Subgroups — C₂×#21445725 #29678114

Order 15616.

1046 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519713	C₂	1	2	yes	maximal
2	C₂×C₄×#11758194	C₂	1	2	yes	maximal
3	#21445725	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839527	C₂	1	2	yes	maximal
5	C₂²×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	4	yes	—
7	C₂×#14519713	C₄	1	16	yes	complemented
8	C₂×#14519713	C₂²	1	8	yes	—
9	C₄×#11758194	—	2	4	no	—
10	C₄×#11758194	C₂²	1	8	yes	—
11	C₂×#14519688	—	2	2	no	—
12	C₂×C₄×#92418	—	2	2	no	—
13	#16839527	C₂²	1	8	yes	—
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₁	C₄	1	4	yes	—
18	C₂×#97×C₆₁	C₄	1	4	yes	—
19	C₂×#97×C₆₁	C₂²	1	4	yes	—
20	C₂×#11758194	—	2	4	no	—
21	C₂×#11758194	—	2	8	no	—
22	C₂×#11758194	C₂×C₄	1	16	yes	—
23	C₂×#11758194	C₂³	1	8	yes	—
24	C₂²×#92418	—	2	4	no	—
25	#14519713	C₂×C₄	1	32	yes	complemented
26	#14519688	—	2	8	no	—
27	C₄×#92418	—	2	8	no	—
28	C₂³×C₄×C₆₁	Q₈	1	2	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	—	2	4	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
38	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₁	D₄	1	2	yes	—
41	C₂×C₄²×C₆₁	D₄	1	2	yes	—
42	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
43	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
44	#97×C₆₁	C₂×C₄	1	16	yes	—
45	C₂⁴×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	—	4	4	no	—
47	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	#41	1	4	yes	—
57	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₁	—	2	4	no	—
59	C₂×C₈×C₆₁	—	2	4	no	—
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	C₄²	1	8	yes	—
62	C₂×C₈×C₆₁	C₄²	1	8	yes	—
63	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
66	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
67	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
68	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
69	C₄²×C₆₁	C₂×D₄	1	4	yes	—
70	#11758194	—	2	16	no	—
71	#11758194	—	2	8	no	—
72	#11758194	C₂²×C₄	1	16	yes	—
73	C₂×#92418	—	2	16	no	—
74	C₂×#92418	—	2	8	no	—
75	C₂³×C₆₁	—	2	4	no	—
76	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₆₁	C₄×Q₈	1	4	yes	—
79	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	8	no	—
81	C₂×C₄×C₆₁	—	4	4	no	—
82	C₂×C₄×C₆₁	—	4	4	no	—
83	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	8	yes	—
86	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₁	#117	1	2	yes	—
89	C₂×C₄×C₆₁	#117	1	2	yes	—
90	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₆₁	C₄×Q₈	1	4	yes	—
93	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
97	C₈×C₆₁	—	2	4	no	—
98	C₈×C₆₁	C₂×C₄²	1	8	yes	—
99	#92418	—	2	16	no	—
100	C₂×#1778	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	—	2	2	no	—
102	C₂²×C₆₁	—	2	2	no	—
103	C₂²×C₆₁	—	2	8	no	—
104	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
105	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
106	C₂²×C₆₁	#430	1	4	yes	—
107	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₆₁	C₂×C₄×Q₈	1	2	yes	—
109	C₄×C₆₁	—	2	8	no	—
110	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
112	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
113	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
114	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
115	C₂²×#108	—	61	2	no	—
116	C₂×C₄×M₁₆	—	61	2	no	—
117	C₂²×#97	D₆₁	1	1	yes	characteristic
118	#1778	—	61	8	no	—
119	C₂×#341	—	61	2	no	—
120	C₂×C₆₁	—	2	2	no	—
121	C₂×C₆₁	—	2	2	no	—
122	C₂×C₆₁	#1778	1	4	yes	complemented
123	C₂×C₆₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₆₁	C₂×#430	1	2	yes	—
125	C₂²×M₁₆	—	61	4	no	—
126	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
127	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
128	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
129	C₂×#108	—	61	16	no	—
130	C₂×#108	—	61	8	no	—
131	C₄×M₁₆	—	61	8	no	—
132	C₄×M₁₆	—	122	4	no	—
133	C₂×#97	—	122	2	no	—
134	C₂×#97	Dic₆₁	1	4	yes	—
135	C₂×#97	Dic₆₁	1	4	yes	—
136	C₂×#97	C₂×D₆₁	1	4	yes	—
137	C₂×C₄×C₈	—	122	2	no	—
138	#341	—	61	8	no	—
139	C₆₁	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	Dic₁₂₂	1	2	yes	—
141	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
142	C₂×M₁₆	—	61	16	no	—
143	C₂×M₁₆	—	61	8	no	—
144	C₂×M₁₆	—	122	4	no	—
145	C₂×M₁₆	—	122	8	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	—	122	4	no	—
148	C₂²×C₈	C₄×D₆₁	1	8	yes	—
149	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
150	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
151	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
152	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
153	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
154	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
155	C₂×C₄²	Dic₁₂₂	1	2	yes	—
156	C₂×C₄²	D₂₄₄	1	2	yes	—
157	C₂×C₄²	D₂₄₄	1	2	yes	—
158	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
159	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
160	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
161	#108	—	61	32	no	—
162	#97	—	122	8	no	—
163	#97	C₂×Dic₆₁	1	16	yes	—
164	C₄×C₈	—	122	8	no	—
165	C₂⁴	C₂×Dic₁₂₂	1	1	yes	characteristic
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₄⋊Dic₆₁	1	4	yes	—
168	C₂²×C₄	C₄⋊Dic₆₁	1	4	yes	—
169	C₂²×C₄	C₂×Dic₁₂₂	1	4	yes	—
170	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₆₁	1	4	yes	—
172	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
174	C₂²×C₄	#11758222	1	4	yes	—
175	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
177	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
178	M₁₆	—	61	16	no	—
179	M₁₆	—	122	8	no	—
180	M₁₆	—	122	16	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	122	8	no	—
185	C₂×C₈	—	122	16	no	—
186	C₂×C₈	C₄×Dic₆₁	1	8	yes	—
187	C₂×C₈	C₄×Dic₆₁	1	8	yes	—
188	C₂×C₈	C₂×C₄×D₆₁	1	8	yes	—
189	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
190	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
191	C₄²	C₄⋊Dic₆₁	1	4	yes	—
192	C₄²	C₄⋊Dic₆₁	1	4	yes	—
193	C₄²	C₂×D₂₄₄	1	4	yes	—
194	C₄²	C₂²×Dic₆₁	1	4	yes	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
198	C₂³	C₄×Dic₁₂₂	1	4	yes	—
199	C₂³	C₂²×Dic₁₂₂	1	1	yes	characteristic
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₆₁	1	8	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
208	C₂×C₄	#14519736	1	2	yes	—
209	C₂×C₄	#14519736	1	2	yes	—
210	C₂×C₄	C₄×D₂₄₄	1	4	yes	—
211	C₂×C₄	C₄×D₂₄₄	1	4	yes	—
212	C₂×C₄	C₄×Dic₁₂₂	1	4	yes	—
213	C₂×C₄	C₂²×C₄×D₆₁	1	2	yes	—
214	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
216	C₂×C₄	C₂×#11758222	1	2	yes	—
217	C₈	—	2	4	no	—
218	C₈	—	122	16	no	—
219	C₈	C₂×C₄×Dic₆₁	1	8	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄×C₄⋊Dic₆₁	1	2	yes	—
224	C₂²	C₄×C₄⋊Dic₆₁	1	2	yes	—
225	C₂²	#16839811	1	4	yes	—
226	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
227	C₂²	C₂×C₄×Dic₁₂₂	1	2	yes	—
228	C₄	—	2	8	no	—
229	C₄	C₄×C₄⋊Dic₆₁	1	2	yes	—
230	C₄	C₄×C₄⋊Dic₆₁	1	2	yes	—
231	C₄	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
232	C₄	C₂×#14519736	1	1	yes	characteristic
233	C₄	C₂×C₄×D₂₄₄	1	2	yes	—
234	C₂	—	2	2	no	—
235	C₂	—	2	2	no	—
236	C₂	#21445725	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
238	C₂	C₂×#16839811	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂²×#97×C₆₁	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₂×Q₈
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂²×#97	D₆₁
hypocenter	C₆₁	C₂×#1778
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #21445725	C₂	direct (N × H)
7. C₂×#14519713	C₄	semidirect
25. #14519713	C₂×C₄	semidirect
122. C₂×C₆₁	#1778	semidirect
139. C₆₁	C₂×#1778	semidirect
236. C₂	#21445725	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14519713	2
C₂×C₄×#11758194	2
#21445725	2
C₂×#16839527	2
C₂²×#97×C₆₁	2
C₂×#1778	61

Subgroups — C2×#21445725 #29678114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445725 #29678114