Subgroups — #15761701

Order 3328.

618 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Q₈⋊#1055	C₂	1	2	yes	maximal, complemented
4	C₂×D₄⋊#1055	C₂	1	2	yes	maximal
5	C₄×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#391	C₄	1	1	yes	characteristic
10	C₁₃×#390	C₄	1	1	yes	characteristic
11	C₁₃×#388	—	2	2	no	—
12	C₁₃×#382	C₄	1	1	yes	characteristic
13	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	C₂×C₄×#1055	—	2	2	no	—
17	C₂×C₄×#1055	C₂²	1	2	yes	—
18	Q₈⋊#1055	—	2	4	no	—
19	D₄⋊#1055	—	2	4	no	—
20	#11718116	—	2	2	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	1	no	—
24	C₁₃×#124	—	2	2	no	—
25	C₁₃×#124	C₈	1	4	yes	complemented
26	C₄×Q₈×C₁₃	—	2	2	no	—
27	C₄×Q₈×C₁₃	—	2	2	no	—
28	C₄×D₄×C₁₃	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₁₃	—	2	1	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₁₃×#95	—	2	2	no	—
45	C₁₃×#95	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄⋊#1055	—	2	2	no	—
48	C₄×#1055	—	2	8	no	—
49	C₄×#1055	—	2	2	no	—
50	C₄×#1055	D₄	1	4	yes	—
51	#9639	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	—	2	1	no	—
53	C₂×Q₈×C₁₃	—	2	1	no	—
54	C₂×Q₈×C₁₃	—	2	2	no	—
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	1	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	—	4	4	no	—
67	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	—	2	2	no	—
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₂²⋊C₄	—	2	8	no	—
76	C₁₃×C₂²⋊C₄	—	4	4	no	—
77	C₁₃×C₂²⋊C₄	—	4	2	no	—
78	C₄²×C₁₃	—	2	4	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄²×C₁₃	—	4	1	no	—
82	C₄²×C₁₃	—	4	1	no	—
83	C₄²×C₁₃	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	M₁₆	1	2	yes	—
85	C₄²×C₁₃	C₂×D₄	1	2	yes	—
86	C₄²×C₁₃	#41	1	2	yes	—
87	C₂×#1055	—	2	8	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂×#1055	—	4	4	no	—
91	C₄×#124	D₁₃	1	1	yes	characteristic
92	#1831	—	13	1	no	—
93	C₄²×C₈	—	13	1	no	—
94	C₂×Q₈⋊C₈	—	13	2	no	—
95	C₂×D₄⋊C₈	—	13	2	no	—
96	C₂³×C₁₃	—	2	2	no	—
97	C₂³×C₁₃	—	4	1	no	—
98	C₂³×C₁₃	—	4	1	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₁₃	—	2	4	no	—
103	Q₈×C₁₃	—	4	2	no	—
104	D₄×C₁₃	—	2	4	no	—
105	D₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	8	2	no	—
119	C₂×C₄×C₁₃	—	8	2	no	—
120	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
121	C₂×C₄×C₁₃	#104	1	4	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	#121	1	2	yes	—
124	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#131	1	2	yes	—
129	C₂×C₄×C₁₃	#135	1	4	yes	—
130	#1055	—	2	8	no	—
131	#1055	—	8	2	no	—
132	C₂×#124	Dic₁₃	1	1	yes	characteristic
133	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
134	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	13	2	no	—
136	C₂×C₄×C₈	—	13	2	no	—
137	C₂×C₄×C₈	—	26	2	no	—
138	#391	Dic₁₃	1	1	yes	characteristic
139	#390	Dic₁₃	1	1	yes	characteristic
140	#388	—	2	2	no	—
141	#382	Dic₁₃	1	1	yes	characteristic
142	C₄×C₂²⋊C₄	—	2	2	no	—
143	C₄³	C₂×D₁₃	1	1	yes	characteristic
144	#336	—	26	2	no	—
145	Q₈⋊C₈	—	26	4	no	—
146	D₄⋊C₈	—	26	4	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	—	4	4	no	—
149	C₂²×C₁₃	—	8	1	no	—
150	C₂²×C₁₃	—	8	1	no	—
151	C₂²×C₁₃	—	8	2	no	—
152	C₂²×C₁₃	#390	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₂²×C₁₃	#408	1	1	yes	characteristic
155	C₂²×C₁₃	#416	1	2	yes	—
156	C₂²×C₁₃	C₂×#104	1	2	yes	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	2	4	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	4	2	no	—
161	C₄×C₁₃	—	8	1	no	—
162	C₄×C₁₃	—	8	1	no	—
163	C₄×C₁₃	C₈×D₄	1	2	yes	—
164	C₄×C₁₃	#436	1	2	yes	—
165	C₄×C₁₃	#488	1	2	yes	—
166	C₄×C₁₃	#495	1	2	yes	—
167	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂²×D₄	C₂×Dic₁₃	1	1	yes	characteristic
169	C₂³×C₄	—	2	1	no	—
170	C₂²×C₈	—	26	2	no	—
171	C₂²×C₈	—	26	2	no	—
172	#124	—	2	2	no	—
173	#124	#1055	1	4	yes	complemented
174	C₄×Q₈	—	2	2	no	—
175	C₄×Q₈	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	2	1	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
188	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
189	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
190	C₄⋊C₈	—	26	2	no	—
191	C₄⋊C₈	—	26	2	no	—
192	C₄×C₈	—	13	4	no	—
193	C₄×C₈	—	26	8	no	—
194	C₄×C₈	—	26	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₂×C₁₃	—	8	2	no	—
198	C₂×C₁₃	#1476	1	2	yes	—
199	C₂×C₁₃	#1494	1	2	yes	—
200	C₂×C₁₃	#1954	1	1	yes	characteristic
201	C₂×C₁₃	#1958	1	1	yes	characteristic
202	C₂×C₁₃	#1962	1	1	yes	characteristic
203	C₂⁴	—	2	1	no	—
204	C₂×Q₈	—	2	1	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	C₂×#1055	1	2	yes	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂×#1055	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
219	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂×C₈	—	26	8	no	—
224	C₂×C₈	—	26	4	no	—
225	C₂×C₈	—	52	2	no	—
226	C₂×C₈	—	52	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	C₂×#1055	1	2	yes	—
239	C₄²	#6099	1	2	yes	—
240	C₄²	D₄×D₁₃	1	2	yes	—
241	C₄²	#6129	1	2	yes	—
242	C₁₃	#9639	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	2	2	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	4	2	no	—
251	D₄	—	2	4	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	C₂²⋊#1055	1	4	yes	—
268	C₂×C₄	#88077	1	4	yes	—
269	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
270	C₂×C₄	C₂×#6099	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
272	C₂×C₄	#88190	1	1	yes	characteristic
273	C₂×C₄	#88192	1	2	yes	—
274	C₂×C₄	#88194	1	1	yes	characteristic
275	C₂×C₄	#88202	1	2	yes	—
276	C₂×C₄	#88204	1	4	yes	—
277	C₈	—	26	8	no	—
278	C₈	—	104	2	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	8	1	no	—
282	C₂²	—	8	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
285	C₂²	#11718882	1	1	yes	characteristic
286	C₂²	#11718894	1	2	yes	—
287	C₂²	#11718895	1	1	yes	characteristic
288	C₂²	C₂×#88077	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	1	no	—
294	C₄	—	8	1	no	—
295	C₄	D₄×#1055	1	2	yes	—
296	C₄	#11718673	1	2	yes	—
297	C₄	#11718717	1	2	yes	—
298	C₄	#11718720	1	2	yes	—
299	C₂	—	8	2	no	—
300	C₂	#13113218	1	2	yes	—
301	C₂	#13113296	1	2	yes	—
302	C₂	#13115631	1	1	yes	characteristic
303	C₂	#13115648	1	1	yes	characteristic
304	C₂	#13115665	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#124	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₃	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#124	D₁₃
hypocenter	C₁₃	#9639
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #13114953	C₂	semidirect
2. C₄²×#1055	C₂	semidirect
3. C₂×Q₈⋊#1055	C₂	semidirect
25. C₁₃×#124	C₈	semidirect
173. #124	#1055	semidirect
242. C₁₃	#9639	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
C₄²×#1055	2
C₂×Q₈⋊#1055	2
C₂×D₄⋊#1055	2
C₄×C₁₃×#124	2
#9639	13