Subgroups — C₂×#20241451 #26127819

Order 12032.

286 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241451	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
6	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
10	C₈×C₄₇⋊C₈	C₄	1	2	yes	—
11	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₄₇	—	2	1	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×C₄⋊C₁₆	—	2	1	no	—
15	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₈	—	2	1	no	—
17	C₂²×C₄₇⋊C₈	C₈	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₈	1	1	yes	characteristic
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	C₂×C₄₇⋊C₁₆	—	4	2	no	—
21	C₄×C₄₇⋊C₈	—	2	1	no	—
22	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	—	2	2	no	—
29	C₂×C₁₆×C₄₇	—	4	2	no	—
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
42	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
47	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄₇⋊C₈	—	2	4	no	—
49	C₂×C₄₇⋊C₈	—	2	2	no	—
50	C₂×C₄₇⋊C₈	—	2	2	no	—
51	C₂×C₄₇⋊C₈	C₁₆	1	4	yes	complemented
52	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
53	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
54	C₁₆×C₄₇	—	4	2	no	—
55	C₄₇⋊C₁₆	—	4	2	no	—
56	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#106	1	2	yes	—
62	C₂×C₄×C₄₇	#108	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₄₇	—	2	1	no	—
69	C₈×C₄₇	—	2	1	no	—
70	C₈×C₄₇	#106	1	2	yes	—
71	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₄₇⋊C₈	—	2	4	no	—
74	C₄₇⋊C₈	—	2	2	no	—
75	C₄₇⋊C₈	C₂×C₁₆	1	4	yes	complemented
76	C₂×#1367	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	#334	1	2	yes	—
78	C₂²×C₄₇	C₄⋊C₁₆	1	2	yes	—
79	C₂²×C₄₇	#364	1	2	yes	—
80	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	#334	1	2	yes	—
84	C₄×C₄₇	C₄⋊C₁₆	1	2	yes	—
85	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
86	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
87	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
88	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
89	C₂×C₄⋊C₁₆	—	47	1	no	—
90	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
91	C₂×C₈²	—	47	1	no	—
92	#1367	—	47	4	no	—
93	C₂×C₄₇	#1367	1	4	yes	complemented
94	C₂×C₄₇	C₂×#334	1	1	yes	characteristic
95	C₂×C₄₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
96	C₂×C₄₇	C₂×#364	1	1	yes	characteristic
97	C₂²×C₁₆	—	2	1	no	—
98	C₂²×C₁₆	—	94	1	no	—
99	C₂×C₄×C₈	—	47	1	no	—
100	C₂×C₄×C₈	—	47	1	no	—
101	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
102	C₄⋊C₁₆	—	2	1	no	—
103	C₄⋊C₁₆	—	47	2	no	—
104	C₄⋊C₁₆	—	94	1	no	—
105	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
106	C₈²	—	47	2	no	—
107	C₈²	—	47	2	no	—
108	C₄₇	C₂×#1367	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	47	1	no	—
110	C₂²×C₈	—	47	1	no	—
111	C₂²×C₈	—	94	1	no	—
112	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	4	2	no	—
117	C₂×C₁₆	—	94	2	no	—
118	C₂×C₁₆	—	188	2	no	—
119	C₄×C₈	—	47	2	no	—
120	C₄×C₈	—	47	2	no	—
121	C₄×C₈	—	47	2	no	—
122	C₄×C₈	—	94	1	no	—
123	C₄×C₈	Dic₉₄	1	1	yes	characteristic
124	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
128	C₂²×C₄	#10625324	1	1	yes	characteristic
129	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	47	4	no	—
132	C₂×C₈	—	47	2	no	—
133	C₂×C₈	—	47	2	no	—
134	C₂×C₈	—	94	4	no	—
135	C₂×C₈	—	94	2	no	—
136	C₂×C₈	—	94	2	no	—
137	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
138	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#10625335	1	2	yes	—
140	C₂×C₈	#10625336	1	2	yes	—
141	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
143	C₄²	C₈×D₄₇	1	1	yes	characteristic
144	C₄²	#10625324	1	1	yes	characteristic
145	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
147	C₁₆	—	4	2	no	—
148	C₁₆	—	188	2	no	—
149	C₂³	Dic₄₇⋊C₈	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
152	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
153	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
154	C₂×C₄	#13093955	1	2	yes	—
155	C₂×C₄	#13093959	1	2	yes	—
156	C₂×C₄	C₁₆×D₄₇	1	2	yes	—
157	C₂×C₄	#13093963	1	2	yes	—
158	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
160	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	47	4	no	—
164	C₈	—	94	4	no	—
165	C₈	—	94	2	no	—
166	C₈	#13093955	1	2	yes	—
167	C₈	C₂×#10625336	1	1	yes	characteristic
168	C₈	C₂×#10625335	1	1	yes	characteristic
169	C₂²	#15688873	1	2	yes	—
170	C₂²	Dic₄₇⋊C₁₆	1	2	yes	—
171	C₂²	#15688943	1	2	yes	—
172	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#15688873	1	2	yes	—
176	C₄	Dic₄₇⋊C₁₆	1	2	yes	—
177	C₄	C₂×#13093955	1	1	yes	characteristic
178	C₄	C₂×#13093959	1	1	yes	characteristic
179	C₄	C₂×C₁₆×D₄₇	1	1	yes	characteristic
180	C₄	C₂×#13093963	1	1	yes	characteristic
181	C₂	#20241451	1	4	yes	direct product factor
182	C₂	C₂×#15688873	1	1	yes	characteristic
183	C₂	C₂×Dic₄₇⋊C₁₆	1	1	yes	characteristic
184	C₂	C₂×#15688943	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#1367
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20241451	C₂	direct (N × H)
51. C₂×C₄₇⋊C₈	C₁₆	semidirect
75. C₄₇⋊C₈	C₂×C₁₆	semidirect
93. C₂×C₄₇	#1367	semidirect
108. C₄₇	C₂×#1367	semidirect
181. C₂	#20241451	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×C₄⋊(C₄₇⋊C₁₆)	2
C₂×C₈×C₄₇⋊C₈	2
#20241451	2
C₂×C₄₇×C₄⋊C₁₆	2
C₂×#1367	47

Subgroups — C2×#20241451 #26127819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241451 #26127819