Subgroups — C₂²×#11718346 #16726624

Order 3328.

3086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718346	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88054	C₂	1	2	yes	maximal, complemented
4	C₂²×#88060	C₂	1	2	yes	maximal
5	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₁₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₁₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂³×#1055	C₂²	1	2	yes	—
16	C₂×C₄²×D₁₃	C₄	1	12	yes	—
17	C₂×C₄²×D₁₃	C₂²	1	12	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
19	C₂×#88060	C₂²	1	24	yes	—
20	C₂×#88054	C₂²	1	24	yes	complemented
21	C₂×C₄×#1055	C₂²	1	12	yes	complemented
22	#11718346	C₂²	1	64	yes	direct product factor
23	C₂⁴×D₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₃	—	2	8	no	—
31	C₂²×C₄×D₁₃	—	2	8	no	—
32	C₂²×C₄×D₁₃	C₈	1	16	yes	complemented
33	C₂²×C₄×D₁₃	C₂×C₄	1	6	yes	—
34	C₂²×C₄×D₁₃	C₂×C₄	1	12	yes	—
35	C₂²×C₄×D₁₃	C₂×C₄	1	12	yes	—
36	C₂²×C₄×D₁₃	C₂³	1	6	yes	—
37	C₂²×C₈×C₁₃	—	2	8	no	—
38	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
42	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
43	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
45	C₂²×#1055	—	2	8	no	—
46	C₂²×#1055	C₂³	1	12	yes	—
47	C₄²×D₁₃	C₂×C₄	1	48	yes	—
48	C₄²×D₁₃	C₂³	1	16	yes	—
49	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
50	#88060	C₂³	1	32	yes	—
51	#88054	C₂³	1	32	yes	complemented
52	C₄×#1055	C₂³	1	16	yes	complemented
53	C₂²×#431	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂³×D₁₃	—	2	4	no	—
56	C₂³×D₁₃	—	2	4	no	—
57	C₂³×D₁₃	C₂×C₈	1	8	yes	—
58	C₂³×D₁₃	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
60	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₁₃	#41	1	8	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
66	C₂²×Dic₁₃	—	2	4	no	—
67	C₂²×Dic₁₃	—	2	4	no	—
68	C₂²×Dic₁₃	C₂×C₈	1	8	yes	—
69	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
70	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
71	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
72	C₂×C₄×D₁₃	—	2	24	no	—
73	C₂×C₄×D₁₃	—	2	48	no	—
74	C₂×C₄×D₁₃	—	2	24	no	—
75	C₂×C₄×D₁₃	C₂×C₈	1	96	yes	complemented
76	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	16	yes	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	16	yes	—
79	C₂×C₄×D₁₃	C₂⁴	1	4	yes	—
80	C₂×C₈×C₁₃	—	2	24	no	—
81	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
83	C₄²×C₁₃	C₂⁴	1	4	yes	—
84	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
85	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
86	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
87	C₄×Dic₁₃	C₂⁴	1	4	yes	—
88	C₂×#1055	—	2	24	no	—
89	C₂×#1055	C₂⁴	1	8	yes	—
90	C₂³×C₄²	—	13	1	no	—
91	C₂×#431	—	13	24	no	—
92	C₂²×C₄⋊C₈	—	13	2	no	—
93	C₂²×C₂²⋊C₈	—	13	2	no	—
94	C₂²×C₄×C₈	—	13	1	no	—
95	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic, complemented
96	C₂³×C₁₃	#117	1	4	yes	—
97	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
98	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
99	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
100	C₂²×D₁₃	—	2	12	no	—
101	C₂²×D₁₃	—	2	12	no	—
102	C₂²×D₁₃	—	2	32	no	—
103	C₂²×D₁₃	C₂²×C₈	1	24	yes	—
104	C₂²×D₁₃	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₁₃	#117	1	12	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	24	yes	—
107	C₂×C₄×C₁₃	C₂×M₁₆	1	24	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
110	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
111	C₂×C₄×C₁₃	C₂×#41	1	12	yes	—
112	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	12	no	—
114	C₂×Dic₁₃	—	2	12	no	—
115	C₂×Dic₁₃	C₂²×C₈	1	24	yes	—
116	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
117	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
118	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
119	C₄×D₁₃	—	2	24	no	—
120	C₄×D₁₃	—	2	32	no	—
121	C₄×D₁₃	—	2	8	no	—
122	C₄×D₁₃	C₂²×C₈	1	64	yes	complemented
123	C₈×C₁₃	—	2	8	no	—
124	#1055	—	2	8	no	—
125	C₂⁴×C₄	—	13	1	no	—
126	C₂⁴×C₄	—	13	1	no	—
127	C₂⁴×C₄	—	13	1	no	—
128	C₂³×C₈	—	13	2	no	—
129	C₂³×C₈	C₂×D₁₃	1	2	yes	—
130	C₂²×C₄²	—	13	1	no	—
131	C₂²×C₄²	—	13	1	no	—
132	C₂²×C₄²	—	13	1	no	—
133	C₂²×C₄²	—	13	12	no	—
134	C₂²×C₄²	—	13	12	no	—
135	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
136	#431	—	13	64	no	—
137	C₂×C₄⋊C₈	—	13	24	no	—
138	C₂×C₂²⋊C₈	—	13	24	no	—
139	C₂×C₄×C₈	—	13	12	no	—
140	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	complemented
141	C₂²×C₁₃	#431	1	16	yes	complemented
142	C₂²×C₁₃	C₂×#117	1	6	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	6	yes	—
144	C₂²×C₁₃	C₂²×M₁₆	1	6	yes	—
145	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
146	C₂×D₁₃	—	2	4	no	—
147	C₂×D₁₃	—	2	48	no	—
148	C₂×D₁₃	—	2	4	no	—
149	C₂×D₁₃	C₂³×C₈	1	8	yes	—
150	C₄×C₁₃	C₂×#117	1	6	yes	—
151	C₄×C₁₃	C₂³×C₈	1	8	yes	—
152	C₄×C₁₃	C₂²×M₁₆	1	8	yes	—
153	C₄×C₁₃	C₂²×#41	1	2	yes	—
154	Dic₁₃	—	2	4	no	—
155	Dic₁₃	—	2	4	no	—
156	Dic₁₃	C₂³×C₈	1	8	yes	—
157	C₂⁵	—	13	1	no	—
158	C₂³×C₄	—	13	1	no	—
159	C₂³×C₄	—	13	16	no	—
160	C₂³×C₄	—	13	12	no	—
161	C₂³×C₄	—	13	1	no	—
162	C₂³×C₄	—	13	1	no	—
163	C₂³×C₄	—	13	6	no	—
164	C₂³×C₄	—	13	12	no	—
165	C₂³×C₄	—	13	6	no	—
166	C₂³×C₄	—	26	8	no	—
167	C₂³×C₄	—	26	8	no	—
168	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
169	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂²×C₈	—	2	8	no	—
172	C₂²×C₈	—	13	12	no	—
173	C₂²×C₈	—	26	8	no	—
174	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
175	C₂×C₄²	—	13	6	no	—
176	C₂×C₄²	—	13	12	no	—
177	C₂×C₄²	—	13	12	no	—
178	C₂×C₄²	—	13	6	no	—
179	C₂×C₄²	—	13	48	no	—
180	C₂×C₄²	—	13	16	no	—
181	C₂×C₄²	C₄×D₁₃	1	6	yes	—
182	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
183	C₄⋊C₈	—	13	32	no	—
184	C₂²⋊C₈	—	13	32	no	—
185	C₄×C₈	—	13	16	no	—
186	C₄×C₈	C₂²×D₁₃	1	16	yes	complemented
187	C₂×C₁₃	C₂×#431	1	12	yes	complemented
188	C₂×C₁₃	C₂²×#117	1	1	yes	characteristic
189	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
190	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
191	D₁₃	—	2	8	no	—
192	C₂⁴	—	13	8	no	—
193	C₂⁴	—	13	6	no	—
194	C₂⁴	—	26	4	no	—
195	C₂⁴	—	26	4	no	—
196	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	—	13	8	no	—
198	C₂²×C₄	—	13	6	no	—
199	C₂²×C₄	—	13	96	no	—
200	C₂²×C₄	—	13	16	no	—
201	C₂²×C₄	—	13	6	no	—
202	C₂²×C₄	—	13	6	no	—
203	C₂²×C₄	—	13	12	no	—
204	C₂²×C₄	—	13	16	no	—
205	C₂²×C₄	—	13	4	no	—
206	C₂²×C₄	—	26	4	no	—
207	C₂²×C₄	—	26	4	no	—
208	C₂²×C₄	—	26	24	no	—
209	C₂²×C₄	—	26	48	no	—
210	C₂²×C₄	—	26	24	no	—
211	C₂²×C₄	C₈×D₁₃	1	8	yes	—
212	C₂²×C₄	#6094	1	8	yes	—
213	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
214	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
215	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
216	C₂²×C₄	#6127	1	8	yes	—
217	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
218	C₂×C₈	—	2	24	no	—
219	C₂×C₈	—	13	8	no	—
220	C₂×C₈	—	26	24	no	—
221	C₂×C₈	C₂³×D₁₃	1	8	yes	—
222	C₄²	—	13	12	no	—
223	C₄²	—	13	16	no	—
224	C₄²	—	13	16	no	—
225	C₄²	—	13	4	no	—
226	C₄²	C₂×C₄×D₁₃	1	12	yes	—
227	C₄²	C₂³×D₁₃	1	4	yes	—
228	C₁₃	C₂²×#431	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	13	24	no	—
230	C₂³	—	13	4	no	—
231	C₂³	—	26	12	no	—
232	C₂³	—	26	12	no	—
233	C₂³	—	26	32	no	—
234	C₂³	#88126	1	4	yes	—
235	C₂³	C₂×C₈×D₁₃	1	4	yes	—
236	C₂³	C₂×#6094	1	4	yes	—
237	C₂³	C₂²×C₄×D₁₃	1	3	yes	—
238	C₂×C₄	—	13	24	no	—
239	C₂×C₄	—	13	4	no	—
240	C₂×C₄	—	13	64	no	—
241	C₂×C₄	—	13	4	no	—
242	C₂×C₄	—	13	4	no	—
243	C₂×C₄	—	26	12	no	—
244	C₂×C₄	—	26	12	no	—
245	C₂×C₄	—	26	24	no	—
246	C₂×C₄	—	26	32	no	—
247	C₂×C₄	—	26	8	no	—
248	C₂×C₄	#88126	1	12	yes	—
249	C₂×C₄	C₂×C₈×D₁₃	1	24	yes	—
250	C₂×C₄	C₂×#6094	1	24	yes	—
251	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
252	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
254	C₂×C₄	C₂×#6127	1	12	yes	—
255	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
256	C₈	—	2	8	no	—
257	C₈	—	26	8	no	—
258	C₂²	—	13	8	no	—
259	C₂²	—	26	4	no	—
260	C₂²	—	26	4	no	—
261	C₂²	—	26	48	no	—
262	C₂²	#11718346	1	16	yes	direct product factor
263	C₂²	C₂×#88126	1	6	yes	—
264	C₂²	C₂²×C₈×D₁₃	1	6	yes	—
265	C₂²	C₂²×#6094	1	6	yes	—
266	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
267	C₄	—	13	8	no	—
268	C₄	—	26	4	no	—
269	C₄	—	26	4	no	—
270	C₄	C₂×#88126	1	6	yes	—
271	C₄	C₂²×C₈×D₁₃	1	8	yes	—
272	C₄	C₂²×#6094	1	8	yes	—
273	C₄	C₂²×#6127	1	2	yes	—
274	C₂	—	26	8	no	—
275	C₂	C₂×#11718346	1	12	yes	direct product factor
276	C₂	C₂²×#88126	1	1	yes	characteristic
277	C₂	C₂³×C₈×D₁₃	1	1	yes	characteristic
278	C₂	C₂³×#6094	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁴×D₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×#431
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718346	C₂	direct (N × H)
3. C₂²×#88054	C₂	semidirect
5. C₂²×C₄×#1055	C₂	semidirect
6. C₂²×C₄×C₈×C₁₃	C₂	semidirect
18. C₂×C₄×C₈×C₁₃	C₂²	semidirect
20. C₂×#88054	C₂²	semidirect
21. C₂×C₄×#1055	C₂²	semidirect
22. #11718346	C₂²	direct (N × H)
32. C₂²×C₄×D₁₃	C₈	semidirect
49. C₄×C₈×C₁₃	C₂³	semidirect
51. #88054	C₂³	semidirect
52. C₄×#1055	C₂³	semidirect
75. C₂×C₄×D₁₃	C₂×C₈	semidirect
95. C₂²×C₄×C₈	D₁₃	semidirect
122. C₄×D₁₃	C₂²×C₈	semidirect
140. C₂×C₄×C₈	C₂×D₁₃	semidirect
141. C₂²×C₁₃	#431	semidirect
186. C₄×C₈	C₂²×D₁₃	semidirect
187. C₂×C₁₃	C₂×#431	semidirect
228. C₁₃	C₂²×#431	semidirect
262. C₂²	#11718346	direct (N × H)
275. C₂	C₂×#11718346	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×D₁₃	2
C₂×#11718346	2
C₂²×#88054	2
C₂²×#88060	2
C₂²×C₄×#1055	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#431	13

Subgroups — C22×#11718346 #16726624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718346 #16726624