Subgroups — C₂²×#11718586 #16726626

Order 3328.

3086 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊#1055	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊#1055	C₂	1	2	yes	maximal
4	C₂²×C₄×#1055	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₄	1	12	yes	—
11	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃	C₂²	1	12	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂³×#1055	C₂²	1	4	yes	—
16	C₂×C₂²⋊#1055	C₂²	1	24	yes	—
17	#11718586	C₂²	1	64	yes	direct product factor
18	C₂×C₄⋊#1055	C₂²	1	24	yes	complemented
19	C₂×C₄×#1055	C₂²	1	24	yes	complemented
20	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	—	2	8	no	—
22	C₂³×C₄×C₁₃	—	2	8	no	—
23	C₂³×C₄×C₁₃	C₈	1	16	yes	complemented
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	12	yes	—
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂×C₄	1	12	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	48	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
42	C₂²×#1055	—	2	16	no	—
43	C₂²×#1055	C₂³	1	24	yes	—
44	C₂²⋊#1055	C₂³	1	32	yes	—
45	C₄⋊#1055	C₂³	1	32	yes	complemented
46	C₄×#1055	C₂³	1	32	yes	complemented
47	C₂²×#431	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	—	2	4	no	—
49	C₂⁴×C₁₃	—	2	4	no	—
50	C₂⁴×C₁₃	C₂×C₈	1	8	yes	—
51	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
52	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	24	no	—
56	C₂²×C₄×C₁₃	—	2	48	no	—
57	C₂²×C₄×C₁₃	—	2	24	no	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂×C₈	1	96	yes	complemented
61	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
71	C₂²×C₄×C₁₃	#41	1	8	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	16	yes	—
75	C₄²×C₁₃	C₂²×C₄	1	16	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
78	C₄²×C₁₃	C₂⁴	1	4	yes	—
79	C₄²×C₁₃	C₂⁴	1	4	yes	—
80	C₂×#1055	—	2	48	no	—
81	C₂×#1055	C₂⁴	1	16	yes	—
82	C₂³×C₄²	D₁₃	1	1	yes	characteristic
83	C₂×#431	—	13	24	no	—
84	C₂²×C₄⋊C₈	—	13	2	no	—
85	C₂²×C₂²⋊C₈	—	13	2	no	—
86	C₂²×C₄×C₈	—	13	2	no	—
87	C₂³×C₁₃	—	2	12	no	—
88	C₂³×C₁₃	—	2	12	no	—
89	C₂³×C₁₃	—	2	32	no	—
90	C₂³×C₁₃	#117	1	4	yes	—
91	C₂³×C₁₃	C₂²×C₈	1	24	yes	—
92	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
93	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
94	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
95	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
96	C₂×C₄×C₁₃	—	2	12	no	—
97	C₂×C₄×C₁₃	—	2	12	no	—
98	C₂×C₄×C₁₃	—	2	24	no	—
99	C₂×C₄×C₁₃	—	2	32	no	—
100	C₂×C₄×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	#117	1	12	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	24	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	24	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	64	yes	complemented
105	C₂×C₄×C₁₃	C₂×M₁₆	1	24	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	4	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	C₂³×C₄	1	4	yes	—
112	C₂×C₄×C₁₃	C₂×#41	1	12	yes	—
113	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
114	#1055	—	2	16	no	—
115	C₂⁴×C₄	Dic₁₃	1	1	yes	characteristic
116	C₂⁴×C₄	Dic₁₃	1	1	yes	characteristic
117	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
118	C₂³×C₈	—	13	4	no	—
119	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
120	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
121	C₂²×C₄²	Dic₁₃	1	12	yes	—
122	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
123	C₂²×C₄²	C₂×D₁₃	1	12	yes	—
124	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
125	#431	—	13	64	no	—
126	C₂×C₄⋊C₈	—	13	24	no	—
127	C₂×C₂²⋊C₈	—	13	24	no	—
128	C₂×C₄×C₈	—	13	24	no	—
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	48	no	—
131	C₂²×C₁₃	—	2	4	no	—
132	C₂²×C₁₃	#431	1	16	yes	complemented
133	C₂²×C₁₃	C₂×#117	1	6	yes	—
134	C₂²×C₁₃	C₂³×C₈	1	8	yes	—
135	C₂²×C₁₃	C₂³×C₈	1	6	yes	—
136	C₂²×C₁₃	C₂²×M₁₆	1	6	yes	—
137	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
138	C₄×C₁₃	—	2	4	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	C₂×#117	1	6	yes	—
141	C₄×C₁₃	C₂³×C₈	1	8	yes	—
142	C₄×C₁₃	C₂³×C₈	1	8	yes	—
143	C₄×C₁₃	C₂²×M₁₆	1	8	yes	—
144	C₄×C₁₃	C₂²×#41	1	2	yes	—
145	C₂⁵	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂³×C₄	—	2	8	no	—
147	C₂³×C₄	—	2	8	no	—
148	C₂³×C₄	#1055	1	16	yes	complemented
149	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
150	C₂³×C₄	C₂×Dic₁₃	1	12	yes	—
151	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
152	C₂³×C₄	C₂×Dic₁₃	1	6	yes	—
153	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₁₃	1	12	yes	—
156	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
158	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
159	C₂²×C₈	—	13	24	no	—
160	C₂²×C₈	—	26	16	no	—
161	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
162	C₂×C₄²	C₂×Dic₁₃	1	48	yes	—
163	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
164	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
165	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
166	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
167	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
168	C₂×C₄²	C₂²×D₁₃	1	16	yes	—
169	C₄⋊C₈	—	13	32	no	—
170	C₂²⋊C₈	—	13	32	no	—
171	C₄×C₈	—	13	32	no	—
172	C₂×C₁₃	—	2	8	no	—
173	C₂×C₁₃	C₂×#431	1	12	yes	complemented
174	C₂×C₁₃	C₂²×#117	1	1	yes	characteristic
175	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
176	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
177	C₂⁴	—	2	4	no	—
178	C₂⁴	—	2	4	no	—
179	C₂⁴	C₂×#1055	1	8	yes	—
180	C₂⁴	C₂²×Dic₁₃	1	6	yes	—
181	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	48	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	2	24	no	—
187	C₂²×C₄	C₂×#1055	1	8	yes	—
188	C₂²×C₄	C₂×#1055	1	8	yes	—
189	C₂²×C₄	C₂×#1055	1	96	yes	complemented
190	C₂²×C₄	#6099	1	8	yes	—
191	C₂²×C₄	#6127	1	8	yes	—
192	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
195	C₂²×C₄	C₂²×Dic₁₃	1	16	yes	—
196	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
197	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	16	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
200	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
201	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
202	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
203	C₂×C₈	—	13	16	no	—
204	C₂×C₈	—	26	48	no	—
205	C₄²	C₂²×Dic₁₃	1	16	yes	—
206	C₄²	C₂²×Dic₁₃	1	16	yes	—
207	C₄²	C₂²×Dic₁₃	1	12	yes	—
208	C₄²	C₂²×Dic₁₃	1	12	yes	—
209	C₄²	C₂³×D₁₃	1	4	yes	—
210	C₄²	C₂³×D₁₃	1	4	yes	—
211	C₁₃	C₂²×#431	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	12	no	—
213	C₂³	—	2	12	no	—
214	C₂³	—	2	32	no	—
215	C₂³	C₂²×#1055	1	24	yes	—
216	C₂³	C₂²×#1055	1	4	yes	—
217	C₂³	C₂×#6099	1	4	yes	—
218	C₂³	#88180	1	4	yes	—
219	C₂³	C₂³×Dic₁₃	1	4	yes	—
220	C₂³	C₂³×Dic₁₃	1	3	yes	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	12	no	—
223	C₂×C₄	—	2	32	no	—
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	C₂²×#1055	1	24	yes	—
227	C₂×C₄	C₂²×#1055	1	24	yes	—
228	C₂×C₄	C₂²×#1055	1	64	yes	complemented
229	C₂×C₄	C₂×#6099	1	24	yes	—
230	C₂×C₄	#88180	1	12	yes	—
231	C₂×C₄	C₂×#6127	1	12	yes	—
232	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
233	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
234	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
235	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
236	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
237	C₂×C₄	C₂³×Dic₁₃	1	3	yes	—
238	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
239	C₈	—	26	16	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	48	no	—
243	C₂²	#11718586	1	16	yes	direct product factor
244	C₂²	C₂³×#1055	1	8	yes	—
245	C₂²	C₂³×#1055	1	6	yes	—
246	C₂²	C₂²×#6099	1	6	yes	—
247	C₂²	C₂×#88180	1	6	yes	—
248	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	2	4	no	—
251	C₄	C₂³×#1055	1	8	yes	—
252	C₄	C₂³×#1055	1	8	yes	—
253	C₄	C₂²×#6099	1	8	yes	—
254	C₄	C₂×#88180	1	6	yes	—
255	C₄	C₂²×#6127	1	2	yes	—
256	C₂	—	2	8	no	—
257	C₂	C₂×#11718586	1	12	yes	direct product factor
258	C₂	C₂²×#88180	1	1	yes	characteristic
259	C₂	C₂⁴×#1055	1	1	yes	characteristic
260	C₂	C₂³×#6099	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂²×#431
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#11718586	C₂	direct (N × H)
2. C₂²×C₄⋊#1055	C₂	semidirect
4. C₂²×C₄×#1055	C₂	semidirect
17. #11718586	C₂²	direct (N × H)
18. C₂×C₄⋊#1055	C₂²	semidirect
19. C₂×C₄×#1055	C₂²	semidirect
23. C₂³×C₄×C₁₃	C₈	semidirect
45. C₄⋊#1055	C₂³	semidirect
46. C₄×#1055	C₂³	semidirect
60. C₂²×C₄×C₁₃	C₂×C₈	semidirect
104. C₂×C₄×C₁₃	C₂²×C₈	semidirect
132. C₂²×C₁₃	#431	semidirect
148. C₂³×C₄	#1055	semidirect
173. C₂×C₁₃	C₂×#431	semidirect
189. C₂²×C₄	C₂×#1055	semidirect
211. C₁₃	C₂²×#431	semidirect
228. C₂×C₄	C₂²×#1055	semidirect
243. C₂²	#11718586	direct (N × H)
257. C₂	C₂×#11718586	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
C₂²×C₄⋊#1055	2
C₂²×C₂²⋊#1055	2
C₂²×C₄×#1055	2
C₂³×C₄²×C₁₃	2
C₂²×#431	13

Subgroups — C22×#11718586 #16726626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718586 #16726626