Subgroups — C₂×#21462154 #30624089

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519851	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11758187	C₂	1	1	yes	maximal, characteristic
3	#21462154	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11758187	C₂	1	4	yes	maximal, complemented
5	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	—	2	4	no	—
7	C₂×#14519851	C₂²	1	4	yes	—
8	C₂²×#11758187	—	2	4	no	—
9	C₂²×#11758187	C₂²	1	2	yes	—
10	C₂²×#11758187	C₂²	1	2	yes	—
11	C₂²×#11758187	C₂²	1	4	yes	—
12	C₄⋊#11758187	C₂²	1	16	yes	complemented
13	C₂²×#11758194	C₄	1	1	yes	characteristic
14	C₂³×#92418	C₂²	1	1	yes	characteristic
15	#16839718	C₄	1	4	yes	—
16	#16839718	C₂²	1	4	yes	—
17	C₂×#14519688	C₄	1	2	yes	—
18	C₂×C₄×#92418	C₂²	1	2	yes	—
19	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
20	#14519851	—	2	8	no	—
21	C₂×#11758187	—	2	4	no	—
22	C₂×#11758187	—	2	8	no	—
23	C₂×#11758187	—	2	4	no	—
24	C₂×#11758187	C₂³	1	4	yes	—
25	C₂×#11758187	C₂³	1	4	yes	—
26	C₂×#11758194	—	2	4	no	—
27	C₂×#11758194	C₂×C₄	1	4	yes	—
28	C₂²×#92418	C₂×C₄	1	2	yes	—
29	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂²×#92418	C₂×C₄	1	2	yes	—
31	C₂²×#92418	D₄	1	4	yes	—
32	C₂²×#92418	Q₈	1	4	yes	—
33	C₂²×#92418	C₂³	1	1	yes	characteristic
34	C₂²×#92418	C₂³	1	2	yes	—
35	#14519688	C₂×C₄	1	8	yes	—
36	C₄×#92418	C₂×C₄	1	4	yes	—
37	C₄×#92418	C₂³	1	4	yes	—
38	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	#117×C₆₁	C₈	1	4	yes	—
40	#117×C₆₁	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
42	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
43	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
44	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
53	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
54	C₄²×C₆₁	C₂×C₈	1	4	yes	—
55	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
56	#11758194	—	2	8	no	—
57	C₂×#92418	—	2	4	no	—
58	C₂×#92418	C₄⋊C₄	1	4	yes	—
59	C₂×#92418	C₄⋊C₄	1	8	yes	—
60	C₂×#92418	C₂²×C₄	1	4	yes	—
61	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
62	C₂×#92418	C₂²×C₄	1	2	yes	—
63	C₂×#92418	C₂×D₄	1	2	yes	—
64	C₂×#92418	C₂×D₄	1	4	yes	—
65	C₂×#92418	C₂×Q₈	1	2	yes	—
66	C₂×#92418	C₂×Q₈	1	4	yes	—
67	C₂×#92418	C₂⁴	1	1	yes	characteristic
68	#11758187	—	2	4	no	—
69	#11758187	—	2	4	no	—
70	C₂³×C₆₁	—	2	4	no	—
71	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
72	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
73	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	2	8	no	—
75	C₂×C₄×C₆₁	C₄⋊C₈	1	8	yes	—
76	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
80	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
84	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
85	#92418	—	2	4	no	—
86	#92418	C₂×C₄⋊C₄	1	2	yes	—
87	#92418	C₂×C₄⋊C₄	1	4	yes	—
88	#92418	C₂²×D₄	1	1	yes	characteristic
89	#92418	C₂²×Q₈	1	1	yes	characteristic
90	C₂×#2151	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	—	2	4	no	—
92	C₂²×C₆₁	—	2	8	no	—
93	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
94	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
95	C₂²×C₆₁	#504	1	4	yes	—
96	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₆₁	—	2	8	no	—
98	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
99	C₄×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
100	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
102	C₂²×M₃₂	—	61	1	no	—
103	C₂³×C₁₆	—	61	1	no	—
104	C₂×#405	—	61	1	no	—
105	#2151	—	61	8	no	—
106	C₂×C₄⋊C₁₆	—	61	4	no	—
107	C₂×C₆₁	—	2	4	no	—
108	C₂×C₆₁	#2151	1	4	yes	complemented
109	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₆₁	C₂×#504	1	2	yes	—
111	C₂²×M₁₆	—	61	1	no	—
112	C₂³×C₈	—	61	1	no	—
113	C₂×#117	#6923	1	1	yes	characteristic
114	C₂×M₃₂	—	61	4	no	—
115	C₂×M₃₂	—	122	4	no	—
116	C₂²×C₁₆	—	61	2	no	—
117	C₂²×C₁₆	—	61	4	no	—
118	C₂²×C₁₆	—	61	2	no	—
119	C₂²×C₁₆	—	122	4	no	—
120	#405	—	61	4	no	—
121	#405	—	61	4	no	—
122	C₂×#97	—	61	2	no	—
123	C₂×C₄×C₈	—	61	2	no	—
124	C₄⋊C₁₆	—	61	16	no	—
125	C₆₁	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
127	C₂×M₁₆	—	61	4	no	—
128	C₂×M₁₆	—	122	4	no	—
129	C₂²×C₈	—	61	4	no	—
130	C₂²×C₈	—	61	4	no	—
131	C₂²×C₈	—	61	2	no	—
132	C₂²×C₈	—	61	2	no	—
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	61	2	no	—
136	#117	#92420	1	4	yes	—
137	#117	C₂×#6923	1	4	yes	—
138	C₂×C₄⋊C₄	#92420	1	2	yes	—
139	C₂×C₂²⋊C₄	#92420	1	2	yes	—
140	C₂×C₄²	C₂×#6923	1	2	yes	—
141	M₃₂	—	122	8	no	—
142	C₂×C₁₆	—	61	4	no	—
143	C₂×C₁₆	—	61	4	no	—
144	C₂×C₁₆	—	122	4	no	—
145	C₂×C₁₆	—	122	8	no	—
146	C₂×C₁₆	—	122	4	no	—
147	#97	—	61	8	no	—
148	C₄×C₈	—	61	4	no	—
149	C₄×C₈	—	61	4	no	—
150	C₂⁴	C₂×#92420	1	1	yes	characteristic
151	C₂²×C₄	#11758213	1	4	yes	—
152	C₂²×C₄	C₄⋊#6923	1	4	yes	—
153	C₂²×C₄	C₂×#92420	1	4	yes	—
154	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
155	C₂²×C₄	C₂×#92420	1	2	yes	—
156	C₂²×C₄	C₂²×#6923	1	2	yes	—
157	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
158	M₁₆	—	122	8	no	—
159	C₂×C₈	—	61	2	no	—
160	C₂×C₈	—	61	4	no	—
161	C₂×C₈	—	61	8	no	—
162	C₂×C₈	—	61	4	no	—
163	C₂×C₈	—	61	4	no	—
164	C₂×C₈	—	61	2	no	—
165	C₂×C₈	—	61	1	no	—
166	C₂×C₈	—	61	1	no	—
167	C₂×C₈	—	61	2	no	—
168	C₂×C₈	—	61	4	no	—
169	C₂×C₈	—	122	4	no	—
170	C₄⋊C₄	C₂×#92420	1	8	yes	—
171	C₂²⋊C₄	C₂×#92420	1	8	yes	—
172	C₄²	C₂×#92420	1	4	yes	—
173	C₄²	C₂²×#6923	1	4	yes	—
174	C₁₆	—	122	4	no	—
175	C₁₆	—	122	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄⋊#92420	1	4	yes	—
178	C₂³	C₂²×#92420	1	2	yes	—
179	C₂³	C₂²×#92420	1	1	yes	characteristic
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	C₄⋊#92420	1	8	yes	—
182	C₂×C₄	C₄⋊#92420	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊#6923	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
185	C₂×C₄	C₂²×#92420	1	8	yes	—
186	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
187	C₂×C₄	C₂²×#92420	1	2	yes	—
188	C₂×C₄	C₂×#11758213	1	2	yes	—
189	C₂×C₄	C₂×#11758213	1	4	yes	—
190	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
191	C₈	—	61	2	no	—
192	C₈	—	61	4	no	—
193	C₈	—	61	1	no	—
194	C₈	—	61	1	no	—
195	C₈	—	122	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂×C₄⋊#92420	1	4	yes	—
199	C₂²	C₂×C₄⋊#92420	1	2	yes	—
200	C₂²	#16840820	1	4	yes	—
201	C₂²	C₂³×#92420	1	1	yes	characteristic
202	C₄	—	2	8	no	—
203	C₄	C₂×C₄⋊#92420	1	4	yes	—
204	C₄	C₂×C₄⋊#92420	1	2	yes	—
205	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
206	C₄	C₂²×#11758213	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#21462154	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊#92420	1	1	yes	characteristic
210	C₂	C₂×#16840820	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×#117×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	C₂×#2151
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #21462154	C₂	direct (N × H)
4. C₂×C₄⋊#11758187	C₂	semidirect
12. C₄⋊#11758187	C₂²	semidirect
108. C₂×C₆₁	#2151	semidirect
125. C₆₁	C₂×#2151	semidirect
208. C₂	#21462154	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×#14519851	2
C₂³×#11758187	2
#21462154	2
C₂×C₄⋊#11758187	2
C₂×#16839718	2
C₂×#2151	61

Subgroups — C2×#21462154 #30624089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462154 #30624089