Subgroups — C₂²×#11719215 #16769683

Order 3328.

12270 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	C₂³×#6127	C₂	1	2	yes	maximal, complemented
2	C₂×#11719215	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#88151	C₂	1	4	yes	maximal, complemented
4	C₂²×#88149	C₂	1	4	yes	maximal, complemented
5	C₂²×#88184	C₂	1	4	yes	maximal, complemented
6	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₂²⋊D₁₃	C₂²	1	4	yes	—
10	C₂²×#6127	—	2	32	no	—
11	C₂²×#6127	C₂²	1	48	yes	complemented
12	C₂³×D₅₂	C₂²	1	2	yes	—
13	C₂³×D₅₂	C₂²	1	4	yes	—
14	C₂³×C₄×D₁₃	C₂²	1	4	yes	—
15	C₂³×Dic₂₆	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	24	yes	complemented
18	C₂×#88184	C₂²	1	96	yes	complemented
19	C₂²×#6103	C₂²	1	8	yes	—
20	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
21	#11719215	C₂²	1	256	yes	direct product factor
22	C₂×#88151	C₂²	1	96	yes	complemented
23	C₂×#88149	C₂²	1	96	yes	complemented
24	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
25	C₂⁴×D₁₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₃	—	4	8	no	—
27	C₂³×C₄×C₁₃	D₄	1	16	yes	complemented
28	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	24	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
32	C₂²×C₂²⋊D₁₃	—	2	32	no	—
33	C₂²×C₂²⋊D₁₃	C₂³	1	48	yes	—
34	C₂³×Dic₁₃	C₂³	1	4	yes	—
35	C₂×#6127	—	2	384	no	—
36	C₂×#6127	C₂³	1	128	yes	complemented
37	C₂²×D₅₂	—	2	16	no	—
38	C₂²×D₅₂	—	4	16	no	—
39	C₂²×D₅₂	C₂³	1	24	yes	—
40	C₂²×D₅₂	C₂³	1	48	yes	—
41	C₂²×C₄×D₁₃	—	2	32	no	—
42	C₂²×C₄×D₁₃	C₂³	1	48	yes	—
43	C₂²×Dic₂₆	—	2	16	no	—
44	C₂²×Dic₂₆	C₂³	1	24	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	64	yes	complemented
47	#88184	C₂³	1	256	yes	complemented
48	C₂×#6103	C₂³	1	96	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
50	#88151	C₂³	1	256	yes	complemented
51	#88149	C₂³	1	256	yes	complemented
52	C₂²×#537	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	—	2	4	no	—
54	C₂⁴×C₁₃	—	2	4	no	—
55	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
56	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
57	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂³×D₁₃	—	2	16	no	—
59	C₂³×D₁₃	—	2	16	no	—
60	C₂³×D₁₃	—	2	16	no	—
61	C₂³×D₁₃	C₂⁴	1	24	yes	—
62	C₂²×C₄×C₁₃	—	2	16	no	—
63	C₂²×C₄×C₁₃	—	4	48	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	16	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	96	yes	complemented
66	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	32	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
70	C₂×C₂²⋊D₁₃	—	2	192	no	—
71	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	—
72	C₂²×Dic₁₃	—	2	16	no	—
73	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
74	#6127	—	2	512	no	—
75	C₂×D₅₂	—	2	96	no	—
76	C₂×D₅₂	—	4	96	no	—
77	C₂×D₅₂	C₂⁴	1	32	yes	—
78	C₂×D₅₂	C₂⁴	1	64	yes	—
79	C₂×C₄×D₁₃	—	2	192	no	—
80	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
81	C₂×Dic₂₆	—	2	96	no	—
82	C₂×Dic₂₆	C₂⁴	1	32	yes	—
83	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
84	#6103	C₂⁴	1	128	yes	—
85	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
86	C₂³×#41	—	13	2	no	—
87	C₂×#537	—	13	48	no	—
88	C₂²×C₂²⋊Q₈	—	13	4	no	—
89	C₂²×#121	—	13	4	no	—
90	C₂²×#121	—	13	4	no	—
91	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
92	C₂³×C₁₃	—	2	32	no	—
93	C₂³×C₁₃	—	2	12	no	—
94	C₂³×C₁₃	—	2	12	no	—
95	C₂³×C₁₃	C₂²×D₄	1	24	yes	—
96	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
97	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
98	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
99	C₂³×C₁₃	C₂⁵	1	4	yes	—
100	C₂³×C₁₃	C₂⁵	1	3	yes	—
101	C₂²×D₁₃	—	2	48	no	—
102	C₂²×D₁₃	—	2	48	no	—
103	C₂²×D₁₃	—	2	48	no	—
104	C₂²×D₁₃	—	4	64	no	—
105	C₂²×D₁₃	C₂⁵	1	16	yes	—
106	C₂×C₄×C₁₃	—	2	48	no	—
107	C₂×C₄×C₁₃	—	4	32	no	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	48	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	64	yes	complemented
110	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
111	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
112	C₂²⋊D₁₃	—	2	128	no	—
113	C₂×Dic₁₃	—	2	48	no	—
114	C₂×Dic₁₃	C₂⁵	1	16	yes	—
115	D₅₂	—	2	64	no	—
116	D₅₂	—	4	64	no	—
117	C₄×D₁₃	—	2	128	no	—
118	Dic₂₆	—	2	64	no	—
119	C₂²×#41	—	13	48	no	—
120	C₂²×#41	—	26	32	no	—
121	C₂³×Q₈	—	13	2	no	—
122	C₂³×D₄	—	13	4	no	—
123	C₂³×D₄	—	13	2	no	—
124	C₂³×D₄	—	13	4	no	—
125	C₂⁴×C₄	—	13	4	no	—
126	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
127	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
128	#537	—	13	256	no	—
129	C₂×C₂²⋊Q₈	—	13	96	no	—
130	C₂×#121	—	13	96	no	—
131	C₂×#121	—	13	96	no	—
132	C₂²×C₄⋊C₄	—	13	4	no	—
133	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
134	C₂²×C₄⋊C₄	C₂×D₁₃	1	24	yes	complemented
135	C₂²×C₂²⋊C₄	—	13	8	no	—
136	C₂²×C₁₃	—	2	4	no	—
137	C₂²×C₁₃	—	2	4	no	—
138	C₂²×C₁₃	—	2	48	no	—
139	C₂²×C₁₃	#537	1	16	yes	complemented
140	C₂²×C₁₃	C₂³×D₄	1	8	yes	—
141	C₂²×C₁₃	C₂³×D₄	1	6	yes	—
142	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	6	yes	—
143	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	6	yes	—
144	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
145	C₂×D₁₃	—	2	16	no	—
146	C₂×D₁₃	—	2	16	no	—
147	C₂×D₁₃	—	2	16	no	—
148	C₂×D₁₃	—	4	96	no	—
149	C₄×C₁₃	—	2	16	no	—
150	C₄×C₁₃	C₂³×D₄	1	16	yes	—
151	Dic₁₃	—	2	16	no	—
152	C₂⁵	—	13	4	no	—
153	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
154	C₂×#41	—	13	128	no	—
155	C₂×#41	—	26	384	no	—
156	C₂²×Q₈	—	13	24	no	—
157	C₂²×Q₈	—	26	16	no	—
158	C₂²×D₄	—	13	48	no	—
159	C₂²×D₄	—	13	24	no	—
160	C₂²×D₄	—	13	48	no	—
161	C₂²×D₄	—	26	32	no	—
162	C₂²×D₄	—	26	16	no	—
163	C₂²×D₄	—	52	16	no	—
164	C₂³×C₄	—	4	8	no	—
165	C₂³×C₄	—	13	4	no	—
166	C₂³×C₄	—	13	48	no	—
167	C₂³×C₄	—	26	32	no	—
168	C₂³×C₄	D₅₂	1	16	yes	complemented
169	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
170	C₂³×C₄	C₂²×D₁₃	1	24	yes	—
171	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
172	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
173	C₂²⋊Q₈	—	13	256	no	—
174	#121	—	13	256	no	—
175	#121	—	13	256	no	—
176	C₂×C₄⋊C₄	—	13	48	no	—
177	C₂×C₄⋊C₄	C₂²×D₁₃	1	48	yes	—
178	C₂×C₄⋊C₄	C₂²×D₁₃	1	64	yes	complemented
179	C₂×C₂²⋊C₄	—	13	96	no	—
180	C₂×C₁₃	—	2	8	no	—
181	C₂×C₁₃	C₂×#537	1	12	yes	complemented
182	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
183	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
185	D₁₃	—	4	16	no	—
186	C₂⁴	—	2	4	no	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	13	24	no	—
189	C₂⁴	—	26	16	no	—
190	C₂⁴	—	26	16	no	—
191	C₂⁴	—	26	16	no	—
192	C₂⁴	C₂×D₅₂	1	8	yes	—
193	C₂⁴	C₂³×D₁₃	1	6	yes	—
194	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
195	#41	—	26	512	no	—
196	C₂×Q₈	—	13	32	no	—
197	C₂×Q₈	—	26	96	no	—
198	C₂×D₄	—	13	64	no	—
199	C₂×D₄	—	13	32	no	—
200	C₂×D₄	—	13	64	no	—
201	C₂×D₄	—	26	192	no	—
202	C₂×D₄	—	26	96	no	—
203	C₂×D₄	—	52	96	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	4	48	no	—
206	C₂²×C₄	—	13	24	no	—
207	C₂²×C₄	—	13	64	no	—
208	C₂²×C₄	—	26	16	no	—
209	C₂²×C₄	—	26	192	no	—
210	C₂²×C₄	C₂×D₅₂	1	16	yes	—
211	C₂²×C₄	C₂×D₅₂	1	96	yes	complemented
212	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
213	C₂²×C₄	C₂³×D₁₃	1	32	yes	—
214	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
215	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
216	C₄⋊C₄	—	13	64	no	—
217	C₄⋊C₄	C₂³×D₁₃	1	64	yes	—
218	C₂²⋊C₄	—	13	128	no	—
219	C₁₃	C₂²×#537	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	2	32	no	—
221	C₂³	—	2	12	no	—
222	C₂³	—	2	12	no	—
223	C₂³	—	13	16	no	—
224	C₂³	—	26	48	no	—
225	C₂³	—	26	48	no	—
226	C₂³	—	26	48	no	—
227	C₂³	—	52	64	no	—
228	C₂³	C₂²×D₅₂	1	24	yes	—
229	C₂³	C₂²×D₅₂	1	4	yes	—
230	C₂³	#88251	1	4	yes	—
231	C₂³	#88254	1	4	yes	—
232	C₂³	C₂⁴×D₁₃	1	4	yes	—
233	C₂³	C₂⁴×D₁₃	1	3	yes	—
234	Q₈	—	26	64	no	—
235	D₄	—	26	128	no	—
236	D₄	—	26	64	no	—
237	D₄	—	52	64	no	—
238	C₂×C₄	—	2	48	no	—
239	C₂×C₄	—	4	32	no	—
240	C₂×C₄	—	13	16	no	—
241	C₂×C₄	—	26	48	no	—
242	C₂×C₄	—	26	128	no	—
243	C₂×C₄	C₂²×D₅₂	1	48	yes	—
244	C₂×C₄	C₂²×D₅₂	1	64	yes	complemented
245	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
246	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	48	no	—
250	C₂²	—	26	16	no	—
251	C₂²	—	26	16	no	—
252	C₂²	—	26	16	no	—
253	C₂²	—	52	96	no	—
254	C₂²	#11719215	1	16	yes	direct product factor
255	C₂²	C₂³×D₅₂	1	8	yes	—
256	C₂²	C₂³×D₅₂	1	6	yes	—
257	C₂²	C₂×#88251	1	6	yes	—
258	C₂²	C₂×#88254	1	6	yes	—
259	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	26	16	no	—
262	C₄	C₂³×D₅₂	1	16	yes	—
263	C₂	—	2	8	no	—
264	C₂	—	52	16	no	—
265	C₂	C₂×#11719215	1	12	yes	direct product factor
266	C₂	C₂⁴×D₅₂	1	1	yes	characteristic
267	C₂	C₂²×#88251	1	1	yes	characteristic
268	C₂	C₂²×#88254	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂²×#537
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#6127	C₂	semidirect
2. C₂×#11719215	C₂	direct (N × H)
3. C₂²×#88151	C₂	semidirect
4. C₂²×#88149	C₂	semidirect
5. C₂²×#88184	C₂	semidirect
6. C₂³×C₁₃×C₄⋊C₄	C₂	semidirect
11. C₂²×#6127	C₂²	semidirect
17. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
18. C₂×#88184	C₂²	semidirect
21. #11719215	C₂²	direct (N × H)
22. C₂×#88151	C₂²	semidirect
23. C₂×#88149	C₂²	semidirect
27. C₂³×C₄×C₁₃	D₄	semidirect
36. C₂×#6127	C₂³	semidirect
46. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
47. #88184	C₂³	semidirect
50. #88151	C₂³	semidirect
51. #88149	C₂³	semidirect
65. C₂²×C₄×C₁₃	C₂×D₄	semidirect
91. C₂³×C₄⋊C₄	D₁₃	semidirect
109. C₂×C₄×C₁₃	C₂²×D₄	semidirect
134. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
139. C₂²×C₁₃	#537	semidirect
168. C₂³×C₄	D₅₂	semidirect
178. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
181. C₂×C₁₃	C₂×#537	semidirect
211. C₂²×C₄	C₂×D₅₂	semidirect
219. C₁₃	C₂²×#537	semidirect
244. C₂×C₄	C₂²×D₅₂	semidirect
254. C₂²	#11719215	direct (N × H)
265. C₂	C₂×#11719215	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6127	2
C₂×#11719215	2
C₂²×#88151	2
C₂²×#88149	2
C₂²×#88184	2
C₂³×C₁₃×C₄⋊C₄	2
C₂²×#537	13

Subgroups — C22×#11719215 #16769683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719215 #16769683