Subgroups — #16016004

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
2	#13116644	C₂	1	4	yes	maximal, complemented
3	#13116641	C₂	1	4	yes	maximal, complemented
4	#13116591	C₂	1	2	yes	maximal, complemented
5	#13116446	C₂	1	2	yes	maximal
6	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#6094	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88126	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88055	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	C₂×#88043	C₄	1	1	yes	characteristic
17	#11718350	C₄	1	2	yes	—
18	#11718350	C₄	1	2	yes	—
19	#11718350	C₂²	1	2	yes	—
20	#11718350	C₂²	1	2	yes	—
21	C₁₃×#346	C₂²	1	2	yes	complemented
22	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
23	#11718099	C₂²	1	8	yes	—
24	#11718093	C₂²	1	8	yes	complemented
25	#11718053	C₂²	1	2	yes	complemented
26	C₄×#6090	C₂²	1	2	yes	complemented
27	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#6103	C₈	1	2	yes	—
32	C₂×#6101	C₈	1	2	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂×#6094	—	2	4	no	—
37	C₂×#6094	C₂×C₄	1	2	yes	—
38	C₂×#6094	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	—	2	4	no	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂³	1	2	yes	—
42	#88126	C₈	1	4	yes	—
43	#88126	C₂×C₄	1	2	yes	—
44	#88126	C₂×C₄	1	2	yes	—
45	C₁₃×M₃₂	—	2	4	no	—
46	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	#88055	C₂×C₄	1	2	yes	—
52	#88055	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₁₃	C₂³	1	2	yes	—
55	#88052	—	2	4	no	—
56	C₂×#6090	C₂³	1	4	yes	—
57	#88043	C₂×C₄	1	2	yes	—
58	#88043	C₂×C₄	1	2	yes	—
59	#33419	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₁₃	—	2	2	no	—
66	C₂×C₄×D₁₃	—	2	2	no	—
67	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	#41	1	4	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×C₈	1	2	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	#6103	C₂×C₈	1	4	yes	—
79	#6103	C₂×C₈	1	4	yes	—
80	#6101	C₂×C₈	1	4	yes	—
81	#6101	C₂×C₈	1	4	yes	—
82	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
83	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	C₂²×C₄	1	2	yes	—
87	C₂×#1055	C₂²×C₄	1	2	yes	—
88	#6094	—	2	4	no	—
89	#6094	—	2	4	no	—
90	C₈×D₁₃	—	2	4	no	—
91	C₈×D₁₃	—	2	4	no	—
92	C₁₃×C₁₆	—	2	4	no	—
93	#6090	—	2	4	no	—
94	C₂×#405	—	13	1	no	—
95	#2164	—	13	4	no	—
96	#2164	—	13	4	no	—
97	#2150	—	13	2	no	—
98	#2112	—	13	2	no	—
99	C₄×M₃₂	—	13	1	no	—
100	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
102	C₂²×D₁₃	—	2	2	no	—
103	C₂²×D₁₃	—	4	2	no	—
104	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	#117	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	2	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
115	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
116	C₄×D₁₃	—	2	2	no	—
117	C₄×D₁₃	—	2	2	no	—
118	C₄×D₁₃	—	2	4	no	—
119	C₈×C₁₃	#117	1	2	yes	—
120	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
121	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
122	C₈×C₁₃	C₂×#41	1	2	yes	—
123	#1055	—	2	2	no	—
124	#1055	—	2	2	no	—
125	C₂²×M₁₆	—	13	1	no	—
126	C₂³×C₈	—	13	1	no	—
127	C₂×#117	—	13	1	no	—
128	C₂×M₃₂	—	13	2	no	—
129	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
130	#405	—	13	2	no	—
131	#405	—	13	2	no	—
132	#405	—	13	2	no	—
133	#405	—	13	2	no	—
134	C₂×#97	—	13	1	no	—
135	C₂×#97	—	13	1	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	C₄⋊C₁₆	—	13	8	no	—
139	C₂²⋊C₁₆	—	13	8	no	—
140	#346	—	13	2	no	—
141	#346	C₂×D₁₃	1	2	yes	complemented
142	C₄×C₁₆	—	13	2	no	—
143	C₄×C₁₆	C₂×D₁₃	1	2	yes	complemented
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	#431	1	2	yes	—
146	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₁₃	—	2	4	no	—
148	C₂×D₁₃	—	2	2	no	—
149	C₂×D₁₃	—	4	4	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	#431	1	2	yes	—
153	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
154	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
155	Dic₁₃	—	2	4	no	—
156	Dic₁₃	—	2	2	no	—
157	C₂³×C₄	—	13	1	no	—
158	C₂×M₁₆	—	13	2	no	—
159	C₂×M₁₆	—	13	2	no	—
160	C₂×M₁₆	—	26	4	no	—
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	—	26	4	no	—
166	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	#117	—	13	4	no	—
169	#117	—	13	2	no	—
170	#117	—	13	2	no	—
171	C₂×C₄⋊C₄	—	13	2	no	—
172	C₂×C₂²⋊C₄	—	13	2	no	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
175	M₃₂	—	2	4	no	—
176	M₃₂	—	26	4	no	—
177	C₂×C₁₆	—	13	4	no	—
178	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
179	#97	—	13	2	no	—
180	#97	—	13	2	no	—
181	#97	—	13	2	no	—
182	#97	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
186	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
191	C₂×C₁₃	#3407	1	2	yes	—
192	D₁₃	—	4	2	no	—
193	C₂⁴	—	13	1	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	—	13	2	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	26	2	no	—
199	C₂²×C₄	—	26	2	no	—
200	C₂²×C₄	C₈×D₁₃	1	2	yes	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	M₁₆	—	26	4	no	—
203	M₁₆	—	26	4	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	26	2	no	—
207	C₂×C₈	—	26	2	no	—
208	C₂×C₈	—	26	4	no	—
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	#6094	1	4	yes	—
211	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₈	#6127	1	4	yes	—
215	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
216	C₄⋊C₄	—	13	4	no	—
217	C₄⋊C₄	—	13	4	no	—
218	C₂²⋊C₄	—	13	4	no	—
219	C₂²⋊C₄	—	13	4	no	—
220	C₄²	—	13	2	no	—
221	C₄²	—	13	2	no	—
222	C₄²	C₈×D₁₃	1	2	yes	—
223	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
225	C₁₆	—	2	4	no	—
226	C₁₆	—	26	4	no	—
227	C₁₃	#33419	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	13	2	no	—
229	C₂³	—	26	2	no	—
230	C₂³	—	52	2	no	—
231	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	13	4	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	26	2	no	—
236	C₂×C₄	—	26	2	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	26	4	no	—
239	C₂×C₄	—	52	2	no	—
240	C₂×C₄	#88126	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#6094	1	2	yes	—
245	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
246	C₈	—	26	2	no	—
247	C₈	—	26	2	no	—
248	C₈	#88126	1	2	yes	—
249	C₈	C₂×#6094	1	2	yes	—
250	C₈	C₂×#6094	1	2	yes	—
251	C₈	C₂×#6127	1	2	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	26	4	no	—
254	C₂²	—	26	2	no	—
255	C₂²	—	52	4	no	—
256	C₂²	#11718346	1	2	yes	—
257	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	26	4	no	—
261	C₄	—	26	2	no	—
262	C₄	#11718346	1	2	yes	—
263	C₄	C₂×#88126	1	1	yes	characteristic
264	C₄	C₂²×#6094	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	52	2	no	—
267	C₂	C₂×#11718346	1	1	yes	characteristic
268	C₂	#13127938	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₄×C₁₃×M₃₂	C₂
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33419
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13116644	C₂	semidirect
3. #13116641	C₂	semidirect
4. #13116591	C₂	semidirect
6. C₄×#88052	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
21. C₁₃×#346	C₂²	semidirect
22. C₄×C₁₃×C₁₆	C₂²	semidirect
24. #11718093	C₂²	semidirect
25. #11718053	C₂²	semidirect
26. C₄×#6090	C₂²	semidirect
100. C₄×M₃₂	D₁₃	semidirect
141. #346	C₂×D₁₃	semidirect
143. C₄×C₁₆	C₂×D₁₃	semidirect
227. C₁₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718350	2
#13116644	2
#13116641	2
#13116591	2
#13116446	2
C₄×#88052	2
C₄×C₁₃×M₃₂	2
#33419	13