Subgroups — C₄×C₂²⋊#6092 #16789284

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#6092	C₂	1	2	yes	maximal
2	C₂×C₄×#6092	C₂	1	2	yes	maximal, complemented
3	#13129874	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	4	yes	—
9	C₂×C₄×#1055	—	2	4	no	—
10	C₂×C₄×#1055	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂²⋊#6092	C₄	1	8	yes	direct product factor
15	C₄×#6092	—	2	4	no	—
16	C₂³×C₄×C₁₃	C₈	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	—	2	2	no	—
20	C₂×C₄²×C₁₃	C₈	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	—	2	2	no	—
27	C₂²×#1055	C₈	1	4	yes	—
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	4	yes	—
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂³	1	1	yes	characteristic
33	C₂×#6092	—	2	4	no	—
34	C₂×#6092	—	2	4	no	—
35	C₂×#6092	C₂×C₄	1	8	yes	complemented
36	C₄×#1055	—	2	2	no	—
37	C₄×#1055	—	2	2	no	—
38	C₄×#1055	—	2	2	no	—
39	C₄×#1055	D₄	1	4	yes	—
40	C₄×C₂²⋊C₁₆	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	C₁₆	1	8	yes	complemented
49	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	—	2	4	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	—	2	1	no	—
58	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₁₃	M₁₆	1	2	yes	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	8	no	—
65	C₂×#1055	—	2	8	no	—
66	C₂×#1055	C₄²	1	2	yes	—
67	C₂×#1055	C₂²⋊C₄	1	2	yes	—
68	C₂×#1055	C₂×C₈	1	2	yes	—
69	C₂×#1055	C₂×C₈	1	2	yes	—
70	C₂×#1055	M₁₆	1	2	yes	—
71	C₂×#1055	C₂²×C₄	1	2	yes	—
72	C₂×#1055	C₂×D₄	1	2	yes	—
73	C₂×#1055	#41	1	2	yes	—
74	#6092	—	2	8	no	—
75	C₂²×C₄×C₈	—	13	1	no	—
76	C₂×C₂²⋊C₁₆	—	13	2	no	—
77	C₂×C₄×C₁₆	—	13	2	no	—
78	#1374	—	13	2	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	C₄×C₈	1	2	yes	—
83	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
84	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	16	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
96	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
97	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
98	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	#1055	—	2	2	no	—
106	#1055	—	2	2	no	—
107	#1055	C₂²⋊C₈	1	4	yes	—
108	#1055	C₄×D₄	1	4	yes	—
109	C₂³×C₈	—	13	1	no	—
110	C₂³×C₈	—	13	1	no	—
111	C₂²×C₄²	#261	1	1	yes	characteristic
112	C₂²×C₁₆	—	13	4	no	—
113	C₂×C₄×C₈	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	C₂×C₄×C₈	—	13	1	no	—
116	C₂×C₄×C₈	—	13	1	no	—
117	C₂×C₄×C₈	—	26	4	no	—
118	C₂²⋊C₁₆	—	13	8	no	—
119	C₄×C₁₆	—	26	4	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	2	4	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	C₄×C₁₆	1	4	yes	complemented
125	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₁₃	#431	1	1	yes	characteristic
128	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
129	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	C₂²⋊C₁₆	1	8	yes	complemented
133	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
135	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₄×C₁₃	#431	1	1	yes	characteristic
137	C₂³×C₄	#1057	1	2	yes	—
138	C₂³×C₄	C₂×#261	1	1	yes	characteristic
139	C₂²×C₈	—	13	1	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	4	no	—
142	C₂²×C₈	—	13	4	no	—
143	C₂²×C₈	—	13	1	no	—
144	C₂²×C₈	—	13	1	no	—
145	C₂²×C₈	—	26	2	no	—
146	C₂²×C₈	—	26	2	no	—
147	C₂²×C₈	—	26	2	no	—
148	C₂²×C₈	—	26	2	no	—
149	C₂×C₄²	—	2	2	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	#1057	1	2	yes	—
152	C₂×C₄²	C₂×#261	1	1	yes	characteristic
153	C₂×C₄²	C₂×#261	1	1	yes	characteristic
154	C₂×C₁₆	—	13	8	no	—
155	C₂×C₁₆	—	26	4	no	—
156	C₂×C₁₆	—	26	4	no	—
157	C₄×C₈	—	13	4	no	—
158	C₄×C₈	—	26	2	no	—
159	C₄×C₈	—	26	2	no	—
160	C₄×C₈	—	26	2	no	—
161	C₂×C₁₃	—	2	4	no	—
162	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
163	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
164	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
165	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	2	yes	—
166	C₂×C₁₃	#2162	1	2	yes	—
167	C₂⁴	C₂×#1057	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	#6092	1	8	yes	complemented
175	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
176	C₂²×C₄	C₄×#261	1	2	yes	—
177	C₂²×C₄	C₂×#1057	1	2	yes	—
178	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
179	C₂²×C₄	C₂×#1057	1	2	yes	—
180	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
181	C₂×C₈	—	13	2	no	—
182	C₂×C₈	—	13	2	no	—
183	C₂×C₈	—	13	2	no	—
184	C₂×C₈	—	13	2	no	—
185	C₂×C₈	—	13	2	no	—
186	C₂×C₈	—	13	2	no	—
187	C₂×C₈	—	13	2	no	—
188	C₂×C₈	—	13	2	no	—
189	C₂×C₈	—	26	8	no	—
190	C₂×C₈	—	26	8	no	—
191	C₂×C₈	—	26	1	no	—
192	C₂×C₈	—	26	1	no	—
193	C₂×C₈	—	26	1	no	—
194	C₂×C₈	—	26	1	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	#6122	1	2	yes	—
199	C₄²	C₂²⋊#261	1	2	yes	—
200	C₁₆	—	26	8	no	—
201	C₁₃	C₄×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	2	1	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	1	no	—
205	C₂³	C₂×#6092	1	4	yes	—
206	C₂³	C₄×#1057	1	2	yes	—
207	C₂³	C₂²×#1057	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	16	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	C₂×#6092	1	4	yes	—
218	C₂×C₄	C₂×#6092	1	4	yes	—
219	C₂×C₄	#88106	1	4	yes	—
220	C₂×C₄	C₄×#1057	1	2	yes	—
221	C₂×C₄	C₂²⋊#1057	1	4	yes	—
222	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
223	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
225	C₂×C₄	#88237	1	1	yes	characteristic
226	C₂×C₄	C₂×#6122	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
228	C₈	—	13	4	no	—
229	C₈	—	13	4	no	—
230	C₈	—	26	2	no	—
231	C₈	—	26	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	C₄×#6092	1	4	yes	complemented
237	C₂²	C₂²×#6092	1	2	yes	—
238	C₂²	C₂×#88106	1	2	yes	—
239	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
240	C₂²	#11719119	1	1	yes	characteristic
241	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	2	2	no	—
244	C₄	C₂²⋊#6092	1	8	yes	direct product factor
245	C₄	C₄×#6122	1	1	yes	characteristic
246	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
247	C₄	#11719125	1	1	yes	characteristic
248	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
249	C₂	—	2	4	no	—
250	C₂	C₄×C₂²⋊#1057	1	1	yes	characteristic
251	C₂	C₂×C₄×#6092	1	1	yes	characteristic
252	C₂	C₄×#88106	1	1	yes	characteristic
253	C₂	C₂×C₂²⋊#6092	1	2	yes	—
254	C₂	#13131193	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₄×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#6092	C₂	semidirect
3. #13129874	C₂	semidirect
14. C₂²⋊#6092	C₄	direct (N × H)
35. C₂×#6092	C₂×C₄	semidirect
48. C₂²×C₄×C₁₃	C₁₆	semidirect
124. C₂²×C₁₃	C₄×C₁₆	semidirect
132. C₄×C₁₃	C₂²⋊C₁₆	semidirect
174. C₂²×C₄	#6092	semidirect
201. C₁₃	C₄×C₂²⋊C₁₆	semidirect
236. C₂²	C₄×#6092	semidirect
244. C₄	C₂²⋊#6092	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×C₂²⋊#6092	2
C₂×C₄×#6092	2
#13129874	2
C₂²×C₄×#1055	2
C₄×C₂²⋊C₁₆	13

Subgroups — C4×C22⋊#6092 #16789284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊#6092 #16789284