Subgroups — C₂×#21443876 #29591294

Order 15616.

350 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
3	#21443876	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519777	—	2	1	no	—
6	#16840167	—	2	1	no	—
7	#16840167	—	2	1	no	—
8	#16840167	C₄	1	2	yes	—
9	#16840167	C₂²	1	2	yes	—
10	C₂×#14519711	—	2	1	no	—
11	C₂×#14519711	C₄	1	2	yes	—
12	C₂×C₄⋊#92418	—	2	1	no	—
13	#16839515	—	2	1	no	—
14	#16839515	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₆₁	—	2	1	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×C₄⋊C₁₆	—	2	1	no	—
18	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×Dic₁₂₂	—	4	1	no	—
20	C₂²×#92418	—	4	1	no	—
21	#14519777	—	2	2	no	—
22	#14519777	—	2	2	no	—
23	#14519777	—	4	1	no	—
24	#14519777	—	4	1	no	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	C₂×C₄⋊Dic₆₁	—	4	1	no	—
27	#14519711	—	2	2	no	—
28	#14519711	—	2	2	no	—
29	#14519711	—	2	2	no	—
30	#14519711	C₂×C₄	1	4	yes	—
31	C₄⋊#92418	—	2	2	no	—
32	C₄⋊#92418	—	4	1	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₁	—	2	2	no	—
37	C₂×C₁₆×C₆₁	—	4	2	no	—
38	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
45	C₂²×Dic₆₁	—	4	2	no	—
46	C₂×Dic₁₂₂	—	4	2	no	—
47	C₂×Dic₁₂₂	—	4	2	no	—
48	C₂×Dic₁₂₂	—	8	4	no	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
53	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₆₁	D₈	1	1	yes	characteristic
58	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
59	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₄⋊Dic₆₁	—	2	4	no	—
61	C₄⋊Dic₆₁	—	4	2	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	C₄⋊Dic₆₁	—	4	2	no	—
64	C₂×#92418	—	4	2	no	—
65	C₂×#92418	—	8	2	no	—
66	C₁₆×C₆₁	—	4	2	no	—
67	C₂³×C₆₁	#102	1	1	yes	characteristic
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	#101	1	2	yes	—
70	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#103	1	2	yes	—
74	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
75	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×Dic₆₁	—	4	4	no	—
80	C₂×Dic₆₁	—	8	4	no	—
81	Dic₁₂₂	—	8	4	no	—
82	Dic₁₂₂	—	8	4	no	—
83	C₈×C₆₁	—	2	1	no	—
84	C₈×C₆₁	—	2	1	no	—
85	C₈×C₆₁	#103	1	2	yes	—
86	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
88	#92418	—	8	2	no	—
89	C₂×#1365	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	#332	1	2	yes	—
91	C₂²×C₆₁	#358	1	2	yes	—
92	C₂²×C₆₁	#361	1	2	yes	—
93	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
94	C₄×C₆₁	—	2	1	no	—
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	#332	1	2	yes	—
97	C₄×C₆₁	#358	1	2	yes	—
98	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
99	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
100	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
101	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
102	Dic₆₁	—	8	4	no	—
103	C₂×#500	—	61	1	no	—
104	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
105	C₂×#335	—	61	1	no	—
106	#1365	—	61	4	no	—
107	C₂×C₆₁	#1365	1	4	yes	complemented
108	C₂×C₆₁	C₂×#332	1	1	yes	characteristic
109	C₂×C₆₁	C₂×#358	1	1	yes	characteristic
110	C₂×C₆₁	C₂×#361	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	122	1	no	—
112	C₂²×C₁₆	—	2	1	no	—
113	#500	—	61	2	no	—
114	#500	—	61	2	no	—
115	#500	—	122	1	no	—
116	#500	—	122	1	no	—
117	C₂×#107	—	61	2	no	—
118	C₂×#107	—	122	1	no	—
119	C₂×C₄⋊C₈	—	122	1	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	C₄⋊C₁₆	—	2	1	no	—
122	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
123	#335	—	61	2	no	—
124	#335	—	122	1	no	—
125	C₆₁	C₂×#1365	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	244	1	no	—
127	C₂²×C₈	—	244	1	no	—
128	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
130	C₄⋊Q₈	—	122	2	no	—
131	C₄⋊Q₈	—	122	2	no	—
132	C₄⋊Q₈	—	244	1	no	—
133	C₄⋊Q₈	—	244	1	no	—
134	C₂×C₄⋊C₄	—	122	2	no	—
135	C₂×C₄⋊C₄	—	244	1	no	—
136	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	4	2	no	—
139	#107	—	61	4	no	—
140	#107	—	122	2	no	—
141	#107	—	122	2	no	—
142	#107	—	122	2	no	—
143	C₄⋊C₈	—	122	2	no	—
144	C₄⋊C₈	—	244	1	no	—
145	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
146	C₄×C₈	D₂₄₄	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
149	C₂×Q₈	—	488	4	no	—
150	C₂×Q₈	—	244	2	no	—
151	C₂×Q₈	—	244	2	no	—
152	C₂²×C₄	—	244	2	no	—
153	C₂²×C₄	#11758190	1	1	yes	characteristic
154	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
155	C₂²×C₄	#11758198	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	488	2	no	—
158	C₂×C₈	—	244	2	no	—
159	C₂×C₈	#11758198	1	1	yes	characteristic
160	C₂×C₈	#11758198	1	1	yes	characteristic
161	C₂×C₈	#11758200	1	2	yes	—
162	C₂×C₈	#11758202	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
165	C₄⋊C₄	—	122	4	no	—
166	C₄⋊C₄	—	244	2	no	—
167	C₄⋊C₄	—	244	2	no	—
168	C₄⋊C₄	—	244	2	no	—
169	C₄²	#11758190	1	1	yes	characteristic
170	C₄²	D₄₈₈	1	1	yes	characteristic
171	C₄²	#11758198	1	1	yes	characteristic
172	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
173	C₁₆	—	4	2	no	—
174	C₂³	#14519699	1	1	yes	characteristic
175	Q₈	—	488	4	no	—
176	Q₈	—	488	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	488	4	no	—
179	C₂×C₄	—	244	4	no	—
180	C₂×C₄	#14519695	1	2	yes	—
181	C₂×C₄	#14519699	1	1	yes	characteristic
182	C₂×C₄	#14519699	1	1	yes	characteristic
183	C₂×C₄	#14519699	1	1	yes	characteristic
184	C₂×C₄	#14519700	1	2	yes	—
185	C₂×C₄	#14519722	1	2	yes	—
186	C₂×C₄	Dic₄₈₈	1	2	yes	—
187	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
189	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	488	2	no	—
193	C₈	#14519700	1	2	yes	—
194	C₈	C₂×#11758200	1	1	yes	characteristic
195	C₈	C₂×#11758202	1	1	yes	characteristic
196	C₂²	#16839507	1	2	yes	—
197	C₂²	#16839567	1	2	yes	—
198	C₂²	#16839576	1	2	yes	—
199	C₂²	C₂×#14519699	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	488	4	no	—
203	C₄	#16839507	1	2	yes	—
204	C₄	#16839567	1	2	yes	—
205	C₄	C₂×#14519695	1	1	yes	characteristic
206	C₄	C₂×#14519700	1	1	yes	characteristic
207	C₄	C₂×#14519722	1	1	yes	characteristic
208	C₄	C₂×Dic₄₈₈	1	1	yes	characteristic
209	C₂	#21443876	1	4	yes	direct product factor
210	C₂	C₂×#16839507	1	1	yes	characteristic
211	C₂	C₂×#16839567	1	1	yes	characteristic
212	C₂	C₂×#16839576	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519699
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₆₁	#102
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	C₂×#1365
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #21443876	C₂	direct (N × H)
107. C₂×C₆₁	#1365	semidirect
125. C₆₁	C₂×#1365	semidirect
209. C₂	#21443876	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×#16840167	2
C₂×#16839515	2
#21443876	2
C₂×C₆₁×C₄⋊C₁₆	2
C₂×#1365	61

Subgroups — C2×#21443876 #29591294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21443876 #29591294