Subgroups — #15861009

Order 3328.

1126 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
3	#13117192	C₂	1	2	yes	maximal
4	#13114733	C₂	1	4	yes	maximal, complemented
5	C₄×#88058	C₂	1	4	yes	maximal, complemented
6	#13114556	C₂	1	2	yes	maximal
7	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂×#88180	C₂²	1	2	yes	—
12	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₁₃×#405	C₄	1	8	yes	complemented
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	#11718767	C₄	1	8	yes	complemented
17	C₂×#88058	C₄	1	8	yes	complemented
18	C₂×#88058	C₂²	1	2	yes	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88057	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₁₃	—	2	4	no	—
23	C₂²⋊C₄×Dic₁₃	—	2	4	no	—
24	C₄×C₄⋊Dic₁₃	C₂²	1	8	yes	—
25	#11718143	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×#117	C₂×C₄	1	4	yes	—
38	C₁₃×#117	Q₈	1	4	yes	—
39	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
40	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
42	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
43	#88180	—	2	4	no	—
44	#88180	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄⋊Dic₁₃	—	2	4	no	—
47	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
50	C₂×C₄×Dic₁₃	—	2	2	no	—
51	C₂×C₄×Dic₁₃	—	2	8	no	—
52	C₁₃×#97	C₂×C₄	1	8	yes	complemented
53	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
54	#88071	—	2	8	no	—
55	#88058	C₂×C₄	1	8	yes	—
56	#88058	C₂×C₄	1	8	yes	—
57	#88058	C₂×C₄	1	8	yes	complemented
58	#88057	C₂×C₄	1	8	yes	complemented
59	#20455	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂²×Dic₁₃	—	2	4	no	—
72	C₂²×Dic₁₃	—	2	8	no	—
73	C₂²×Dic₁₃	—	8	4	no	—
74	C₁₃×M₁₆	—	2	8	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₄²	1	16	yes	complemented
78	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
86	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
89	C₂²⋊Dic₁₃	—	2	8	no	—
90	C₄⋊Dic₁₃	—	2	8	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
94	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
95	C₄×Dic₁₃	—	2	8	no	—
96	C₄×Dic₁₃	—	8	8	no	—
97	C₂×#427	—	13	1	no	—
98	C₂²×#107	—	13	1	no	—
99	C₂×#405	D₁₃	1	1	yes	characteristic
100	#2274	—	13	2	no	—
101	#1776	—	13	4	no	—
102	C₄×#107	—	13	4	no	—
103	#1736	—	13	2	no	—
104	C₂³×C₁₃	—	2	2	no	—
105	C₂³×C₁₃	—	2	2	no	—
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₁₃	C₄×D₄	1	2	yes	—
108	C₂³×C₁₃	C₄×D₄	1	2	yes	—
109	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	#117	1	4	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₁₃	—	2	8	no	—
127	C₂×Dic₁₃	—	2	8	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	8	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	8	8	no	—
132	C₈×C₁₃	—	2	4	no	—
133	C₈×C₁₃	C₂×C₄²	1	4	yes	—
134	C₈×C₁₃	C₂×C₄²	1	4	yes	—
135	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×#117	—	13	2	no	—
138	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	—	13	2	no	—
140	#427	—	13	8	no	—
141	C₂×#107	—	13	8	no	—
142	C₂×#107	—	13	2	no	—
143	C₂×#107	—	13	2	no	—
144	C₂×#107	—	13	2	no	—
145	C₂×#106	—	13	2	no	—
146	#405	Dic₁₃	1	8	yes	complemented
147	C₂×#97	C₂×D₁₃	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
149	C₄×C₄⋊C₄	—	13	8	no	—
150	C₄×C₄⋊C₄	—	26	4	no	—
151	C₄×C₂²⋊C₄	—	26	4	no	—
152	#340	—	13	8	no	—
153	C₂²×C₁₃	—	2	4	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	2	4	no	—
157	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
158	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂²×C₁₃	#576	1	2	yes	—
162	C₂²×C₁₃	#578	1	2	yes	—
163	C₄×C₁₃	—	2	8	no	—
164	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
165	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
167	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
169	Dic₁₃	—	4	8	no	—
170	Dic₁₃	—	4	8	no	—
171	C₂³×C₄	—	26	2	no	—
172	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
176	C₂²×C₈	C₄×D₁₃	1	4	yes	—
177	C₂²×C₈	C₄×D₁₃	1	4	yes	—
178	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
179	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	#117	—	13	8	no	—
183	#117	—	26	4	no	—
184	#117	Dic₂₆	1	4	yes	—
185	#117	C₂×Dic₁₃	1	4	yes	—
186	C₂×C₄⋊C₄	—	13	8	no	—
187	C₂×C₄⋊C₄	—	13	4	no	—
188	C₂×C₄⋊C₄	—	13	4	no	—
189	C₂×C₄⋊C₄	—	26	2	no	—
190	C₂×C₄⋊C₄	—	26	4	no	—
191	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
192	C₂×C₂²⋊C₄	—	26	2	no	—
193	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
194	C₂×C₄²	—	26	2	no	—
195	C₂×C₄²	—	26	8	no	—
196	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
197	#107	—	13	8	no	—
198	#107	—	13	8	no	—
199	#107	—	13	8	no	—
200	#106	—	13	8	no	—
201	#97	C₂×Dic₁₃	1	8	yes	complemented
202	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
203	#95	—	26	8	no	—
204	C₂×C₁₃	—	2	4	no	—
205	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₁₃	#2913	1	2	yes	—
207	C₂×C₁₃	#2948	1	1	yes	characteristic
208	C₂×C₁₃	#2950	1	1	yes	characteristic
209	C₂×C₁₃	#2951	1	2	yes	—
210	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
211	C₂²×C₄	—	26	4	no	—
212	C₂²×C₄	—	26	4	no	—
213	C₂²×C₄	—	26	8	no	—
214	C₂²×C₄	—	104	4	no	—
215	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
218	C₂²×C₄	C₂×D₅₂	1	4	yes	—
219	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
220	C₂²×C₄	#6127	1	2	yes	—
221	C₂²×C₄	#6127	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
223	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
224	M₁₆	—	2	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₄×Dic₁₃	1	16	yes	complemented
228	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
232	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
233	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
234	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
235	C₄⋊C₄	—	13	8	no	—
236	C₄⋊C₄	—	13	8	no	—
237	C₄⋊C₄	—	26	8	no	—
238	C₄⋊C₄	—	26	4	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
241	C₂²⋊C₄	—	26	8	no	—
242	C₂²⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
243	C₄²	—	26	8	no	—
244	C₄²	—	104	8	no	—
245	C₄²	C₂×Dic₂₆	1	4	yes	—
246	C₄²	C₂²×Dic₁₃	1	4	yes	—
247	C₁₃	#20455	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₄×D₅₂	1	2	yes	—
251	C₂³	C₄×D₅₂	1	2	yes	—
252	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
253	C₂³	C₂²×D₅₂	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	26	8	no	—
257	C₂×C₄	—	26	8	no	—
258	C₂×C₄	—	52	4	no	—
259	C₂×C₄	—	52	8	no	—
260	C₂×C₄	—	52	4	no	—
261	C₂×C₄	—	104	8	no	—
262	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
263	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
264	C₂×C₄	C₄×D₅₂	1	2	yes	—
265	C₂×C₄	C₄×D₅₂	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
269	C₂×C₄	#88180	1	4	yes	—
270	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#6127	1	1	yes	characteristic
274	C₂×C₄	C₂×#6127	1	1	yes	characteristic
275	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
278	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₄×C₄⋊Dic₁₃	1	4	yes	—
284	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
285	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
286	C₂²	#11719457	1	2	yes	—
287	C₂²	#11719458	1	2	yes	—
288	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
289	C₄	—	2	8	no	—
290	C₄	—	52	8	no	—
291	C₄	—	52	8	no	—
292	C₄	C₄×C₄⋊Dic₁₃	1	4	yes	—
293	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
294	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
295	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
296	C₄	C₂×#88180	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
299	C₂	#13123436	1	2	yes	—
300	C₂	#13123689	1	1	yes	characteristic
301	C₂	#13123705	1	1	yes	characteristic
302	C₂	#13123713	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#20455
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #13114733	C₂	semidirect
5. C₄×#88058	C₂	semidirect
13. C₁₃×#405	C₄	semidirect
16. #11718767	C₄	semidirect
17. C₂×#88058	C₄	semidirect
52. C₁₃×#97	C₂×C₄	semidirect
53. C₄×C₈×C₁₃	C₂×C₄	semidirect
57. #88058	C₂×C₄	semidirect
58. #88057	C₂×C₄	semidirect
77. C₂×C₈×C₁₃	C₄²	semidirect
146. #405	Dic₁₃	semidirect
201. #97	C₂×Dic₁₃	semidirect
202. C₄×C₈	C₂×Dic₁₃	semidirect
227. C₂×C₈	C₄×Dic₁₃	semidirect
247. C₁₃	#20455	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
C₂²×#88058	2
#13117192	2
#13114733	2
C₄×#88058	2
#13114556	2
C₂×C₁₃×#405	2
#20455	13