Subgroups — C₂×#21443689 #29569905

Order 15616.

602 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	C₂×#16839819	C₂	1	2	yes	maximal
2	#21443689	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	#16839819	—	2	2	no	—
5	#16839819	—	2	2	no	—
6	#16839819	C₄	1	4	yes	—
7	#16839819	C₂²	1	4	yes	—
8	C₂×C₄⋊#92418	—	2	2	no	—
9	C₂×C₂²⋊#92418	—	2	2	no	—
10	C₂×#14519688	C₄	1	4	yes	—
11	C₂³×C₈×C₆₁	C₄	1	2	yes	—
12	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
14	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
17	C₂²×#92418	—	2	4	no	—
18	C₄⋊#92418	—	2	4	no	—
19	C₄⋊#92418	—	2	4	no	—
20	C₂²⋊#92418	—	2	4	no	—
21	C₂²⋊#92418	—	2	4	no	—
22	#14519688	—	2	4	no	—
23	#14519688	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	4	4	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	D₄	1	2	yes	—
36	C₂×C₄²×C₆₁	D₄	1	2	yes	—
37	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₁	—	2	4	no	—
40	C₄⋊C₈×C₆₁	—	2	4	no	—
41	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	—	2	4	no	—
43	C₂²×C₄×C₆₁	—	2	2	no	—
44	C₂²×C₄×C₆₁	—	2	2	no	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	—	4	4	no	—
62	C₂×C₈×C₆₁	—	4	4	no	—
63	C₂×C₈×C₆₁	—	4	4	no	—
64	C₄²×C₆₁	C₄²	1	2	yes	—
65	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
70	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
71	C₄²×C₆₁	C₂×D₄	1	2	yes	—
72	C₂×#92418	—	2	8	no	—
73	C₂×#92418	—	4	8	no	—
74	C₂³×C₆₁	—	2	2	no	—
75	C₂³×C₆₁	—	2	2	no	—
76	C₂³×C₆₁	#95	1	1	yes	characteristic
77	C₂³×C₆₁	#95	1	1	yes	characteristic
78	C₂³×C₆₁	#108	1	4	yes	—
79	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	4	no	—
81	C₂×C₄×C₆₁	—	2	4	no	—
82	C₂×C₄×C₆₁	—	2	4	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	4	1	no	—
94	C₂×C₄×C₆₁	—	4	1	no	—
95	C₂×C₄×C₆₁	#95	1	2	yes	—
96	C₂×C₄×C₆₁	#95	1	2	yes	—
97	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#100	1	2	yes	—
100	C₂×C₄×C₆₁	#101	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₈×C₆₁	—	4	4	no	—
106	#92418	—	4	8	no	—
107	C₂×#1290	—	61	1	no	maximal, Sylow
108	C₂²×C₆₁	—	2	8	no	—
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	2	1	no	—
111	C₂²×C₆₁	—	2	1	no	—
112	C₂²×C₆₁	—	2	1	no	—
113	C₂²×C₆₁	H(3,ℤ/4)	1	2	yes	—
114	C₂²×C₆₁	#341	1	2	yes	—
115	C₂²×C₆₁	#341	1	2	yes	—
116	C₂²×C₆₁	#343	1	2	yes	—
117	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×#108	1	2	yes	—
119	C₄×C₆₁	—	2	2	no	—
120	C₄×C₆₁	—	2	2	no	—
121	C₄×C₆₁	—	2	2	no	—
122	C₄×C₆₁	—	2	2	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	#343	1	2	yes	—
126	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
127	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
128	C₂×#432	—	61	2	no	—
129	C₂²×C₄⋊C₈	D₆₁	1	1	yes	characteristic
130	#1290	—	61	4	no	—
131	C₂×C₆₁	—	2	2	no	—
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	#1290	1	4	yes	complemented
134	C₂×C₆₁	C₂×H(3,ℤ/4)	1	1	yes	characteristic
135	C₂×C₆₁	C₂×#341	1	1	yes	characteristic
136	C₂×C₆₁	C₂×#343	1	1	yes	characteristic
137	C₂³×C₈	Dic₆₁	1	2	yes	—
138	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
139	#432	—	61	4	no	—
140	#432	—	61	4	no	—
141	#432	—	122	2	no	—
142	#432	—	122	2	no	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	—	2	2	no	—
145	C₂×C₄⋊C₈	—	122	2	no	—
146	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
147	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
148	C₂×C₂²⋊C₈	—	122	2	no	—
149	C₂×#97	—	61	4	no	—
150	C₆₁	C₂×#1290	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
152	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
153	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	4	4	no	—
157	C₂²×C₈	—	122	4	no	—
158	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
159	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
160	C₂×C₄²	C₄×D₆₁	1	2	yes	—
161	C₂×C₄²	C₄×D₆₁	1	2	yes	—
162	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
165	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
166	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	—	122	4	no	—
170	C₄⋊C₈	—	122	4	no	—
171	C₂²⋊C₈	—	122	4	no	—
172	C₂²⋊C₈	—	122	4	no	—
173	#97	—	61	8	no	—
174	#97	—	122	4	no	—
175	C₂⁴	C₄⋊Dic₆₁	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
181	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
182	C₂²×C₄	#11758196	1	2	yes	—
183	C₂²×C₄	#11758196	1	2	yes	—
184	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
187	C₂²×C₄	#11758198	1	2	yes	—
188	C₂²×C₄	#11758198	1	2	yes	—
189	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	—	122	8	no	—
199	C₂×C₈	—	244	8	no	—
200	C₄²	C₄×Dic₆₁	1	2	yes	—
201	C₄²	#11758196	1	2	yes	—
202	C₄²	#11758196	1	2	yes	—
203	C₄²	#11758198	1	2	yes	—
204	C₄²	#11758198	1	2	yes	—
205	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
206	C₄²	C₂×C₄×D₆₁	1	2	yes	—
207	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	#14519684	1	1	yes	characteristic
211	C₂³	#14519684	1	1	yes	characteristic
212	C₂³	#14519713	1	4	yes	—
213	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	#14519684	1	2	yes	—
230	C₂×C₄	#14519684	1	2	yes	—
231	C₂×C₄	#14519684	1	1	yes	characteristic
232	C₂×C₄	#14519684	1	1	yes	characteristic
233	C₂×C₄	#14519694	1	2	yes	—
234	C₂×C₄	#14519696	1	2	yes	—
235	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
236	C₂×C₄	C₂×#11758198	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
238	C₂×C₄	C₂×#11758196	1	2	yes	—
239	C₈	—	4	4	no	—
240	C₈	—	244	8	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	#16839519	1	2	yes	—
247	C₂²	#16839527	1	2	yes	—
248	C₂²	#16839527	1	2	yes	—
249	C₂²	#16839531	1	2	yes	—
250	C₂²	C₂×#14519684	1	1	yes	characteristic
251	C₂²	C₂×#14519713	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	#16839531	1	2	yes	—
259	C₄	C₂×#14519694	1	1	yes	characteristic
260	C₄	C₂×#14519696	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	#21443689	1	4	yes	direct product factor
264	C₂	C₂×#16839519	1	1	yes	characteristic
265	C₂	C₂×#16839527	1	1	yes	characteristic
266	C₂	C₂×#16839531	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519684
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂²×C₄⋊C₈×C₆₁	C₂
socle	C₂³×C₆₁	#95
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂²×C₄⋊C₈	D₆₁
hypocenter	C₆₁	C₂×#1290
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #21443689	C₂	direct (N × H)
133. C₂×C₆₁	#1290	semidirect
150. C₆₁	C₂×#1290	semidirect
263. C₂	#21443689	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×#16839819	2
#21443689	2
C₂²×C₄⋊C₈×C₆₁	2
C₂×#1290	61

Subgroups — C2×#21443689 #29569905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21443689 #29569905