Subgroups — C₂²×#11718357 #16732331

Order 3328.

3518 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#6095	C₂	1	2	yes	maximal, complemented
3	C₂³×D₁₀₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88129	C₂	1	2	yes	maximal, complemented
5	C₂×#11718357	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₅₂	C₂²	1	2	yes	—
8	C₂³×Dic₂₆	C₂²	1	2	yes	—
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	4	no	—
13	C₂²×Dic₅₂	—	2	8	no	—
14	C₂²×Dic₅₂	C₂²	1	12	yes	complemented
15	C₂²×D₁₀₄	—	2	8	no	—
16	C₂²×D₁₀₄	C₂²	1	12	yes	—
17	C₂²×#6095	—	2	16	no	—
18	C₂²×#6095	C₂²	1	24	yes	complemented
19	C₂×#88129	—	2	12	no	—
20	C₂×#88129	C₂²	1	24	yes	complemented
21	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
22	#11718357	C₂²	1	64	yes	direct product factor
23	C₂⁴×D₁₃	—	2	2	no	—
24	C₂³×C₄×C₁₃	D₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂³×Dic₁₃	—	2	2	no	—
27	C₂²×D₅₂	—	2	8	no	—
28	C₂²×D₅₂	—	2	8	no	—
29	C₂²×D₅₂	—	2	6	no	—
30	C₂²×D₅₂	C₂³	1	12	yes	—
31	C₂²×Dic₂₆	—	2	8	no	—
32	C₂²×Dic₂₆	—	2	8	no	—
33	C₂²×Dic₂₆	—	2	6	no	—
34	C₂²×Dic₂₆	C₂³	1	12	yes	—
35	C₂²×C₈×C₁₃	D₄	1	8	yes	—
36	C₂²×C₈×C₁₃	D₄	1	8	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
39	C₂×C₄²×C₁₃	D₄	1	6	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
41	C₂×#6103	—	2	48	no	—
42	C₂×Dic₅₂	—	2	48	no	—
43	C₂×Dic₅₂	C₂³	1	16	yes	complemented
44	C₂×D₁₀₄	—	2	48	no	—
45	C₂×D₁₀₄	C₂³	1	16	yes	—
46	C₂×#6095	—	2	96	no	—
47	C₂×#6095	C₂³	1	32	yes	complemented
48	#88129	—	2	48	no	—
49	#88129	C₂³	1	32	yes	complemented
50	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
51	C₂²×#495	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂³×D₁₃	—	2	12	no	—
54	C₂³×D₁₃	—	4	8	no	—
55	C₂³×D₁₃	—	4	16	no	—
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
62	C₂²×Dic₁₃	—	2	12	no	—
63	C₂²×Dic₁₃	—	4	8	no	—
64	C₂×D₅₂	—	2	24	no	—
65	C₂×D₅₂	—	2	24	no	—
66	C₂×D₅₂	—	2	12	no	—
67	C₂×D₅₂	—	4	24	no	—
68	C₂×D₅₂	C₂⁴	1	8	yes	—
69	C₂×Dic₂₆	—	2	24	no	—
70	C₂×Dic₂₆	—	2	24	no	—
71	C₂×Dic₂₆	—	2	12	no	—
72	C₂×Dic₂₆	—	4	24	no	—
73	C₂×Dic₂₆	C₂⁴	1	8	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	24	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	24	yes	—
76	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
78	C₄²×C₁₃	C₂×D₄	1	12	yes	—
79	C₄²×C₁₃	C₂⁴	1	4	yes	—
80	#6103	—	2	64	no	—
81	Dic₅₂	—	2	32	no	—
82	D₁₀₄	—	2	32	no	—
83	#6095	—	2	64	no	—
84	C₂³×Q₁₆	—	13	1	no	—
85	C₂³×SD₁₆	—	13	2	no	—
86	C₂³×D₈	—	13	1	no	—
87	C₂²×#124	—	13	2	no	—
88	C₂×#495	—	13	24	no	—
89	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₁₃	#127	1	4	yes	—
91	C₂³×C₁₃	#135	1	8	yes	—
92	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
93	C₂²×D₁₃	—	2	8	no	—
94	C₂²×D₁₃	—	4	24	no	—
95	C₂²×D₁₃	—	4	48	no	—
96	C₂²×D₁₃	—	8	32	no	—
97	C₂×C₄×C₁₃	—	2	24	no	—
98	C₂×C₄×C₁₃	#127	1	12	yes	—
99	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
101	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
102	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
103	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
104	C₂×Dic₁₃	—	2	8	no	—
105	C₂×Dic₁₃	—	4	24	no	—
106	D₅₂	—	2	8	no	—
107	D₅₂	—	2	8	no	—
108	D₅₂	—	4	24	no	—
109	Dic₂₆	—	2	8	no	—
110	Dic₂₆	—	2	8	no	—
111	Dic₂₆	—	4	24	no	—
112	C₈×C₁₃	C₂²×D₄	1	8	yes	—
113	C₈×C₁₃	C₂²×D₄	1	8	yes	—
114	C₂³×Q₈	—	13	2	no	—
115	C₂³×D₄	—	13	2	no	—
116	C₂²×Q₁₆	—	13	12	no	—
117	C₂²×Q₁₆	—	26	8	no	—
118	C₂²×SD₁₆	—	13	24	no	—
119	C₂²×SD₁₆	—	26	16	no	—
120	C₂²×D₈	—	13	12	no	—
121	C₂²×D₈	—	26	8	no	—
122	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
124	C₂×#124	—	13	24	no	—
125	C₂×#124	—	26	12	no	—
126	C₂²×C₂²⋊C₄	—	26	4	no	—
127	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
128	#495	—	13	64	no	—
129	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	complemented
130	C₂²×C₁₃	#495	1	16	yes	complemented
131	C₂²×C₁₃	C₂×#127	1	6	yes	—
132	C₂²×C₁₃	C₂×#135	1	12	yes	—
133	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
134	C₂×D₁₃	—	4	8	no	—
135	C₂×D₁₃	—	4	16	no	—
136	C₂×D₁₃	—	8	48	no	—
137	C₄×C₁₃	—	2	8	no	—
138	C₄×C₁₃	C₂×#127	1	6	yes	—
139	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
140	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
141	Dic₁₃	—	4	8	no	—
142	C₂⁵	—	26	2	no	—
143	C₂²×Q₈	—	13	12	no	—
144	C₂²×Q₈	—	26	6	no	—
145	C₂²×Q₈	—	26	8	no	—
146	C₂²×Q₈	—	26	8	no	—
147	C₂²×D₄	—	13	12	no	—
148	C₂²×D₄	—	26	8	no	—
149	C₂²×D₄	—	26	8	no	—
150	C₂²×D₄	—	26	6	no	—
151	C₂³×C₄	—	26	2	no	—
152	C₂³×C₄	D₅₂	1	2	yes	—
153	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
154	C₂×Q₁₆	—	13	16	no	—
155	C₂×Q₁₆	—	26	48	no	—
156	C₂×SD₁₆	—	13	32	no	—
157	C₂×SD₁₆	—	26	96	no	—
158	C₂×D₈	—	13	16	no	—
159	C₂×D₈	—	26	48	no	—
160	C₂²×C₈	D₅₂	1	8	yes	—
161	C₂²×C₈	D₅₂	1	8	yes	—
162	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
163	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
164	#124	—	13	32	no	—
165	#124	—	26	48	no	—
166	C₂×C₂²⋊C₄	—	26	48	no	—
167	C₂×C₄²	D₅₂	1	6	yes	—
168	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
169	C₄×C₈	C₂²×D₁₃	1	16	yes	complemented
170	C₂×C₁₃	C₂×#495	1	12	yes	complemented
171	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
172	C₂×C₁₃	C₂²×#135	1	2	yes	—
173	D₁₃	—	8	8	no	—
174	C₂⁴	—	26	12	no	—
175	C₂⁴	—	52	8	no	—
176	C₂⁴	—	52	16	no	—
177	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
178	C₂×Q₈	—	13	8	no	—
179	C₂×Q₈	—	26	12	no	—
180	C₂×Q₈	—	26	24	no	—
181	C₂×Q₈	—	26	24	no	—
182	C₂×Q₈	—	52	24	no	—
183	C₂×D₄	—	13	8	no	—
184	C₂×D₄	—	26	24	no	—
185	C₂×D₄	—	26	24	no	—
186	C₂×D₄	—	26	12	no	—
187	C₂×D₄	—	52	24	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	26	12	no	—
190	C₂²×C₄	—	52	8	no	—
191	C₂²×C₄	C₂×D₅₂	1	12	yes	—
192	C₂²×C₄	C₂×D₅₂	1	3	yes	—
193	C₂²×C₄	C₂×D₅₂	1	4	yes	—
194	C₂²×C₄	C₂×D₅₂	1	4	yes	—
195	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
196	Q₁₆	—	26	32	no	—
197	SD₁₆	—	26	64	no	—
198	D₈	—	26	32	no	—
199	C₂×C₈	C₂×D₅₂	1	24	yes	—
200	C₂×C₈	C₂×D₅₂	1	24	yes	—
201	C₂×C₈	C₂³×D₁₃	1	4	yes	—
202	C₂×C₈	C₂³×D₁₃	1	4	yes	—
203	C₂²⋊C₄	—	26	64	no	—
204	C₄²	C₂×D₅₂	1	12	yes	—
205	C₄²	C₂³×D₁₃	1	4	yes	—
206	C₁₃	C₂²×#495	1	1	yes	characteristic, complemented, Sylow
207	C₂³	—	26	8	no	—
208	C₂³	—	52	24	no	—
209	C₂³	—	52	48	no	—
210	C₂³	—	104	32	no	—
211	C₂³	#88128	1	4	yes	—
212	C₂³	#88158	1	8	yes	—
213	C₂³	C₂²×D₅₂	1	3	yes	—
214	Q₈	—	26	8	no	—
215	Q₈	—	26	8	no	—
216	Q₈	—	52	24	no	—
217	D₄	—	26	8	no	—
218	D₄	—	26	8	no	—
219	D₄	—	52	24	no	—
220	C₂×C₄	—	2	24	no	—
221	C₂×C₄	—	26	8	no	—
222	C₂×C₄	—	52	24	no	—
223	C₂×C₄	#88128	1	12	yes	—
224	C₂×C₄	C₂²×D₅₂	1	8	yes	—
225	C₂×C₄	C₂²×D₅₂	1	3	yes	—
226	C₂×C₄	C₂²×D₅₂	1	6	yes	—
227	C₂×C₄	C₂²×D₅₂	1	6	yes	—
228	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
229	C₈	C₂²×D₅₂	1	8	yes	—
230	C₈	C₂²×D₅₂	1	8	yes	—
231	C₂²	—	52	16	no	—
232	C₂²	—	52	8	no	—
233	C₂²	—	104	48	no	—
234	C₂²	#11718357	1	16	yes	direct product factor
235	C₂²	C₂×#88128	1	6	yes	—
236	C₂²	C₂×#88158	1	12	yes	—
237	C₂²	C₂³×D₅₂	1	1	yes	characteristic
238	C₄	—	2	8	no	—
239	C₄	—	52	8	no	—
240	C₄	C₂×#88128	1	6	yes	—
241	C₄	C₂³×D₅₂	1	1	yes	characteristic
242	C₄	C₂³×D₅₂	1	1	yes	characteristic
243	C₂	—	104	8	no	—
244	C₂	C₂×#11718357	1	12	yes	direct product factor
245	C₂	C₂²×#88128	1	1	yes	characteristic
246	C₂	C₂²×#88158	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂³×D₁₀₄	C₂
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×#495
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₅₂	C₂	semidirect
2. C₂³×#6095	C₂	semidirect
4. C₂²×#88129	C₂	semidirect
5. C₂×#11718357	C₂	direct (N × H)
6. C₂²×C₄×C₈×C₁₃	C₂	semidirect
14. C₂²×Dic₅₂	C₂²	semidirect
18. C₂²×#6095	C₂²	semidirect
20. C₂×#88129	C₂²	semidirect
21. C₂×C₄×C₈×C₁₃	C₂²	semidirect
22. #11718357	C₂²	direct (N × H)
43. C₂×Dic₅₂	C₂³	semidirect
47. C₂×#6095	C₂³	semidirect
49. #88129	C₂³	semidirect
50. C₄×C₈×C₁₃	C₂³	semidirect
89. C₂²×C₄×C₈	D₁₃	semidirect
129. C₂×C₄×C₈	C₂×D₁₃	semidirect
130. C₂²×C₁₃	#495	semidirect
169. C₄×C₈	C₂²×D₁₃	semidirect
170. C₂×C₁₃	C₂×#495	semidirect
206. C₁₃	C₂²×#495	semidirect
234. C₂²	#11718357	direct (N × H)
244. C₂	C₂×#11718357	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×Dic₅₂	2
C₂³×#6095	2
C₂³×D₁₀₄	2
C₂²×#88129	2
C₂×#11718357	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#495	13

Subgroups — C22×#11718357 #16732331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718357 #16732331