Subgroups — #15879635

Order 3328.

1428 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1055	C₂	1	2	yes	maximal, complemented
2	#13115631	C₂	1	6	yes	maximal, complemented
3	#13114816	C₂	1	3	yes	maximal, complemented
4	C₄×C₂²⋊#1055	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₄×C₁₃×#41	—	2	8	no	—
8	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₄	1	3	yes	—
11	C₂×C₄×D₄×C₁₃	C₂²	1	3	yes	—
12	C₂×C₁₃×#117	C₄	1	3	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
14	C₂×C₂²⋊#1055	C₂²	1	6	yes	—
15	C₂×C₂²⋊#1055	C₂²	1	6	yes	—
16	Q₈×#1055	—	2	8	no	—
17	C₂×C₄⋊#1055	C₂²	1	6	yes	—
18	C₂×C₄×#1055	C₂²	1	6	yes	—
19	#11718116	C₂²	1	3	yes	—
20	#11718116	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#41	—	2	4	no	—
22	C₂×C₁₃×#41	—	2	4	no	—
23	C₂×C₁₃×#41	C₈	1	8	yes	complemented
24	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₁₃	C₂×C₄	1	3	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
29	C₄×Q₈×C₁₃	—	2	4	no	—
30	C₄×Q₈×C₁₃	D₄	1	8	yes	—
31	C₄×D₄×C₁₃	—	2	12	no	—
32	C₄×D₄×C₁₃	—	2	12	no	—
33	C₁₃×#117	—	2	12	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
36	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	6	yes	—
37	C₂×C₄²×C₁₃	—	2	12	no	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
42	C₂²×#1055	C₂³	1	6	yes	—
43	C₂²×#1055	C₂³	1	2	yes	—
44	C₂²⋊#1055	—	2	12	no	—
45	C₂²⋊#1055	—	4	6	no	—
46	C₄⋊#1055	—	2	12	no	—
47	C₄×#1055	—	2	12	no	—
48	#22530	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂²×C₄	1	3	yes	—
50	C₁₃×#41	—	2	32	no	—
51	C₂×Q₈×C₁₃	C₂²⋊C₄	1	4	yes	—
52	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
53	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
54	C₂×D₄×C₁₃	—	2	6	no	—
55	C₂×D₄×C₁₃	—	2	6	no	—
56	C₂×D₄×C₁₃	C₂×C₈	1	12	yes	—
57	C₂²×C₄×C₁₃	—	2	6	no	—
58	C₂²×C₄×C₁₃	—	2	6	no	—
59	C₂²×C₄×C₁₃	—	2	6	no	—
60	C₂²×C₄×C₁₃	—	2	6	no	—
61	C₂²×C₄×C₁₃	—	2	6	no	—
62	C₂²×C₄×C₁₃	—	2	6	no	—
63	C₂²×C₄×C₁₃	—	2	3	no	—
64	C₂²×C₄×C₁₃	—	2	3	no	—
65	C₂²×C₄×C₁₃	—	4	3	no	—
66	C₂²×C₄×C₁₃	C₂×C₈	1	12	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	—	2	6	no	—
74	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
75	C₁₃×C₂²⋊C₄	—	2	12	no	—
76	C₄²×C₁₃	—	2	6	no	—
77	C₄²×C₁₃	—	2	6	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	C₂×D₄	1	12	yes	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	—	2	6	no	—
83	C₂×#1055	—	2	6	no	—
84	C₂×#1055	—	2	6	no	—
85	C₂×#1055	#41	1	4	yes	complemented
86	C₂×C₄×#41	D₁₃	1	1	yes	characteristic
87	C₂×C₈×Q₈	—	13	2	no	—
88	#1954	—	13	6	no	—
89	#1797	—	13	3	no	—
90	C₄×C₂²⋊C₈	—	13	3	no	—
91	C₂³×C₁₃	—	2	3	no	—
92	C₂³×C₁₃	—	2	3	no	—
93	C₂³×C₁₃	—	2	3	no	—
94	C₂³×C₁₃	—	2	3	no	—
95	C₂³×C₁₃	—	2	3	no	—
96	C₂³×C₁₃	—	2	3	no	—
97	C₂³×C₁₃	C₂²×C₈	1	6	yes	—
98	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	Q₈×C₁₃	C₂²⋊C₈	1	16	yes	complemented
100	D₄×C₁₃	—	2	24	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	3	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	3	no	—
107	C₂×C₄×C₁₃	—	2	24	no	—
108	C₂×C₄×C₁₃	—	2	3	no	—
109	C₂×C₄×C₁₃	—	2	3	no	—
110	C₂×C₄×C₁₃	—	2	3	no	—
111	C₂×C₄×C₁₃	—	4	12	no	—
112	C₂×C₄×C₁₃	—	4	6	no	—
113	C₂×C₄×C₁₃	—	4	6	no	—
114	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
118	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
119	C₂×C₄×C₁₃	#131	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
122	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
124	#1055	—	2	4	no	—
125	#1055	—	4	6	no	—
126	C₂²×#41	Dic₁₃	1	1	yes	characteristic
127	C₄×#41	—	2	8	no	—
128	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
129	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂×C₄×D₄	Dic₁₃	1	3	yes	—
131	C₂×C₄×D₄	C₂×D₁₃	1	3	yes	—
132	C₂×#117	Dic₁₃	1	3	yes	—
133	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
134	C₈×Q₈	—	26	8	no	—
135	C₂×C₄⋊C₈	—	13	6	no	—
136	C₂×C₂²⋊C₈	—	13	6	no	—
137	C₂×C₂²⋊C₈	—	13	6	no	—
138	C₂×C₄×C₈	—	13	6	no	—
139	#336	—	13	3	no	—
140	#336	—	13	1	no	—
141	C₂²×C₁₃	—	2	6	no	—
142	C₂²×C₁₃	—	2	6	no	—
143	C₂²×C₁₃	—	2	6	no	—
144	C₂²×C₁₃	—	4	3	no	—
145	C₂²×C₁₃	—	4	3	no	—
146	C₂²×C₁₃	—	4	3	no	—
147	C₂²×C₁₃	—	4	3	no	—
148	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
150	C₂²×C₁₃	#520	1	1	yes	characteristic
151	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
152	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
154	C₂²×C₁₃	#568	1	1	yes	characteristic
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	6	no	—
158	C₄×C₁₃	—	4	3	no	—
159	C₄×C₁₃	C₂×C₂²⋊C₈	1	12	yes	—
160	C₄×C₁₃	#548	1	2	yes	—
161	C₄×C₁₃	#550	1	2	yes	—
162	C₂×#41	—	2	4	no	—
163	C₂×#41	—	2	4	no	—
164	C₂×#41	#1055	1	8	yes	complemented
165	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
166	C₂²×D₄	C₂×Dic₁₃	1	3	yes	—
167	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
168	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
169	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
170	C₂²×C₈	—	13	2	no	—
171	C₂²×C₈	—	13	6	no	—
172	C₄×Q₈	—	2	4	no	—
173	C₄×Q₈	C₂²⋊D₁₃	1	8	yes	—
174	C₄×D₄	—	2	12	no	—
175	C₄×D₄	—	2	12	no	—
176	#117	—	2	12	no	—
177	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
178	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
179	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	6	yes	—
180	C₂×C₄²	—	2	12	no	—
181	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
182	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
183	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
185	C₄⋊C₈	—	26	12	no	—
186	C₂²⋊C₈	—	26	12	no	—
187	C₂²⋊C₈	—	52	6	no	—
188	C₄×C₈	—	26	12	no	—
189	C₂×C₁₃	—	4	6	no	—
190	C₂×C₁₃	#2341	1	1	yes	characteristic
191	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
192	C₂×C₁₃	#2880	1	1	yes	characteristic
193	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
194	C₂×C₁₃	#2971	1	1	yes	characteristic
195	C₂×C₁₃	#2988	1	1	yes	characteristic
196	C₂×C₁₃	#2991	1	1	yes	characteristic
197	C₂⁴	C₂²×Dic₁₃	1	3	yes	—
198	#41	—	2	32	no	—
199	C₂×Q₈	C₂×#1055	1	4	yes	—
200	C₂×Q₈	#6099	1	4	yes	—
201	C₂×Q₈	C₂²⋊Dic₁₃	1	4	yes	—
202	C₂×D₄	—	2	6	no	—
203	C₂×D₄	—	2	6	no	—
204	C₂×D₄	C₂×#1055	1	12	yes	—
205	C₂²×C₄	—	2	6	no	—
206	C₂²×C₄	—	2	6	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	6	no	—
210	C₂²×C₄	—	2	3	no	—
211	C₂²×C₄	—	2	3	no	—
212	C₂²×C₄	—	2	6	no	—
213	C₂²×C₄	—	4	3	no	—
214	C₂²×C₄	C₂×#1055	1	12	yes	—
215	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
216	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
219	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
220	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
221	C₂×C₈	—	13	4	no	—
222	C₂×C₈	—	26	2	no	—
223	C₂×C₈	—	26	2	no	—
224	C₂×C₈	—	26	6	no	—
225	C₂×C₈	—	26	6	no	—
226	C₂×C₈	—	26	6	no	—
227	C₄⋊C₄	—	2	6	no	—
228	C₄⋊C₄	C₂²⋊Dic₁₃	1	12	yes	—
229	C₂²⋊C₄	—	2	12	no	—
230	C₄²	—	2	6	no	—
231	C₄²	—	2	6	no	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
234	C₁₃	#22530	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	3	no	—
236	C₂³	—	2	3	no	—
237	C₂³	—	2	3	no	—
238	C₂³	—	2	3	no	—
239	C₂³	—	2	3	no	—
240	C₂³	—	2	3	no	—
241	C₂³	C₂²×#1055	1	6	yes	—
242	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
243	Q₈	C₂²⋊#1055	1	16	yes	complemented
244	D₄	—	2	24	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	3	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	3	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	4	12	no	—
256	C₂×C₄	—	4	6	no	—
257	C₂×C₄	—	4	6	no	—
258	C₂×C₄	C₂²×#1055	1	2	yes	—
259	C₂×C₄	C₂²×#1055	1	6	yes	—
260	C₂×C₄	C₂×#6099	1	6	yes	—
261	C₂×C₄	#88202	1	2	yes	—
262	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
264	C₂×C₄	#41×D₁₃	1	2	yes	—
265	C₂×C₄	#88256	1	1	yes	characteristic
266	C₂×C₄	#88257	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₂²⋊D₁₃	1	2	yes	—
268	C₈	—	26	4	no	—
269	C₈	—	52	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	6	no	—
273	C₂²	—	4	3	no	—
274	C₂²	—	4	3	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	3	no	—
277	C₂²	C₂³×#1055	1	1	yes	characteristic
278	C₂²	C₂²×#6099	1	1	yes	characteristic
279	C₂²	C₂×#88202	1	1	yes	characteristic
280	C₂²	#11719524	1	1	yes	characteristic
281	C₂²	#41×Dic₁₃	1	1	yes	characteristic
282	C₂²	#11719532	1	1	yes	characteristic
283	C₂²	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	6	no	—
287	C₄	—	4	3	no	—
288	C₄	C₂×C₂²⋊#1055	1	12	yes	—
289	C₄	#11719536	1	2	yes	—
290	C₄	#11719539	1	2	yes	—
291	C₂	—	4	6	no	—
292	C₂	#13117677	1	1	yes	characteristic
293	C₂	C₂²×C₂²⋊#1055	1	1	yes	characteristic
294	C₂	#13123285	1	1	yes	characteristic
295	C₂	#41×#1055	1	1	yes	characteristic
296	C₂	#13123856	1	1	yes	characteristic
297	C₂	#13123947	1	1	yes	characteristic
298	C₂	#13123968	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₄×C₁₃×#41	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×C₁₃×#41	C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×#41	D₁₃
hypocenter	C₁₃	#22530
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×#1055	C₂	semidirect
2. #13115631	C₂	semidirect
3. #13114816	C₂	semidirect
4. C₄×C₂²⋊#1055	C₂	semidirect
23. C₂×C₁₃×#41	C₈	semidirect
85. C₂×#1055	#41	semidirect
99. Q₈×C₁₃	C₂²⋊C₈	semidirect
164. C₂×#41	#1055	semidirect
234. C₁₃	#22530	semidirect
243. Q₈	C₂²⋊#1055	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1055	2
#13115631	2
#13114816	2
C₄×C₂²⋊#1055	2
C₂×C₄×C₁₃×#41	2
#22530	13