Subgroups — C₂×#20251775 #26422302

Order 12032.

2078 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251775	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689021	C₂	1	4	yes	maximal
3	C₂×#15689007	C₂	1	8	yes	maximal
4	C₂×#15688989	C₂	1	2	yes	maximal
5	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
7	C₂²×Dic₄₇⋊C₄	C₂²	1	8	yes	—
8	#15689021	C₂²	1	32	yes	—
9	#15689007	C₂²	1	64	yes	—
10	#15688989	C₂²	1	16	yes	—
11	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
12	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄³×C₄₇	C₂²	1	8	yes	—
17	C₂³×Dic₄₇	C₂³	1	8	yes	—
18	C₂×C₄⋊Dic₄₇	—	2	16	no	—
19	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
20	C₂×Dic₄₇⋊C₄	—	2	32	no	—
21	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	—
22	C₄₇⋊#95	C₂³	1	32	yes	—
23	C₄₇⋊#95	C₂³	1	32	yes	—
24	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	—	2	16	no	—
28	C₂×C₄²×C₄₇	—	2	8	no	—
29	C₂×C₄²×C₄₇	Q₈	1	8	yes	—
30	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
33	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	—	2	8	no	—
35	C₂²×C₄×C₄₇	—	2	8	no	—
36	C₂²×C₄×C₄₇	—	2	8	no	—
37	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
38	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
39	C₂²×C₄×C₄₇	#41	1	16	yes	—
40	C₂²×C₄×C₄₇	#41	1	8	yes	—
41	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
43	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
44	C₂²×Dic₄₇	—	2	32	no	—
45	C₂²×Dic₄₇	—	2	16	no	—
46	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
47	C₄²×C₄₇	—	2	32	no	—
48	C₄²×C₄₇	—	2	16	no	—
49	C₄²×C₄₇	C₂×Q₈	1	16	yes	—
50	C₄⋊Dic₄₇	—	2	32	no	—
51	Dic₄₇⋊C₄	—	2	64	no	—
52	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
53	C₂³×C₄₇	C₂×#41	1	4	yes	—
54	C₂³×C₄₇	C₂×#41	1	8	yes	—
55	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	32	no	—
57	C₂×C₄×C₄₇	—	2	16	no	—
58	C₂×C₄×C₄₇	—	2	8	no	—
59	C₂×C₄×C₄₇	—	2	8	no	—
60	C₂×C₄×C₄₇	—	2	8	no	—
61	C₂×C₄×C₄₇	#125	1	16	yes	—
62	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
63	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
64	C₂×C₄×C₄₇	C₂×#41	1	16	yes	—
65	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
66	C₂×Dic₄₇	—	2	32	no	—
67	C₂×Dic₄₇	—	2	16	no	—
68	C₂×Dic₄₇	—	4	32	no	—
69	C₂×#2864	—	47	1	no	maximal, Sylow
70	C₂²×C₄₇	C₂×#125	1	4	yes	—
71	C₂²×C₄₇	#529	1	16	yes	—
72	C₂²×C₄₇	#533	1	8	yes	—
73	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
74	C₂²×C₄₇	C₂²×#41	1	2	yes	—
75	C₂²×C₄₇	C₂²×#41	1	4	yes	—
76	C₄×C₄₇	—	2	16	no	—
77	C₄×C₄₇	—	2	8	no	—
78	C₄×C₄₇	C₂×#125	1	8	yes	—
79	Dic₄₇	—	4	16	no	—
80	#2864	—	47	16	no	—
81	C₂×#389	—	47	4	no	—
82	C₂×#387	—	47	8	no	—
83	C₂×#382	—	47	2	no	—
84	C₂×C₄³	D₄₇	1	1	yes	characteristic
85	C₂×C₄₇	#2864	1	8	yes	complemented
86	C₂×C₄₇	C₂²×#125	1	1	yes	characteristic
87	C₂×C₄₇	C₂×#529	1	4	yes	—
88	C₂×C₄₇	C₂×#533	1	2	yes	—
89	C₂²×C₄⋊C₄	—	47	8	no	—
90	C₂²×C₄⋊C₄	—	47	4	no	—
91	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
92	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
93	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	#389	—	47	32	no	—
95	#387	—	47	64	no	—
96	#382	—	47	16	no	—
97	C₂×#95	—	47	8	no	—
98	C₂×#95	—	47	8	no	—
99	C₄³	C₂×D₄₇	1	8	yes	—
100	C₄₇	C₂×#2864	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₄	—	47	8	no	—
102	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
103	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
104	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	47	32	no	—
106	C₂×C₄⋊C₄	—	47	16	no	—
107	C₂×C₄⋊C₄	—	94	32	no	—
108	C₂×C₄⋊C₄	—	94	16	no	—
109	C₂×C₄²	—	2	16	no	—
110	C₂×C₄²	—	2	8	no	—
111	C₂×C₄²	Dic₉₄	1	8	yes	—
112	C₂×C₄²	C₂²×D₄₇	1	16	yes	—
113	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
114	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
115	#95	—	47	32	no	—
116	#95	—	47	32	no	—
117	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	2	8	no	—
121	C₂²×C₄	—	47	16	no	—
122	C₂²×C₄	—	94	16	no	—
123	C₂²×C₄	—	94	32	no	—
124	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
125	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
126	C₂²×C₄	#10625350	1	16	yes	—
127	C₂²×C₄	#10625350	1	8	yes	—
128	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
129	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
130	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
131	C₄⋊C₄	—	94	64	no	—
132	C₄⋊C₄	—	94	32	no	—
133	C₄²	—	2	32	no	—
134	C₄²	—	2	16	no	—
135	C₄²	C₂×Dic₉₄	1	16	yes	—
136	C₂³	C₂²×Dic₉₄	1	2	yes	—
137	C₂³	C₂×#10625350	1	4	yes	—
138	C₂³	C₂×#10625350	1	8	yes	—
139	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
140	C₂×C₄	—	2	32	no	—
141	C₂×C₄	—	2	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	94	16	no	—
146	C₂×C₄	—	94	32	no	—
147	C₂×C₄	—	188	32	no	—
148	C₂×C₄	#13094016	1	16	yes	—
149	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
150	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
151	C₂×C₄	C₂×#10625350	1	16	yes	—
152	C₂×C₄	C₂×#10625350	1	8	yes	—
153	C₂²	C₂×#13094016	1	4	yes	—
154	C₂²	#15689657	1	16	yes	—
155	C₂²	#15689680	1	8	yes	—
156	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
157	C₂²	C₂²×#10625350	1	2	yes	—
158	C₂²	C₂²×#10625350	1	4	yes	—
159	C₄	—	2	16	no	—
160	C₄	—	2	8	no	—
161	C₄	—	188	16	no	—
162	C₄	C₂×#13094016	1	8	yes	—
163	C₂	#20251775	1	8	yes	direct product factor
164	C₂	C₂²×#13094016	1	1	yes	characteristic
165	C₂	C₂×#15689657	1	4	yes	—
166	C₂	C₂×#15689680	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	C₂×#2864
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. #20251775	C₂	direct (N × H)
85. C₂×C₄₇	#2864	semidirect
100. C₄₇	C₂×#2864	semidirect
163. C₂	#20251775	direct (N × H)

Maximal subgroups

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

subgroup	index
#20251775	2
C₂×#15689021	2
C₂×#15689007	2
C₂×#15688989	2
C₂×C₄³×C₄₇	2
C₂×#2864	47

Subgroups — C2×#20251775 #26422302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251775 #26422302