Subgroups — C₂²×C₄×#6122 #16808000

Order 3328.

3950 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6122	C₂	1	2	yes	maximal
2	C₂×C₄×#6122	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88110	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×#1057	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
7	C₂⁴×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6122	C₄	1	32	yes	direct product factor
10	C₂²×#6122	C₂²	1	24	yes	—
11	C₂³×#1057	C₂²	1	4	yes	—
12	C₂²×C₄×Dic₁₃	C₄	1	12	yes	—
13	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×Dic₁₃	C₂²	1	12	yes	—
18	C₄×#6122	C₂²	1	64	yes	direct product factor
19	C₂×#88110	C₂²	1	24	yes	complemented
20	C₂×C₄×#1057	C₂²	1	24	yes	complemented
21	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	—	2	8	no	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	12	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₁₃	—	2	8	no	—
27	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₁₃	C₂×C₄	1	12	yes	—
29	C₂³×Dic₁₃	C₂×C₄	1	6	yes	—
30	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×Dic₁₃	C₂×C₄	1	16	yes	—
32	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
34	C₂³×Dic₁₃	C₂³	1	6	yes	—
35	C₂×#6122	C₂×C₄	1	192	yes	direct product factor
36	C₂×#6122	C₂³	1	32	yes	—
37	C₂²×#1057	C₂×C₄	1	32	yes	complemented
38	C₂²×#1057	C₂³	1	24	yes	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	48	yes	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
42	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
43	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
46	C₂×C₄×Dic₁₃	C₂³	1	16	yes	—
47	#88110	C₂³	1	32	yes	complemented
48	C₄×#1057	C₂³	1	32	yes	complemented
49	C₂²×C₄×M₁₆	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	—	2	4	no	—
51	C₂⁴×C₁₃	—	2	4	no	—
52	C₂⁴×C₁₃	C₄²	1	8	yes	—
53	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
54	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	48	no	—
57	C₂²×C₄×C₁₃	M₁₆	1	8	yes	complemented
58	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
61	C₂²×Dic₁₃	—	2	4	no	—
62	C₂²×Dic₁₃	—	2	48	no	—
63	C₂²×Dic₁₃	C₄²	1	48	yes	—
64	C₂²×Dic₁₃	C₄²	1	8	yes	—
65	C₂²×Dic₁₃	M₁₆	1	8	yes	complemented
66	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
67	C₂²×Dic₁₃	C₂²×C₄	1	16	yes	—
68	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
69	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
70	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
71	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
72	C₂²×Dic₁₃	C₂²×C₄	1	48	yes	—
73	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
74	C₂²×Dic₁₃	C₂⁴	1	3	yes	—
75	C₂²×Dic₁₃	C₂⁴	1	4	yes	—
76	#6122	C₂²×C₄	1	128	yes	direct product factor
77	C₂×#1057	C₂²×C₄	1	96	yes	complemented
78	C₂×#1057	C₂⁴	1	16	yes	—
79	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
80	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
81	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
82	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
83	C₄×Dic₁₃	C₂⁴	1	4	yes	—
84	C₄×Dic₁₃	C₂⁴	1	4	yes	—
85	C₂³×M₁₆	—	13	2	no	—
86	C₂³×C₄²	—	13	1	no	—
87	C₂×C₄×M₁₆	—	13	24	no	—
88	C₂²×#97	—	13	2	no	—
89	C₂²×C₄×C₈	—	13	2	no	—
90	C₂³×C₁₃	—	2	12	no	—
91	C₂³×C₁₃	—	2	12	no	—
92	C₂³×C₁₃	—	2	32	no	—
93	C₂³×C₁₃	C₂×C₄²	1	24	yes	—
94	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
95	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
96	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
97	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
98	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
99	C₂×C₄×C₁₃	—	2	12	no	—
100	C₂×C₄×C₁₃	—	2	32	no	—
101	C₂×C₄×C₁₃	C₂×M₁₆	1	24	yes	complemented
102	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
104	C₂×Dic₁₃	—	2	12	no	—
105	C₂×Dic₁₃	—	2	32	no	—
106	C₂×Dic₁₃	C₂×C₄²	1	48	yes	—
107	C₂×Dic₁₃	C₂×C₄²	1	12	yes	—
108	C₂×Dic₁₃	C₂×C₄²	1	24	yes	—
109	C₂×Dic₁₃	C₂×M₁₆	1	24	yes	complemented
110	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
111	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
112	C₂×Dic₁₃	C₂³×C₄	1	12	yes	—
113	C₂×Dic₁₃	C₂³×C₄	1	16	yes	—
114	C₂×Dic₁₃	C₂³×C₄	1	3	yes	—
115	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
116	C₂×Dic₁₃	C₂⁵	1	1	yes	characteristic
117	#1057	C₂³×C₄	1	32	yes	complemented
118	C₂⁴×C₄	—	13	1	no	—
119	C₂⁴×C₄	—	13	1	no	—
120	C₂⁴×C₄	#261	1	1	yes	characteristic
121	C₂²×M₁₆	—	13	24	no	—
122	C₂²×M₁₆	—	13	32	no	—
123	C₂³×C₈	—	13	4	no	—
124	C₂²×C₄²	—	13	1	no	—
125	C₂²×C₄²	—	13	12	no	—
126	C₂²×C₄²	—	13	1	no	—
127	C₂²×C₄²	—	13	12	no	—
128	C₂²×C₄²	—	13	1	no	—
129	C₂²×C₄²	—	13	1	no	—
130	C₄×M₁₆	—	13	64	no	—
131	C₂×#97	—	13	24	no	—
132	C₂×C₄×C₈	—	13	24	no	—
133	C₂²×C₁₃	—	2	48	no	—
134	C₂²×C₁₃	—	2	4	no	—
135	C₂²×C₁₃	—	2	4	no	—
136	C₂²×C₁₃	C₄×M₁₆	1	16	yes	complemented
137	C₂²×C₁₃	C₂²×C₄²	1	8	yes	—
138	C₂²×C₁₃	C₂²×C₄²	1	6	yes	—
139	C₂²×C₁₃	C₂²×M₁₆	1	6	yes	—
140	C₂²×C₁₃	C₂²×M₁₆	1	6	yes	—
141	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	C₂²×M₁₆	1	8	yes	complemented
144	Dic₁₃	—	2	4	no	—
145	Dic₁₃	C₂²×C₄²	1	6	yes	—
146	Dic₁₃	C₂²×C₄²	1	8	yes	—
147	Dic₁₃	C₂²×M₁₆	1	8	yes	complemented
148	Dic₁₃	C₂⁴×C₄	1	2	yes	—
149	C₂⁵	C₂×#261	1	1	yes	characteristic
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	—	13	16	no	—
152	C₂³×C₄	—	13	1	no	—
153	C₂³×C₄	—	13	6	no	—
154	C₂³×C₄	—	13	1	no	—
155	C₂³×C₄	—	13	6	no	—
156	C₂³×C₄	—	13	1	no	—
157	C₂³×C₄	—	13	12	no	—
158	C₂³×C₄	—	13	1	no	—
159	C₂³×C₄	—	26	8	no	—
160	C₂³×C₄	C₂×#261	1	1	yes	characteristic
161	C₂³×C₄	C₂×#261	1	12	yes	—
162	C₂³×C₄	C₂×#261	1	1	yes	characteristic
163	C₂×M₁₆	—	13	32	no	—
164	C₂×M₁₆	—	13	192	no	—
165	C₂²×C₈	—	13	24	no	—
166	C₂²×C₈	—	13	32	no	—
167	C₂×C₄²	—	13	12	no	—
168	C₂×C₄²	—	13	6	no	—
169	C₂×C₄²	—	13	6	no	—
170	C₂×C₄²	—	13	48	no	—
171	C₂×C₄²	—	13	12	no	—
172	C₂×C₄²	—	13	6	no	—
173	C₂×C₄²	—	13	6	no	—
174	C₂×C₄²	—	13	16	no	—
175	#97	—	13	32	no	—
176	C₄×C₈	—	13	32	no	—
177	C₂×C₁₃	—	2	8	no	—
178	C₂×C₁₃	C₂×C₄×M₁₆	1	12	yes	complemented
179	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
180	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
181	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
182	C₂⁴	—	2	4	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	C₄×#261	1	8	yes	—
185	C₂⁴	C₂²×#261	1	6	yes	—
186	C₂⁴	C₂²×#261	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	48	no	—
189	C₂²×C₄	—	13	8	no	—
190	C₂²×C₄	—	13	8	no	—
191	C₂²×C₄	—	13	48	no	—
192	C₂²×C₄	—	13	48	no	—
193	C₂²×C₄	—	13	8	no	—
194	C₂²×C₄	—	13	3	no	—
195	C₂²×C₄	—	13	4	no	—
196	C₂²×C₄	—	13	6	no	—
197	C₂²×C₄	—	13	12	no	—
198	C₂²×C₄	—	13	3	no	—
199	C₂²×C₄	—	13	6	no	—
200	C₂²×C₄	—	13	16	no	—
201	C₂²×C₄	—	13	6	no	—
202	C₂²×C₄	—	26	48	no	—
203	C₂²×C₄	—	26	4	no	—
204	C₂²×C₄	#6122	1	8	yes	direct product factor
205	C₂²×C₄	C₂²×#261	1	6	yes	—
206	C₂²×C₄	C₂²×#261	1	16	yes	—
207	C₂²×C₄	C₂²×#261	1	6	yes	—
208	M₁₆	—	13	128	no	—
209	C₂×C₈	—	13	16	no	—
210	C₂×C₈	—	13	96	no	—
211	C₄²	—	13	4	no	—
212	C₄²	—	13	4	no	—
213	C₄²	—	13	16	no	—
214	C₄²	—	13	12	no	—
215	C₄²	—	13	12	no	—
216	C₄²	—	13	16	no	—
217	C₁₃	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	2	12	no	—
219	C₂³	—	2	32	no	—
220	C₂³	—	2	12	no	—
221	C₂³	D₁₃⋊M₁₆	1	4	yes	—
222	C₂³	C₂×C₄×#261	1	24	yes	—
223	C₂³	C₂×C₄×#261	1	4	yes	—
224	C₂³	C₂×#6122	1	4	yes	—
225	C₂³	C₂³×#261	1	4	yes	—
226	C₂³	C₂³×#261	1	3	yes	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	13	24	no	—
230	C₂×C₄	—	13	12	no	—
231	C₂×C₄	—	13	16	no	—
232	C₂×C₄	—	13	48	no	—
233	C₂×C₄	—	13	12	no	—
234	C₂×C₄	—	13	24	no	—
235	C₂×C₄	—	13	1	no	—
236	C₂×C₄	—	13	3	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	4	no	—
240	C₂×C₄	—	26	32	no	—
241	C₂×C₄	—	26	12	no	—
242	C₂×C₄	C₂×#6122	1	24	yes	direct product factor
243	C₂×C₄	C₂³×#261	1	4	yes	—
244	C₂×C₄	C₂³×#261	1	4	yes	—
245	C₈	—	13	32	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	48	no	—
249	C₂²	C₄×#6122	1	16	yes	direct product factor
250	C₂²	C₂×D₁₃⋊M₁₆	1	6	yes	—
251	C₂²	C₂²×C₄×#261	1	8	yes	—
252	C₂²	C₂²×C₄×#261	1	6	yes	—
253	C₂²	C₂²×#6122	1	6	yes	—
254	C₂²	C₂⁴×#261	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	13	2	no	—
257	C₄	—	13	6	no	—
258	C₄	—	13	8	no	—
259	C₄	—	13	8	no	—
260	C₄	—	26	4	no	—
261	C₄	C₂²×#6122	1	8	yes	direct product factor
262	C₂	—	2	8	no	—
263	C₂	C₂×C₄×#6122	1	12	yes	direct product factor
264	C₂	C₂³×C₄×#261	1	1	yes	characteristic
265	C₂	C₂³×#6122	1	1	yes	characteristic
266	C₂	C₂²×D₁₃⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#261
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂²	C₂⁴×#261
Fitting	C₂⁴×C₄×C₁₃	C₄
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×Dic₁₃	C₂⁵
hypercenter	C₂⁴×C₄	#261
hypocenter	C₁₃	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#6122	C₂	direct (N × H)
3. C₂²×#88110	C₂	semidirect
4. C₂²×C₄×#1057	C₂	semidirect
9. C₂²×#6122	C₄	direct (N × H)
18. C₄×#6122	C₂²	direct (N × H)
19. C₂×#88110	C₂²	semidirect
20. C₂×C₄×#1057	C₂²	semidirect
35. C₂×#6122	C₂×C₄	direct (N × H)
37. C₂²×#1057	C₂×C₄	semidirect
47. #88110	C₂³	semidirect
48. C₄×#1057	C₂³	semidirect
57. C₂²×C₄×C₁₃	M₁₆	semidirect
65. C₂²×Dic₁₃	M₁₆	semidirect
76. #6122	C₂²×C₄	direct (N × H)
77. C₂×#1057	C₂²×C₄	semidirect
101. C₂×C₄×C₁₃	C₂×M₁₆	semidirect
109. C₂×Dic₁₃	C₂×M₁₆	semidirect
117. #1057	C₂³×C₄	semidirect
136. C₂²×C₁₃	C₄×M₁₆	semidirect
143. C₄×C₁₃	C₂²×M₁₆	semidirect
147. Dic₁₃	C₂²×M₁₆	semidirect
178. C₂×C₁₃	C₂×C₄×M₁₆	semidirect
204. C₂²×C₄	#6122	direct (N × H)
217. C₁₃	C₂²×C₄×M₁₆	semidirect
242. C₂×C₄	C₂×#6122	direct (N × H)
249. C₂²	C₄×#6122	direct (N × H)
261. C₄	C₂²×#6122	direct (N × H)
263. C₂	C₂×C₄×#6122	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×#6122	2
C₂×C₄×#6122	2
C₂²×#88110	2
C₂²×C₄×#1057	2
C₂³×C₄×Dic₁₃	2
C₂²×C₄×M₁₆	13

Subgroups — C22×C4×#6122 #16808000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#6122 #16808000