Subgroups — C₂×#20243521 #26216474

Order 12032.

1054 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20243521	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	—	2	2	no	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×#13093935	C₄	1	8	yes	complemented
10	C₂×#13093935	C₂²	1	4	yes	—
11	#15688993	—	2	2	no	—
12	#15688993	C₂²	1	4	yes	—
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	—	2	8	no	—
27	C₂×C₄⋊Dic₄₇	—	4	8	no	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
29	#13093935	C₂×C₄	1	16	yes	complemented
30	C₂³×C₄×C₄₇	—	2	2	no	—
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
40	#95×C₄₇	—	2	8	no	—
41	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	4	no	—
45	C₂²×C₄×C₄₇	—	4	4	no	—
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	—	4	4	no	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₄₇	#41	1	4	yes	—
54	C₂²×C₄×C₄₇	#41	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	2	4	no	—
57	C₂²×Dic₄₇	—	4	4	no	—
58	C₂²×Dic₄₇	—	4	8	no	—
59	C₄⋊C₄×C₄₇	—	2	8	no	—
60	C₄⋊C₄×C₄₇	—	4	4	no	—
61	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
62	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
63	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
64	C₄⋊Dic₄₇	—	2	8	no	—
65	C₄⋊Dic₄₇	—	4	16	no	—
66	C₄⋊Dic₄₇	—	4	4	no	—
67	C₂×C₄₇⋊C₈	—	2	16	no	—
68	C₂×C₄₇⋊C₈	—	2	4	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
71	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₄₇	#124	1	2	yes	—
73	C₂³×C₄₇	#124	1	2	yes	—
74	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
75	C₂³×C₄₇	#137	1	4	yes	—
76	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	16	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	8	8	no	—
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₄₇	#117	1	8	yes	—
87	C₂×C₄×C₄₇	#124	1	2	yes	—
88	C₂×C₄×C₄₇	#124	1	2	yes	—
89	C₂×C₄×C₄₇	#125	1	8	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
94	C₂×Dic₄₇	—	4	4	no	—
95	C₂×Dic₄₇	—	4	8	no	—
96	C₂×Dic₄₇	—	8	8	no	—
97	C₄₇⋊C₈	—	2	8	no	—
98	C₂×#1836	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	#382	1	4	yes	—
100	C₂²×C₄₇	#428	1	8	yes	—
101	C₂²×C₄₇	#489	1	8	yes	—
102	C₂²×C₄₇	#490	1	4	yes	—
103	C₂²×C₄₇	#491	1	4	yes	—
104	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
108	C₂²×C₄₇	C₂×#137	1	2	yes	—
109	C₄×C₄₇	—	2	8	no	—
110	C₄×C₄₇	—	8	4	no	—
111	C₄×C₄₇	#382	1	4	yes	—
112	C₄×C₄₇	C₂×#117	1	2	yes	—
113	C₄×C₄₇	C₂×#125	1	2	yes	—
114	Dic₄₇	—	8	4	no	—
115	C₂²×#97	—	47	1	no	—
116	C₂×#384	—	47	1	no	—
117	C₂×#382	D₄₇	1	1	yes	characteristic
118	#1836	—	47	8	no	—
119	C₂×#340	—	47	4	no	—
120	C₂×C₄₇	#1836	1	8	yes	complemented
121	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
122	C₂×C₄₇	C₂×#428	1	2	yes	—
123	C₂×C₄₇	C₂×#489	1	2	yes	—
124	C₂×C₄₇	C₂×#490	1	1	yes	characteristic
125	C₂×C₄₇	C₂×#491	1	1	yes	characteristic
126	C₂³×C₈	—	47	2	no	—
127	C₂²×C₄⋊C₄	—	47	2	no	—
128	C₂²×C₄⋊C₄	—	94	2	no	—
129	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
130	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
131	C₂×#97	—	47	8	no	—
132	C₂×#97	—	47	4	no	—
133	#384	—	47	4	no	—
134	#384	—	94	2	no	—
135	#382	—	2	2	no	—
136	#382	C₂×D₄₇	1	4	yes	—
137	C₂×#95	—	2	2	no	—
138	#340	—	47	16	no	—
139	C₄₇	C₂×#1836	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	—	2	2	no	—
141	C₂³×C₄	—	94	2	no	—
142	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
143	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
144	C₂²×C₈	—	47	8	no	—
145	C₂²×C₈	—	47	4	no	—
146	C₂²×C₈	—	94	4	no	—
147	C₂²×C₈	—	94	4	no	—
148	C₂×C₄⋊C₄	—	2	8	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	47	4	no	—
151	C₂×C₄⋊C₄	—	94	8	no	—
152	C₂×C₄⋊C₄	—	94	2	no	—
153	C₂×C₄⋊C₄	—	94	8	no	—
154	C₂×C₄⋊C₄	—	188	8	no	—
155	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
156	C₂×C₄²	Dic₉₄	1	4	yes	—
157	C₂×C₄²	C₄×D₄₇	1	4	yes	—
158	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
159	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
160	#97	—	47	16	no	—
161	#95	—	2	8	no	—
162	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	—	94	4	no	—
170	C₂²×C₄	—	188	4	no	—
171	C₂²×C₄	—	188	8	no	—
172	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
173	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
175	C₂²×C₄	#10625352	1	4	yes	—
176	C₂²×C₄	#10625354	1	4	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
180	C₂×C₈	—	47	8	no	—
181	C₂×C₈	—	94	16	no	—
182	C₂×C₈	—	94	4	no	—
183	C₂×C₈	—	94	4	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	94	8	no	—
187	C₄⋊C₄	—	188	16	no	—
188	C₄⋊C₄	—	188	4	no	—
189	C₄²	Dic₄₇⋊C₄	1	8	yes	—
190	C₄²	C₂×Dic₉₄	1	4	yes	—
191	C₄²	C₂×C₄×D₄₇	1	4	yes	—
192	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
193	C₂³	#13094011	1	2	yes	—
194	C₂³	#13094013	1	2	yes	—
195	C₂³	#13094049	1	4	yes	—
196	C₂³	#13094054	1	4	yes	—
197	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	16	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	8	no	—
205	C₂×C₄	—	376	8	no	—
206	C₂×C₄	—	188	4	no	—
207	C₂×C₄	—	188	8	no	—
208	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
209	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
210	C₂×C₄	#13093982	1	8	yes	—
211	C₂×C₄	#13094011	1	2	yes	—
212	C₂×C₄	#13094013	1	2	yes	—
213	C₂×C₄	#13094015	1	8	yes	—
214	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
216	C₂×C₄	C₂×#10625352	1	2	yes	—
217	C₂×C₄	C₂×#10625354	1	2	yes	—
218	C₈	—	94	8	no	—
219	C₂²	#15688988	1	4	yes	—
220	C₂²	#15689166	1	8	yes	—
221	C₂²	#15689482	1	8	yes	—
222	C₂²	#15689489	1	4	yes	—
223	C₂²	#15689493	1	4	yes	—
224	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
225	C₂²	C₂×#13094011	1	1	yes	characteristic
226	C₂²	C₂×#13094013	1	1	yes	characteristic
227	C₂²	C₂×#13094049	1	2	yes	—
228	C₂²	C₂×#13094054	1	2	yes	—
229	C₄	—	2	8	no	—
230	C₄	—	8	4	no	—
231	C₄	—	376	4	no	—
232	C₄	#15688988	1	4	yes	—
233	C₄	C₂×#13093982	1	2	yes	—
234	C₄	C₂×#13094015	1	2	yes	—
235	C₂	#20243521	1	8	yes	direct product factor
236	C₂	C₂×#15688988	1	1	yes	characteristic
237	C₂	C₂×#15689166	1	2	yes	—
238	C₂	C₂×#15689482	1	2	yes	—
239	C₂	C₂×#15689489	1	1	yes	characteristic
240	C₂	C₂×#15689493	1	1	yes	characteristic

	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×#1836
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. #20243521	C₂	direct (N × H)
9. C₂×#13093935	C₄	semidirect
29. #13093935	C₂×C₄	semidirect
120. C₂×C₄₇	#1836	semidirect
139. C₄₇	C₂×#1836	semidirect
235. C₂	#20243521	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×#13093935	2
C₂×#15688993	2
#20243521	2
C₂×#15688885	2
C₂×C₄₇×#382	2
C₂×#1836	47

Subgroups — C2×#20243521 #26216474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243521 #26216474