Subgroups — C₂²×#11718372 #16726965

Order 3328.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₅₂	C₂	1	2	yes	maximal, complemented
3	C₂×#11718372	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88057	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88061	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₅₂	C₂²	1	2	yes	—
8	C₂³×C₄×D₁₃	—	2	2	no	—
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6103	—	2	2	no	—
13	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₂²×D₁₀₄	C₄	1	16	yes	complemented
15	C₂²×D₁₀₄	C₂²	1	12	yes	—
16	C₂×C₄×D₅₂	—	2	12	no	—
17	C₂×C₄×D₅₂	C₂²	1	24	yes	complemented
18	C₂×C₁₃×#97	C₂²	1	12	yes	complemented
19	C₂×#88061	C₂²	1	24	yes	complemented
20	C₂×#88057	C₂²	1	12	yes	complemented
21	#11718372	C₂²	1	64	yes	direct product factor
22	C₂⁴×D₁₃	—	2	2	no	—
23	C₂³×C₄×C₁₃	D₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₂²×D₅₂	—	2	6	no	—
27	C₂²×D₅₂	—	2	8	no	—
28	C₂²×D₅₂	C₂×C₄	1	16	yes	complemented
29	C₂²×D₅₂	C₂³	1	12	yes	—
30	C₂²×C₄×D₁₃	—	2	24	no	—
31	C₂²×C₄×D₁₃	—	4	16	no	—
32	C₂²×C₈×C₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
34	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
36	C₂×C₄²×C₁₃	D₄	1	6	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
38	C₂×#6103	—	2	24	no	—
39	C₂×C₄⋊Dic₁₃	—	2	6	no	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
41	C₂×D₁₀₄	C₂×C₄	1	96	yes	complemented
42	C₂×D₁₀₄	C₂³	1	16	yes	—
43	C₄×D₅₂	—	2	48	no	—
44	C₄×D₅₂	C₂³	1	32	yes	complemented
45	C₁₃×#97	C₂³	1	16	yes	complemented
46	#88061	C₂³	1	32	yes	complemented
47	#88057	C₂³	1	16	yes	complemented
48	C₂²×#442	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂³×D₁₃	—	2	16	no	—
51	C₂³×D₁₃	—	2	12	no	—
52	C₂³×D₁₃	—	4	8	no	—
53	C₂³×D₁₃	—	4	8	no	—
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₁₃	#41	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
60	C₂²×Dic₁₃	—	2	12	no	—
61	C₂²×Dic₁₃	—	4	8	no	—
62	C₂×D₅₂	—	2	48	no	—
63	C₂×D₅₂	—	2	12	no	—
64	C₂×D₅₂	—	2	24	no	—
65	C₂×D₅₂	C₂²×C₄	1	48	yes	complemented
66	C₂×D₅₂	C₂⁴	1	8	yes	—
67	C₂×C₄×D₁₃	—	2	32	no	—
68	C₂×C₄×D₁₃	—	4	96	no	—
69	C₂×C₈×C₁₃	—	2	12	no	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	complemented
71	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
73	C₄²×C₁₃	C₂×D₄	1	12	yes	—
74	C₄²×C₁₃	C₂⁴	1	4	yes	—
75	#6103	—	2	32	no	—
76	C₄⋊Dic₁₃	—	2	12	no	—
77	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
78	D₁₀₄	C₂²×C₄	1	64	yes	complemented
79	C₂³×D₈	—	13	1	no	—
80	C₂²×C₄×D₄	—	13	2	no	—
81	C₂×#442	—	13	24	no	—
82	C₂²×#106	—	13	1	no	—
83	C₂²×#102	—	13	2	no	—
84	C₂²×#97	D₁₃	1	1	yes	characteristic, complemented
85	C₂³×C₁₃	C₄×D₄	1	4	yes	—
86	C₂³×C₁₃	AGL(1,ℤ/8)	1	8	yes	—
87	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
88	C₂²×D₁₃	—	2	48	no	—
89	C₂²×D₁₃	—	2	8	no	—
90	C₂²×D₁₃	—	4	24	no	—
91	C₂²×D₁₃	—	4	24	no	—
92	C₂²×D₁₃	—	4	64	no	—
93	C₂×C₄×C₁₃	—	2	24	no	—
94	C₂×C₄×C₁₃	C₄×D₄	1	12	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
96	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
98	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
99	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
100	C₂×Dic₁₃	—	2	8	no	—
101	C₂×Dic₁₃	—	4	24	no	—
102	D₅₂	—	2	48	no	—
103	D₅₂	—	2	8	no	—
104	D₅₂	C₂³×C₄	1	16	yes	complemented
105	C₄×D₁₃	—	4	64	no	—
106	C₈×C₁₃	—	2	4	no	—
107	C₈×C₁₃	C₂³×C₄	1	8	yes	complemented
108	C₂³×D₄	—	13	2	no	—
109	C₂⁴×C₄	—	26	2	no	—
110	C₂²×D₈	—	13	16	no	—
111	C₂²×D₈	—	13	12	no	—
112	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
113	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
114	C₂×C₄×D₄	—	13	24	no	—
115	C₂×C₄×D₄	—	26	12	no	—
116	C₂²×C₄⋊C₄	—	13	2	no	—
117	C₂²×C₂²⋊C₄	—	26	2	no	—
118	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
119	#442	—	13	64	no	—
120	C₂×#106	—	13	12	no	—
121	C₂×#102	—	13	24	no	—
122	C₂×#97	C₂×D₁₃	1	12	yes	complemented
123	C₂²×C₁₃	#442	1	16	yes	complemented
124	C₂²×C₁₃	C₂×C₄×D₄	1	6	yes	—
125	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
126	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
127	C₂×D₁₃	—	2	16	no	—
128	C₂×D₁₃	—	4	96	no	—
129	C₂×D₁₃	—	4	8	no	—
130	C₂×D₁₃	—	4	8	no	—
131	C₄×C₁₃	—	2	8	no	—
132	C₄×C₁₃	C₂×C₄×D₄	1	6	yes	—
133	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
135	Dic₁₃	—	4	8	no	—
136	C₂⁵	—	26	2	no	—
137	C₂²×D₄	—	13	16	no	—
138	C₂²×D₄	—	13	12	no	—
139	C₂²×D₄	—	26	6	no	—
140	C₂²×D₄	—	26	8	no	—
141	C₂³×C₄	—	26	2	no	—
142	C₂³×C₄	—	26	24	no	—
143	C₂³×C₄	—	52	16	no	—
144	C₂³×C₄	D₅₂	1	2	yes	—
145	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
146	C₂×D₈	—	13	96	no	—
147	C₂×D₈	—	13	16	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₄×D₁₃	1	8	yes	complemented
150	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
151	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
152	C₄×D₄	—	13	32	no	—
153	C₄×D₄	—	26	48	no	—
154	C₂×C₄⋊C₄	—	13	12	no	—
155	C₂×C₄⋊C₄	—	26	6	no	—
156	C₂×C₂²⋊C₄	—	26	24	no	—
157	C₂×C₄²	D₅₂	1	6	yes	—
158	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
159	#106	—	13	16	no	—
160	#102	—	13	32	no	—
161	#97	C₂²×D₁₃	1	16	yes	complemented
162	C₂×C₁₃	C₂×#442	1	12	yes	complemented
163	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
164	C₂×C₁₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
165	D₁₃	—	4	16	no	—
166	C₂⁴	—	26	16	no	—
167	C₂⁴	—	26	12	no	—
168	C₂⁴	—	52	8	no	—
169	C₂⁴	—	52	8	no	—
170	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
171	C₂×D₄	—	13	48	no	—
172	C₂×D₄	—	13	8	no	—
173	C₂×D₄	—	26	48	no	—
174	C₂×D₄	—	26	12	no	—
175	C₂×D₄	—	26	24	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	26	12	no	—
178	C₂²×C₄	—	26	32	no	—
179	C₂²×C₄	—	52	96	no	—
180	C₂²×C₄	—	52	8	no	—
181	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
182	C₂²×C₄	C₂×D₅₂	1	12	yes	—
183	C₂²×C₄	C₂×D₅₂	1	3	yes	—
184	C₂²×C₄	#6127	1	4	yes	—
185	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
186	D₈	—	13	64	no	—
187	C₂×C₈	—	2	12	no	—
188	C₂×C₈	C₂×C₄×D₁₃	1	24	yes	complemented
189	C₂×C₈	C₂³×D₁₃	1	4	yes	—
190	C₂×C₈	C₂³×D₁₃	1	4	yes	—
191	C₄⋊C₄	—	13	8	no	—
192	C₄⋊C₄	—	26	12	no	—
193	C₂²⋊C₄	—	26	32	no	—
194	C₄²	C₂×D₅₂	1	12	yes	—
195	C₄²	C₂³×D₁₃	1	4	yes	—
196	C₁₃	C₂²×#442	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	26	48	no	—
198	C₂³	—	26	8	no	—
199	C₂³	—	52	24	no	—
200	C₂³	—	52	24	no	—
201	C₂³	—	52	64	no	—
202	C₂³	C₄×D₅₂	1	4	yes	—
203	C₂³	#88162	1	8	yes	—
204	C₂³	C₂²×D₅₂	1	3	yes	—
205	D₄	—	13	16	no	—
206	D₄	—	26	48	no	—
207	D₄	—	26	8	no	—
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	26	8	no	—
210	C₂×C₄	—	52	64	no	—
211	C₂×C₄	—	52	24	no	—
212	C₂×C₄	C₄×D₅₂	1	12	yes	—
213	C₂×C₄	C₂²×C₄×D₁₃	1	6	yes	—
214	C₂×C₄	C₂²×D₅₂	1	8	yes	—
215	C₂×C₄	C₂²×D₅₂	1	3	yes	—
216	C₂×C₄	C₂×#6127	1	6	yes	—
217	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	C₂²×C₄×D₁₃	1	8	yes	complemented
220	C₂²	—	26	16	no	—
221	C₂²	—	52	96	no	—
222	C₂²	—	52	8	no	—
223	C₂²	—	52	8	no	—
224	C₂²	#11718372	1	16	yes	direct product factor
225	C₂²	C₂×C₄×D₅₂	1	6	yes	—
226	C₂²	C₂×#88162	1	12	yes	—
227	C₂²	C₂³×D₅₂	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	52	8	no	—
230	C₄	C₂×C₄×D₅₂	1	6	yes	—
231	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
232	C₄	C₂²×#6127	1	1	yes	characteristic
233	C₂	—	52	16	no	—
234	C₂	C₂×#11718372	1	12	yes	direct product factor
235	C₂	C₂²×C₄×D₅₂	1	1	yes	characteristic
236	C₂	C₂²×#88162	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#97	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂³×D₁₀₄	C₂
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×#97	D₁₃
hypocenter	C₁₃	C₂²×#442
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₅₂	C₂	semidirect
3. C₂×#11718372	C₂	direct (N × H)
4. C₂²×#88057	C₂	semidirect
5. C₂²×#88061	C₂	semidirect
6. C₂²×C₁₃×#97	C₂	semidirect
14. C₂²×D₁₀₄	C₄	semidirect
17. C₂×C₄×D₅₂	C₂²	semidirect
18. C₂×C₁₃×#97	C₂²	semidirect
19. C₂×#88061	C₂²	semidirect
20. C₂×#88057	C₂²	semidirect
21. #11718372	C₂²	direct (N × H)
28. C₂²×D₅₂	C₂×C₄	semidirect
33. C₂²×C₈×C₁₃	C₂×C₄	semidirect
41. C₂×D₁₀₄	C₂×C₄	semidirect
44. C₄×D₅₂	C₂³	semidirect
45. C₁₃×#97	C₂³	semidirect
46. #88061	C₂³	semidirect
47. #88057	C₂³	semidirect
65. C₂×D₅₂	C₂²×C₄	semidirect
70. C₂×C₈×C₁₃	C₂²×C₄	semidirect
78. D₁₀₄	C₂²×C₄	semidirect
84. C₂²×#97	D₁₃	semidirect
104. D₅₂	C₂³×C₄	semidirect
107. C₈×C₁₃	C₂³×C₄	semidirect
122. C₂×#97	C₂×D₁₃	semidirect
123. C₂²×C₁₃	#442	semidirect
149. C₂²×C₈	C₄×D₁₃	semidirect
161. #97	C₂²×D₁₃	semidirect
162. C₂×C₁₃	C₂×#442	semidirect
188. C₂×C₈	C₂×C₄×D₁₃	semidirect
196. C₁₃	C₂²×#442	semidirect
219. C₈	C₂²×C₄×D₁₃	semidirect
224. C₂²	#11718372	direct (N × H)
234. C₂	C₂×#11718372	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×D₁₀₄	2
C₂²×C₄×D₅₂	2
C₂×#11718372	2
C₂²×#88057	2
C₂²×#88061	2
C₂²×C₁₃×#97	2
C₂²×#442	13

Subgroups — C22×#11718372 #16726965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718372 #16726965