Subgroups — C₂×#21448095 #29835492

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840164	C₂	1	1	yes	maximal, characteristic
2	#21448095	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16839589	C₂	1	2	yes	maximal
4	C₂×#16839587	C₂	1	2	yes	maximal
5	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₁×#499	C₂	1	1	yes	maximal, characteristic
7	C₂×#14519769	—	2	1	no	—
8	C₂²×#11758185	C₂²	1	1	yes	characteristic
9	C₂²×#11758185	C₂²	1	1	yes	characteristic
10	#16840164	—	2	2	no	—
11	#16840164	C₂²	1	4	yes	—
12	C₂×#14519711	C₂²	1	2	yes	—
13	C₂×#14519709	—	2	1	no	—
14	#16839589	C₂²	1	8	yes	—
15	#16839587	C₂²	1	8	yes	—
16	C₄×#11758185	C₂²	1	4	yes	—
17	C₂×#125×C₆₁	—	2	1	no	—
18	C₆₁×#499	—	2	2	no	—
19	C₆₁×#499	C₂²	1	4	yes	—
20	C₂×#107×C₆₁	C₂²	1	2	yes	—
21	C₂×#106×C₆₁	—	2	1	no	—
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	#14519769	—	2	4	no	—
24	#14519769	—	4	2	no	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	C₂×#11758196	—	4	2	no	—
27	C₂×#11758185	Q₈	1	4	yes	—
28	C₂×#11758185	Q₈	1	4	yes	—
29	C₂×#11758185	C₂³	1	2	yes	—
30	C₂×#11758185	C₂³	1	2	yes	—
31	#14519711	—	2	2	no	—
32	#14519711	C₂³	1	4	yes	—
33	#14519709	—	2	4	no	—
34	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	#125×C₆₁	—	2	4	no	—
37	#125×C₆₁	—	4	2	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	#107×C₆₁	—	2	2	no	—
42	#107×C₆₁	C₂³	1	4	yes	—
43	#106×C₆₁	—	2	4	no	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₄×C₈×C₆₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₆₁	—	4	2	no	—
47	C₂²×C₄×C₆₁	D₈	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
51	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₆₁	—	2	4	no	—
57	C₄⋊C₄×C₆₁	—	4	2	no	—
58	C₄⋊C₄×C₆₁	—	4	8	no	—
59	C₄²×C₆₁	D₈	1	2	yes	—
60	C₄²×C₆₁	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₆₁	—	2	4	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	#11758196	—	4	8	no	—
64	#11758185	C₂×Q₈	1	4	yes	—
65	#11758185	C₂×Q₈	1	4	yes	—
66	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	—	4	4	no	—
69	C₂×C₄×C₆₁	—	8	4	no	—
70	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
74	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₆₁	—	4	4	no	—
76	C₂×Dic₆₁	—	8	4	no	—
77	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
78	C₈×C₆₁	C₂×Q₁₆	1	4	yes	—
79	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂×#2253	—	61	1	no	maximal, Sylow
82	C₂²×C₆₁	#500	1	2	yes	—
83	C₂²×C₆₁	#508	1	4	yes	—
84	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
85	C₄×C₆₁	—	2	4	no	—
86	C₄×C₆₁	—	8	4	no	—
87	C₄×C₆₁	#500	1	2	yes	—
88	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
90	Dic₆₁	—	8	4	no	—
91	C₂×#499	—	61	1	no	—
92	C₂×#499	D₆₁	1	1	yes	characteristic
93	#2253	—	61	8	no	—
94	C₂×#367	—	61	2	no	—
95	C₂×#366	—	61	2	no	—
96	C₂×C₄×C₁₆	—	61	1	no	—
97	C₂×C₆₁	#2253	1	4	yes	complemented
98	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
99	C₂×C₆₁	C₂×#508	1	2	yes	—
100	C₂×#125	—	2	1	no	—
101	C₂×#125	—	122	1	no	—
102	C₂²×C₁₆	—	61	1	no	—
103	C₂²×C₁₆	—	61	1	no	—
104	#499	—	2	2	no	—
105	#499	—	61	4	no	—
106	#499	—	122	2	no	—
107	#499	C₂×D₆₁	1	4	yes	—
108	C₂×#107	—	61	2	no	—
109	C₂×#107	C₂×D₆₁	1	2	yes	—
110	C₂×#106	—	2	1	no	—
111	C₂×#106	—	122	1	no	—
112	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
113	#367	—	61	8	no	—
114	#366	—	61	8	no	—
115	C₄×C₁₆	—	61	4	no	—
116	C₆₁	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
119	#125	—	2	4	no	—
120	#125	—	4	2	no	—
121	#125	—	122	4	no	—
122	#125	—	244	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	2	no	—
125	C₂×C₄⋊C₄	—	122	2	no	—
126	C₂×C₄⋊C₄	—	244	2	no	—
127	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
128	C₂×C₁₆	—	61	4	no	—
129	C₂×C₁₆	—	61	4	no	—
130	C₂×C₁₆	—	61	2	no	—
131	C₂×C₁₆	—	61	2	no	—
132	#107	—	2	2	no	—
133	#107	—	61	4	no	—
134	#107	—	122	2	no	—
135	#107	C₂²×D₆₁	1	4	yes	—
136	#106	—	2	4	no	—
137	#106	—	122	4	no	—
138	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
139	C₄×C₈	C₂²×D₆₁	1	2	yes	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	244	2	no	—
142	C₂²×C₄	#11758199	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
144	C₂×C₈	#11758202	1	4	yes	—
145	C₂×C₈	Q₈×D₆₁	1	2	yes	—
146	C₂×C₈	Q₈×D₆₁	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
149	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	122	4	no	—
154	C₄⋊C₄	—	244	2	no	—
155	C₄⋊C₄	—	244	8	no	—
156	C₄²	#11758199	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
158	C₁₆	—	61	4	no	—
159	C₁₆	—	61	4	no	—
160	C₂³	C₂×#11758199	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	488	4	no	—
165	C₂×C₄	—	244	4	no	—
166	C₂×C₄	#14519775	1	2	yes	—
167	C₂×C₄	C₂×#11758199	1	4	yes	—
168	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
169	C₂×C₄	C₂×#11758202	1	2	yes	—
170	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
171	C₈	#14519775	1	2	yes	—
172	C₈	C₂×#11758202	1	4	yes	—
173	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
174	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
175	C₂²	#16840165	1	2	yes	—
176	C₂²	#16840192	1	4	yes	—
177	C₂²	C₂²×#11758199	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	488	4	no	—
181	C₄	#16840165	1	2	yes	—
182	C₄	C₂×#14519775	1	1	yes	characteristic
183	C₄	C₂²×#11758202	1	1	yes	characteristic
184	C₂	#21448095	1	4	yes	direct product factor
185	C₂	C₂×#16840165	1	1	yes	characteristic
186	C₂	C₂×#16840192	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#499	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂×#499	D₆₁
hypocenter	C₆₁	C₂×#2253
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #21448095	C₂	direct (N × H)
97. C₂×C₆₁	#2253	semidirect
116. C₆₁	C₂×#2253	semidirect
184. C₂	#21448095	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×#16840164	2
#21448095	2
C₂×#16839589	2
C₂×#16839587	2
C₂×C₄×#11758185	2
C₂×C₆₁×#499	2
C₂×#2253	61

Subgroups — C2×#21448095 #29835492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21448095 #29835492