3	yes	—
292	C₂×C₄	C₂×#6103	1	3	yes	—
293	C₂×C₄	C₂×#6103	1	3	yes	—
294	C₂×C₄	C₂×#6103	1	3	yes	—
295	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
296	C₂×C₄	C₂×#6131	1	1	yes	characteristic
297	C₂×C₄	#88254	1	1	yes	characteristic
298	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
299	C₂×C₄	#88256	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	4	3	no	—
303	C₈	—	26	2	no	—
304	C₈	—	52	3	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	C₄×#6131	1	1	yes	characteristic
308	C₂²	#11719279	1	1	yes	characteristic
309	C₂²	C₂²×#6103	1	1	yes	characteristic
310	C₄	—	2	3	no	—
311	C₄	—	2	3	no	—
312	C₄	—	2	3	no	—
313	C₄	—	52	6	no	—
314	C₄	—	52	2	no	—
315	C₄	C₂×#88064	1	6	yes	—
316	C₄	#11719281	1	1	yes	characteristic
317	C₄	#11719282	1	1	yes	characteristic
318	C₄	#11719522	1	1	yes	characteristic
319	C₄	#11719539	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	#13117674	1	1	yes	characteristic
323	C₂	C₂²×#88064	1	1	yes	characteristic
324	C₂	#13123318	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6103
commutator, prime radical	C₂³×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	Q₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	Q₈×M₁₆	D₁₃
hypocenter	C₁₃	#22538
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
124. Q₈×C₁₃	#101	semidirect
276. C₁₃	#22538	semidirect
283. Q₈	#88064	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719275	2
Q₈×#6099	2
#13115680	2
#13115679	2
#13114829	2
C₄×#88064	2
Q₈×C₁₃×M₁₆	2
#22538	13