Subgroups — #16015906

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
3	#13116596	C₂	1	4	yes	maximal, complemented
4	#13116594	C₂	1	4	yes	maximal, complemented
5	#13116591	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₄	1	1	yes	characteristic
9	C₂²×#6099	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	—	2	4	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
14	#11718762	C₄	1	2	yes	—
15	#11718762	C₄	1	2	yes	—
16	#11718762	C₂²	1	2	yes	—
17	#11718762	C₂²	1	2	yes	—
18	C₂×#88055	C₄	1	2	yes	—
19	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
20	C₂×#88052	—	2	4	no	—
21	C₂×#88052	C₂²	1	2	yes	—
22	C₂×#88052	C₂²	1	2	yes	—
23	C₂²×#6090	C₂²	1	2	yes	—
24	#11718093	C₂²	1	16	yes	complemented
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	4	yes	—
30	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₃	C₈	1	2	yes	—
35	#88180	C₈	1	2	yes	—
36	#88180	C₈	1	2	yes	—
37	#88180	C₂×C₄	1	2	yes	—
38	#88180	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
41	C₂×#6099	—	2	4	no	—
42	C₂×#6099	C₂×C₄	1	2	yes	—
43	C₂×#6099	C₂×C₄	1	2	yes	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₁₃×M₃₂	—	2	4	no	—
46	C₁₃×M₃₂	—	2	4	no	—
47	C₂×C₁₃×C₁₆	—	2	4	no	—
48	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
49	#88055	C₂×C₄	1	4	yes	—
50	#88055	C₂×C₄	1	4	yes	—
51	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
52	C₈×Dic₁₃	C₂³	1	4	yes	—
53	#88052	—	2	4	no	—
54	#88052	—	2	4	no	—
55	C₂×#6090	—	2	4	no	—
56	C₂×#6090	C₂³	1	4	yes	—
57	#33397	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
82	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
83	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
84	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
85	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
86	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
87	#6099	—	2	4	no	—
88	#6099	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	C₂²×C₄	1	4	yes	—
91	C₁₃×C₁₆	—	2	4	no	—
92	#6090	—	2	4	no	—
93	C₂²×M₃₂	—	13	1	no	—
94	C₂²×M₃₂	D₁₃	1	1	yes	characteristic
95	C₂×#405	—	13	1	no	—
96	#2151	—	13	4	no	—
97	#2151	—	13	4	no	—
98	#2150	—	13	4	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
101	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
105	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
106	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
119	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	#1055	—	2	4	no	—
125	C₂²×M₁₆	—	13	1	no	—
126	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂×#117	—	13	1	no	—
128	C₂×M₃₂	—	2	4	no	—
129	C₂×M₃₂	—	13	2	no	—
130	C₂×M₃₂	—	13	2	no	—
131	C₂×M₃₂	—	26	4	no	—
132	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
133	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
134	C₂²×C₁₆	—	13	2	no	—
135	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
136	#405	—	13	2	no	—
137	#405	—	13	2	no	—
138	#405	—	13	2	no	—
139	#405	—	13	2	no	—
140	C₂×#97	—	13	2	no	—
141	C₂×C₄×C₈	—	13	2	no	—
142	C₄⋊C₁₆	—	13	16	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	4	4	no	—
147	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
148	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
149	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
150	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
151	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
152	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
153	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
154	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
156	Dic₁₃	—	2	8	no	—
157	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
158	C₂×M₁₆	—	13	2	no	—
159	C₂×M₁₆	—	13	2	no	—
160	C₂×M₁₆	—	26	4	no	—
161	C₂²×C₈	—	13	2	no	—
162	C₂²×C₈	Dic₂₆	1	4	yes	—
163	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
170	#117	—	13	2	no	—
171	#117	—	13	2	no	—
172	#117	—	13	2	no	—
173	#117	—	13	2	no	—
174	C₂×C₄⋊C₄	—	13	2	no	—
175	C₂×C₂²⋊C₄	—	13	2	no	—
176	C₂×C₄²	—	13	2	no	—
177	M₃₂	—	2	4	no	—
178	M₃₂	—	2	4	no	—
179	M₃₂	—	26	4	no	—
180	M₃₂	—	26	4	no	—
181	C₂×C₁₆	—	2	4	no	—
182	C₂×C₁₆	—	13	4	no	—
183	C₂×C₁₆	—	26	4	no	—
184	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
185	#97	—	13	4	no	—
186	#97	—	13	4	no	—
187	C₄×C₈	—	13	4	no	—
188	C₄×C₈	—	13	4	no	—
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
193	C₂×C₁₃	#3407	1	2	yes	—
194	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	—	13	4	no	—
196	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
199	C₂²×C₄	#6094	1	4	yes	—
200	C₂²×C₄	#6101	1	4	yes	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	M₁₆	—	26	4	no	—
205	M₁₆	—	26	4	no	—
206	C₂×C₈	—	13	4	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	#6101	1	4	yes	—
209	C₂×C₈	#6101	1	4	yes	—
210	C₂×C₈	#6101	1	4	yes	—
211	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
212	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
213	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
217	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
219	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
220	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
221	C₄⋊C₄	—	13	4	no	—
222	C₄⋊C₄	—	13	4	no	—
223	C₂²⋊C₄	—	13	4	no	—
224	C₂²⋊C₄	—	13	4	no	—
225	C₄²	—	13	4	no	—
226	C₄²	—	13	4	no	—
227	C₁₆	—	2	4	no	—
228	C₁₆	—	26	4	no	—
229	C₁₃	#33397	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	4	2	no	—
231	C₂³	#88054	1	4	yes	—
232	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
233	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
234	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
235	C₂×C₄	—	13	8	no	—
236	C₂×C₄	—	52	4	no	—
237	C₂×C₄	#88054	1	4	yes	—
238	C₂×C₄	#88054	1	4	yes	—
239	C₂×C₄	#88054	1	4	yes	—
240	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂×#6094	1	2	yes	—
244	C₂×C₄	C₂×#6094	1	2	yes	—
245	C₂×C₄	C₂×#6094	1	2	yes	—
246	C₂×C₄	C₂×#6101	1	2	yes	—
247	C₂×C₄	C₂×#6101	1	2	yes	—
248	C₂×C₄	C₂×#6101	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
250	C₈	—	26	4	no	—
251	C₈	C₂×#6101	1	2	yes	—
252	C₈	C₂×#6101	1	2	yes	—
253	C₈	C₂×#6101	1	2	yes	—
254	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
255	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₂×#88054	1	2	yes	—
261	C₂²	C₂×#88054	1	2	yes	—
262	C₂²	C₂×#88054	1	2	yes	—
263	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
264	C₄	—	26	8	no	—
265	C₄	C₂×#88054	1	2	yes	—
266	C₄	C₂×#88054	1	2	yes	—
267	C₄	C₂×#88054	1	2	yes	—
268	C₄	C₂²×#6094	1	1	yes	characteristic
269	C₄	C₂²×#6101	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	C₂²×#88054	1	1	yes	characteristic
274	C₂	#13127938	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂²×C₁₃×M₃₂	C₂
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	#33397
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13116596	C₂	semidirect
4. #13116594	C₂	semidirect
5. #13116591	C₂	semidirect
24. #11718093	C₂²	semidirect
229. C₁₃	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
C₂×#11718762	2
#13116596	2
#13116594	2
#13116591	2
C₂²×C₁₃×M₃₂	2
#33397	13