Subgroups — C₂×#20243507 #26216424

Order 12032.

1006 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#15688969	C₂	1	1	yes	maximal, characteristic
3	#20243507	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688877	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₈×C₄₇	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)	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	#15688969	C₄	1	4	yes	—
11	#15688969	C₂²	1	4	yes	—
12	C₂×C₄₇⋊#95	C₄	1	2	yes	—
13	C₂×C₄₇⋊#95	C₄	1	2	yes	—
14	#15688877	C₂²	1	16	yes	—
15	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
19	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₂²×C₄₇⋊C₈	—	2	8	no	—
22	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₄×Dic₄₇	—	2	4	no	—
24	C₂×C₄×Dic₄₇	—	2	4	no	—
25	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
27	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
28	C₄₇⋊#95	C₂×C₄	1	8	yes	—
29	C₄₇⋊#95	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	—	2	4	no	—
34	C₂²×C₈×C₄₇	—	2	8	no	—
35	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	4	yes	—
38	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
40	C₄⋊C₈×C₄₇	—	2	8	no	—
41	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	#41	1	4	yes	—
52	C₂²×C₄×C₄₇	#41	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	8	no	—
55	C₂²×Dic₄₇	—	2	8	no	—
56	C₂²×Dic₄₇	—	2	8	no	—
57	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₄₇	—	2	4	no	—
59	C₂×C₈×C₄₇	—	2	8	no	—
60	C₂×C₈×C₄₇	—	4	8	no	—
61	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
62	C₄²×C₄₇	C₂×D₄	1	4	yes	—
63	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
64	C₄×Dic₄₇	—	2	8	no	—
65	C₄×Dic₄₇	—	2	4	no	—
66	C₄×Dic₄₇	—	2	4	no	—
67	C₂×C₄₇⋊C₈	—	2	4	no	—
68	C₂×C₄₇⋊C₈	—	2	8	no	—
69	C₂×C₄₇⋊C₈	—	4	8	no	—
70	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₄₇	#117	1	4	yes	—
72	C₂³×C₄₇	#131	1	4	yes	—
73	C₂³×C₄₇	#131	1	4	yes	—
74	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	2	8	no	—
78	C₂×C₄×C₄₇	—	2	8	no	—
79	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₄₇	#117	1	4	yes	—
83	C₂×C₄×C₄₇	#124	1	4	yes	—
84	C₂×C₄×C₄₇	#124	1	4	yes	—
85	C₂×C₄×C₄₇	#125	1	8	yes	—
86	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
90	C₂×Dic₄₇	—	2	8	no	—
91	C₂×Dic₄₇	—	2	8	no	—
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	4	16	no	—
94	C₈×C₄₇	—	4	4	no	—
95	C₄₇⋊C₈	—	4	4	no	—
96	C₂×#1832	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	#382	1	4	yes	—
98	C₂²×C₄₇	#424	1	4	yes	—
99	C₂²×C₄₇	#424	1	4	yes	—
100	C₂²×C₄₇	#433	1	16	yes	—
101	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×#117	1	2	yes	—
103	C₂²×C₄₇	C₂×#131	1	2	yes	—
104	C₂²×C₄₇	C₂×#131	1	2	yes	—
105	C₄×C₄₇	—	2	4	no	—
106	C₄×C₄₇	—	2	4	no	—
107	C₄×C₄₇	#382	1	4	yes	—
108	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
109	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
110	C₄×C₄₇	C₂×#125	1	2	yes	—
111	Dic₄₇	—	4	8	no	—
112	C₂²×C₄⋊C₈	—	47	1	no	—
113	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
114	C₂×#376	—	47	1	no	—
115	#1832	—	47	8	no	—
116	C₂×#336	—	47	4	no	—
117	C₂×C₄₇	#1832	1	8	yes	complemented
118	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
119	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
121	C₂×C₄₇	C₂×#433	1	4	yes	—
122	C₂³×C₈	—	47	2	no	—
123	C₂³×C₈	C₂×D₄₇	1	2	yes	—
124	C₂²×C₄²	—	47	2	no	—
125	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	2	4	no	—
127	C₂×C₄⋊C₈	—	47	4	no	—
128	C₂×C₄⋊C₈	—	94	4	no	—
129	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
130	#376	—	47	4	no	—
131	#376	—	47	4	no	—
132	C₂×#95	—	47	2	no	—
133	C₂×#95	—	47	2	no	—
134	#336	—	47	16	no	—
135	C₄₇	C₂×#1832	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	—	47	4	no	—
137	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
138	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
139	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
140	C₂²×C₈	—	2	8	no	—
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	47	4	no	—
143	C₂²×C₈	—	94	8	no	—
144	C₂²×C₈	—	94	4	no	—
145	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
146	C₂×C₄²	—	47	4	no	—
147	C₂×C₄²	—	47	4	no	—
148	C₂×C₄²	—	94	4	no	—
149	C₂×C₄²	—	94	4	no	—
150	C₂×C₄²	Dic₉₄	1	4	yes	—
151	C₂×C₄²	C₄×D₄₇	1	2	yes	—
152	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
153	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
154	C₄⋊C₈	—	2	8	no	—
155	C₄⋊C₈	—	94	8	no	—
156	#95	—	47	8	no	—
157	#95	—	47	8	no	—
158	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	47	8	no	—
162	C₂²×C₄	—	94	8	no	—
163	C₂²×C₄	—	94	8	no	—
164	C₂²×C₄	—	94	8	no	—
165	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
166	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
167	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
170	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₄	#10625352	1	4	yes	—
172	C₂²×C₄	#10625354	1	4	yes	—
173	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	4	8	no	—
178	C₂×C₈	—	94	4	no	—
179	C₂×C₈	—	94	8	no	—
180	C₂×C₈	—	188	8	no	—
181	C₄²	—	94	8	no	—
182	C₄²	—	94	4	no	—
183	C₄²	—	94	4	no	—
184	C₄²	Dic₄₇⋊C₄	1	8	yes	—
185	C₄²	C₂×Dic₉₄	1	4	yes	—
186	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
187	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
188	C₂³	#13093982	1	4	yes	—
189	C₂³	#13094030	1	4	yes	—
190	C₂³	#13094032	1	4	yes	—
191	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	94	8	no	—
197	C₂×C₄	—	94	8	no	—
198	C₂×C₄	—	94	8	no	—
199	C₂×C₄	—	188	16	no	—
200	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
201	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
202	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
203	C₂×C₄	#13093982	1	4	yes	—
204	C₂×C₄	#13094011	1	4	yes	—
205	C₂×C₄	#13094013	1	4	yes	—
206	C₂×C₄	#13094015	1	8	yes	—
207	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
209	C₂×C₄	C₂×#10625352	1	2	yes	—
210	C₂×C₄	C₂×#10625354	1	2	yes	—
211	C₈	—	4	4	no	—
212	C₈	—	188	4	no	—
213	C₂²	#15688988	1	4	yes	—
214	C₂²	#15689149	1	4	yes	—
215	C₂²	#15689151	1	4	yes	—
216	C₂²	#15689183	1	16	yes	—
217	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
218	C₂²	C₂×#13093982	1	2	yes	—
219	C₂²	C₂×#13094030	1	2	yes	—
220	C₂²	C₂×#13094032	1	2	yes	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	188	8	no	—
224	C₄	#15688988	1	4	yes	—
225	C₄	C₂×#13094011	1	1	yes	characteristic
226	C₄	C₂×#13094013	1	1	yes	characteristic
227	C₄	C₂×#13094015	1	2	yes	—
228	C₂	#20243507	1	8	yes	direct product factor
229	C₂	C₂×#15688988	1	1	yes	characteristic
230	C₂	C₂×#15689149	1	1	yes	characteristic
231	C₂	C₂×#15689151	1	1	yes	characteristic
232	C₂	C₂×#15689183	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#1832
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20243507	C₂	direct (N × H)
117. C₂×C₄₇	#1832	semidirect
135. C₄₇	C₂×#1832	semidirect
228. C₂	#20243507	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×#15688969	2
#20243507	2
C₂×#15688877	2
C₂²×C₄⋊C₈×C₄₇	2
C₂×#1832	47

Subgroups — C2×#20243507 #26216424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243507 #26216424