Subgroups — #15768747

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718150	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	C₂²×#6099	C₂²	1	2	yes	—
8	C₂³×#1055	C₂²	1	2	yes	—
9	C₁₃×#423	—	2	2	no	—
10	C₁₃×#423	C₄	1	4	yes	—
11	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
12	#11718585	—	2	2	no	—
13	#11718585	C₄	1	4	yes	—
14	C₂×C₄⋊#1055	—	2	2	no	—
15	C₂×C₄×#1055	—	2	2	no	—
16	C₂×C₄×#1055	—	2	2	no	—
17	C₂×C₄×#1055	C₄	1	2	yes	—
18	C₂×C₄×#1055	C₄	1	2	yes	—
19	#11718150	—	2	8	no	—
20	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	4	no	—
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	—	2	4	no	—
36	C₂×#6099	—	2	4	no	—
37	C₂×#6099	C₂×C₄	1	4	yes	—
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	C₂×C₄	1	4	yes	—
44	C₂²×#1055	D₄	1	4	yes	—
45	C₂²×#1055	Q₈	1	4	yes	—
46	C₁₃×C₄⋊C₈	—	2	8	no	—
47	C₄⋊#1055	—	2	8	no	—
48	C₄×#1055	—	2	4	no	—
49	C₄×#1055	—	2	4	no	—
50	#10844	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₁₃×M₁₆	—	2	8	no	—
68	C₁₃×M₁₆	—	4	4	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₄²×C₁₃	C₄²	1	2	yes	—
73	C₄²×C₁₃	C₄²	1	2	yes	—
74	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
77	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
78	#6099	—	2	8	no	—
79	#6099	—	4	4	no	—
80	C₂×#1055	—	2	4	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	4	2	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×#1055	C₄⋊C₄	1	8	yes	—
90	C₂×#423	—	13	1	no	—
91	C₂×#423	D₁₃	1	1	yes	characteristic
92	C₂²×C₄×C₈	—	13	1	no	—
93	C₂×#341	—	13	4	no	—
94	C₂³×C₁₃	—	2	4	no	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₃	#125	1	2	yes	—
99	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	8	no	—
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	#95	1	4	yes	—
104	C₂×C₄×C₁₃	#95	1	4	yes	—
105	C₂×C₄×C₁₃	#108	1	8	yes	—
106	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃	#117	1	2	yes	—
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₃	#121	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
124	C₈×C₁₃	—	4	4	no	—
125	#1055	—	2	8	no	—
126	#1055	—	4	4	no	—
127	C₂²×M₁₆	—	13	2	no	—
128	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
129	C₂³×C₈	—	13	2	no	—
130	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
131	#423	—	2	2	no	—
132	#423	—	13	4	no	—
133	#423	—	26	2	no	—
134	#423	Dic₁₃	1	4	yes	—
135	C₂×C₄⋊C₈	—	2	2	no	—
136	C₂×C₄⋊C₈	—	26	2	no	—
137	C₂×C₄×C₈	—	13	2	no	—
138	C₂×C₄×C₈	—	13	2	no	—
139	C₂×C₄×C₈	—	26	2	no	—
140	C₂×C₄×C₈	—	26	2	no	—
141	#341	—	26	8	no	—
142	C₂²×C₁₃	—	2	4	no	—
143	C₂²×C₁₃	—	2	4	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
147	C₂²×C₁₃	#382	1	1	yes	characteristic
148	C₂²×C₁₃	#384	1	1	yes	characteristic
149	C₂²×C₁₃	#389	1	2	yes	—
150	C₂²×C₁₃	#389	1	2	yes	—
151	C₂²×C₁₃	C₂×#108	1	4	yes	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	#341	1	8	yes	complemented
154	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
155	C₄×C₁₃	#380	1	2	yes	—
156	C₄×C₁₃	#390	1	1	yes	characteristic
157	C₄×C₁₃	#390	1	1	yes	characteristic
158	C₄×C₁₃	#391	1	1	yes	characteristic
159	C₄×C₁₃	#391	1	1	yes	characteristic
160	C₂³×C₄	Dic₂₆	1	2	yes	—
161	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	13	4	no	—
166	C₂×M₁₆	—	26	2	no	—
167	C₂×M₁₆	—	26	4	no	—
168	C₂×M₁₆	—	26	4	no	—
169	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	13	4	no	—
172	C₂²×C₈	—	13	4	no	—
173	C₂²×C₈	—	13	4	no	—
174	C₂²×C₈	—	26	2	no	—
175	C₂²×C₈	—	26	2	no	—
176	C₂²×C₈	—	26	2	no	—
177	C₂²×C₈	—	26	2	no	—
178	C₂²×C₈	—	26	2	no	—
179	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₁₃	1	2	yes	—
182	C₂×C₄²	D₅₂	1	2	yes	—
183	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
186	C₄⋊C₈	—	2	8	no	—
187	C₄⋊C₈	—	26	8	no	—
188	C₄×C₈	—	26	4	no	—
189	C₄×C₈	—	26	4	no	—
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
193	C₂×C₁₃	C₂×#341	1	2	yes	—
194	C₂×C₁₃	#1837	1	2	yes	—
195	C₂×C₁₃	#1848	1	2	yes	—
196	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	#6101	1	2	yes	—
199	C₂²×C₄	#6101	1	4	yes	—
200	C₂²×C₄	#6101	1	2	yes	—
201	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
202	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₁₃	1	2	yes	—
206	C₂²×C₄	#6129	1	2	yes	—
207	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
208	C₂²×C₄	#6131	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
212	M₁₆	—	2	8	no	—
213	M₁₆	—	4	4	no	—
214	M₁₆	—	26	8	no	—
215	M₁₆	—	52	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	13	8	no	—
220	C₂×C₈	—	26	4	no	—
221	C₂×C₈	—	26	4	no	—
222	C₂×C₈	—	26	4	no	—
223	C₂×C₈	—	26	4	no	—
224	C₂×C₈	—	26	4	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	—	26	4	no	—
227	C₂×C₈	—	52	2	no	—
228	C₂×C₈	—	52	2	no	—
229	C₄²	C₄×Dic₁₃	1	2	yes	—
230	C₄²	C₄×Dic₁₃	1	2	yes	—
231	C₄²	#6103	1	2	yes	—
232	C₄²	#6103	1	2	yes	—
233	C₄²	#6103	1	4	yes	—
234	C₄²	C₂²⋊Dic₁₃	1	4	yes	—
235	C₁₃	#10844	1	1	yes	characteristic, complemented, Sylow
236	C₂³	—	2	4	no	—
237	C₂³	#88142	1	2	yes	—
238	C₂³	#88144	1	2	yes	—
239	C₂³	C₂×#6101	1	1	yes	characteristic
240	C₂³	C₂×#6101	1	1	yes	characteristic
241	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	#88062	1	8	yes	—
246	C₂×C₄	#88071	1	4	yes	—
247	C₂×C₄	#88071	1	4	yes	—
248	C₂×C₄	#88141	1	2	yes	—
249	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
250	C₂×C₄	#88146	1	2	yes	—
251	C₂×C₄	#88147	1	2	yes	—
252	C₂×C₄	#88149	1	2	yes	—
253	C₂×C₄	#88151	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#6103	1	1	yes	characteristic
256	C₂×C₄	C₂×#6103	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
258	C₂×C₄	#88190	1	1	yes	characteristic
259	C₂×C₄	#88192	1	2	yes	—
260	C₂×C₄	#88194	1	1	yes	characteristic
261	C₂×C₄	#88198	1	1	yes	characteristic
262	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
263	C₂×C₄	#88200	1	2	yes	—
264	C₂×C₄	#88201	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	26	8	no	—
268	C₈	—	52	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	#11718634	1	2	yes	—
274	C₂²	#11718637	1	1	yes	characteristic
275	C₂²	#11718638	1	1	yes	characteristic
276	C₂²	#11718646	1	2	yes	—
277	C₂²	C₂×#88062	1	4	yes	—
278	C₂²	C₂×#88071	1	1	yes	characteristic
279	C₄	—	2	4	no	—
280	C₄	#11718150	1	8	yes	complemented
281	C₄	#11718635	1	1	yes	characteristic
282	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
283	C₄	#11718648	1	2	yes	—
284	C₄	#11718649	1	1	yes	characteristic
285	C₄	#11718882	1	1	yes	characteristic
286	C₄	#11718891	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#13113065	1	1	yes	characteristic
290	C₂	C₂×#11718150	1	2	yes	—
291	C₂	#13114973	1	2	yes	—
292	C₂	#13115007	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#423	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#423	D₁₃
hypocenter	C₁₃	#10844
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
153. C₄×C₁₃	#341	semidirect
235. C₁₃	#10844	semidirect
280. C₄	#11718150	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
C₂²×C₄×#1055	2
C₂×#11718150	2
C₂×C₁₃×#423	2
#10844	13