Subgroups — C₂×#20243518 #26216465

Order 12032.

1006 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
3	#20243518	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	4	yes	maximal
5	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
10	#15688989	—	2	2	no	—
11	#15688989	C₂²	1	4	yes	—
12	C₂×C₄₇⋊#95	—	2	2	no	—
13	#15688885	C₂²	1	16	yes	—
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#382	—	2	2	no	—
17	C₄₇×#382	C₂²	1	4	yes	—
18	C₂×#95×C₄₇	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₂²×C₄₇⋊C₈	—	2	4	no	—
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
23	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
24	C₂×C₄⋊Dic₄₇	—	2	8	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
27	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
28	C₄₇⋊#95	—	2	8	no	—
29	C₂³×C₄×C₄₇	—	2	2	no	—
30	C₂³×C₄×C₄₇	D₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
39	#95×C₄₇	—	2	8	no	—
40	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	8	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	4	4	no	—
44	C₂²×C₄×C₄₇	—	4	8	no	—
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₄₇	#41	1	4	yes	—
51	C₂²×C₄×C₄₇	#41	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	4	no	—
54	C₂²×Dic₄₇	—	4	4	no	—
55	C₂²×Dic₄₇	—	4	8	no	—
56	C₄⋊C₄×C₄₇	—	2	8	no	—
57	C₄⋊C₄×C₄₇	—	4	4	no	—
58	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
59	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
60	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
61	C₄⋊Dic₄₇	—	2	8	no	—
62	C₄⋊Dic₄₇	—	4	4	no	—
63	C₂×C₄₇⋊C₈	—	2	16	no	—
64	C₂×C₄₇⋊C₈	—	2	4	no	—
65	C₂×C₄₇⋊C₈	—	2	4	no	—
66	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
67	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₄₇	#124	1	2	yes	—
69	C₂³×C₄₇	#124	1	2	yes	—
70	C₂³×C₄₇	#135	1	8	yes	—
71	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	—	2	8	no	—
73	C₂×C₄×C₄₇	—	2	16	no	—
74	C₂×C₄×C₄₇	—	4	4	no	—
75	C₂×C₄×C₄₇	—	4	8	no	—
76	C₂×C₄×C₄₇	—	8	8	no	—
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
78	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₄₇	#117	1	8	yes	—
80	C₂×C₄×C₄₇	#124	1	2	yes	—
81	C₂×C₄×C₄₇	#124	1	2	yes	—
82	C₂×C₄×C₄₇	#125	1	8	yes	—
83	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
87	C₂×Dic₄₇	—	4	4	no	—
88	C₂×Dic₄₇	—	4	8	no	—
89	C₂×Dic₄₇	—	8	8	no	—
90	C₄₇⋊C₈	—	2	8	no	—
91	C₂×#1835	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	#382	1	4	yes	—
93	C₂²×C₄₇	#427	1	8	yes	—
94	C₂²×C₄₇	#488	1	8	yes	—
95	C₂²×C₄₇	#488	1	8	yes	—
96	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
98	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
99	C₂²×C₄₇	C₂×#135	1	4	yes	—
100	C₄×C₄₇	—	2	8	no	—
101	C₄×C₄₇	—	8	4	no	—
102	C₄×C₄₇	#382	1	4	yes	—
103	C₄×C₄₇	C₂×#117	1	2	yes	—
104	C₄×C₄₇	C₂×#125	1	2	yes	—
105	Dic₄₇	—	8	4	no	—
106	C₂²×C₄×C₈	—	47	1	no	—
107	C₂×#382	—	47	1	no	—
108	C₂×#382	D₄₇	1	1	yes	characteristic
109	#1835	—	47	8	no	—
110	C₂×#340	—	47	4	no	—
111	C₂×C₄₇	#1835	1	8	yes	complemented
112	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
113	C₂×C₄₇	C₂×#427	1	2	yes	—
114	C₂×C₄₇	C₂×#488	1	2	yes	—
115	C₂×C₄₇	C₂×#488	1	2	yes	—
116	C₂³×C₈	—	47	2	no	—
117	C₂²×C₄⋊C₄	—	47	2	no	—
118	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
119	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
120	C₂×C₄×C₈	—	47	8	no	—
121	C₂×C₄×C₈	—	47	4	no	—
122	#382	—	2	2	no	—
123	#382	—	47	4	no	—
124	#382	—	94	2	no	—
125	#382	C₂×D₄₇	1	4	yes	—
126	C₂×#95	—	2	2	no	—
127	C₂×#95	—	94	2	no	—
128	#340	—	47	16	no	—
129	C₄₇	C₂×#1835	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	—	2	2	no	—
131	C₂³×C₄	—	94	2	no	—
132	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
133	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	C₂²×C₈	—	47	8	no	—
135	C₂²×C₈	—	47	4	no	—
136	C₂²×C₈	—	94	4	no	—
137	C₂²×C₈	—	94	4	no	—
138	C₂×C₄⋊C₄	—	2	8	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	47	4	no	—
141	C₂×C₄⋊C₄	—	94	8	no	—
142	C₂×C₄⋊C₄	—	94	2	no	—
143	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
144	C₂×C₄²	Dic₉₄	1	4	yes	—
145	C₂×C₄²	C₄×D₄₇	1	4	yes	—
146	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
147	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
148	C₄×C₈	—	47	16	no	—
149	#95	—	2	8	no	—
150	#95	—	94	8	no	—
151	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	—	4	8	no	—
156	C₂²×C₄	—	94	4	no	—
157	C₂²×C₄	—	188	4	no	—
158	C₂²×C₄	—	188	8	no	—
159	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
160	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
161	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
162	C₂²×C₄	#10625352	1	4	yes	—
163	C₂²×C₄	#10625354	1	4	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
166	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
167	C₂×C₈	—	47	8	no	—
168	C₂×C₈	—	94	16	no	—
169	C₂×C₈	—	94	4	no	—
170	C₂×C₈	—	94	4	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	94	8	no	—
174	C₄⋊C₄	—	188	4	no	—
175	C₄²	Dic₄₇⋊C₄	1	8	yes	—
176	C₄²	C₂×Dic₉₄	1	4	yes	—
177	C₄²	C₂×C₄×D₄₇	1	4	yes	—
178	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
179	C₂³	#13094011	1	2	yes	—
180	C₂³	#13094013	1	2	yes	—
181	C₂³	#13094044	1	8	yes	—
182	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	8	8	no	—
188	C₂×C₄	—	376	8	no	—
189	C₂×C₄	—	188	4	no	—
190	C₂×C₄	—	188	8	no	—
191	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
192	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
193	C₂×C₄	#13093982	1	8	yes	—
194	C₂×C₄	#13094011	1	2	yes	—
195	C₂×C₄	#13094013	1	2	yes	—
196	C₂×C₄	#13094015	1	8	yes	—
197	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
198	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
199	C₂×C₄	C₂×#10625352	1	2	yes	—
200	C₂×C₄	C₂×#10625354	1	2	yes	—
201	C₈	—	94	8	no	—
202	C₂²	#15688988	1	4	yes	—
203	C₂²	#15689159	1	8	yes	—
204	C₂²	#15689477	1	8	yes	—
205	C₂²	#15689480	1	8	yes	—
206	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
207	C₂²	C₂×#13094011	1	1	yes	characteristic
208	C₂²	C₂×#13094013	1	1	yes	characteristic
209	C₂²	C₂×#13094044	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	—	8	4	no	—
212	C₄	—	376	4	no	—
213	C₄	#15688988	1	4	yes	—
214	C₄	C₂×#13093982	1	2	yes	—
215	C₄	C₂×#13094015	1	2	yes	—
216	C₂	#20243518	1	8	yes	direct product factor
217	C₂	C₂×#15688988	1	1	yes	characteristic
218	C₂	C₂×#15689159	1	2	yes	—
219	C₂	C₂×#15689477	1	2	yes	—
220	C₂	C₂×#15689480	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#382	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	C₂×#1835
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
3. #20243518	C₂	direct (N × H)
8. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
27. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
111. C₂×C₄₇	#1835	semidirect
129. C₄₇	C₂×#1835	semidirect
216. C₂	#20243518	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×C₄×C₄₇⋊C₈	2
C₂×#15688989	2
#20243518	2
C₂×#15688885	2
C₂×C₄₇×#382	2
C₂×#1835	47

Subgroups — C2×#20243518 #26216465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243518 #26216465