Subgroups — C₂×#20243245 #26214154

Order 12032.

1246 conjugacy classes of subgroups.

The subgroup lattice has 248 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	2	yes	maximal
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20243245	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688878	C₂	1	2	yes	maximal
6	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	16	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
11	C₂×#13093935	—	2	4	no	—
12	C₂×#13093935	C₂²	1	4	yes	—
13	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
15	#15688878	C₂²	1	8	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#376	C₄	1	4	yes	—
20	C₄₇×#376	C₂²	1	4	yes	—
21	C₂×#95×C₄₇	C₄	1	4	yes	—
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	4	no	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	32	yes	complemented
31	#13093935	—	2	8	no	—
32	C₄×C₄₇⋊C₈	—	2	8	no	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	—	2	4	no	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	4	yes	—
43	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	#95×C₄₇	C₂×C₄	1	16	yes	—
48	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	8	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₄²×C₄₇	—	2	4	no	—
65	C₄²×C₄₇	—	2	4	no	—
66	C₄²×C₄₇	C₄²	1	8	yes	—
67	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
69	C₄²×C₄₇	C₂×D₄	1	4	yes	—
70	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
71	C₂×C₄₇⋊C₈	—	2	16	no	—
72	C₂×C₄₇⋊C₈	—	2	16	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
78	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
79	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
80	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₄₇	#117	1	2	yes	—
82	C₂³×C₄₇	#131	1	8	yes	—
83	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	2	16	no	—
87	C₂×C₄×C₄₇	—	2	16	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	4	8	no	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	#117	1	2	yes	—
96	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
97	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₄₇⋊C₈	—	2	8	no	—
103	C₄₇⋊C₈	—	2	8	no	—
104	C₂×#1768	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
106	C₂²×C₄₇	#405	1	8	yes	—
107	C₂²×C₄₇	#424	1	8	yes	—
108	C₂²×C₄₇	#433	1	8	yes	—
109	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
110	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#131	1	4	yes	—
113	C₄×C₄₇	—	2	8	no	—
114	C₄×C₄₇	—	2	8	no	—
115	C₄×C₄₇	—	4	4	no	—
116	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
117	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
118	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
119	C₂²×C₄⋊C₈	—	47	2	no	—
120	C₂²×#97	—	47	1	no	—
121	C₂²×C₄×C₈	—	47	1	no	—
122	C₂×#376	D₄₇	1	1	yes	characteristic
123	#1768	—	47	8	no	—
124	C₂×#336	—	47	2	no	—
125	C₂×C₄₇	#1768	1	8	yes	complemented
126	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄₇	C₂×#405	1	2	yes	—
128	C₂×C₄₇	C₂×#424	1	2	yes	—
129	C₂×C₄₇	C₂×#433	1	2	yes	—
130	C₂³×C₈	—	47	2	no	—
131	C₂³×C₈	—	47	2	no	—
132	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
133	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	47	16	no	—
136	C₂×C₄⋊C₈	—	47	8	no	—
137	C₂×#97	—	47	4	no	—
138	C₂×#97	—	94	4	no	—
139	C₂×C₄×C₈	—	47	4	no	—
140	C₂×C₄×C₈	—	94	4	no	—
141	#376	Dic₄₇	1	4	yes	—
142	#376	C₂×D₄₇	1	4	yes	—
143	C₂×#95	Dic₄₇	1	4	yes	—
144	#336	—	47	8	no	—
145	C₄₇	C₂×#1768	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
147	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
148	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
149	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂²×C₈	—	47	8	no	—
151	C₂²×C₈	—	47	8	no	—
152	C₂²×C₈	—	47	4	no	—
153	C₂²×C₈	—	47	4	no	—
154	C₂²×C₈	—	94	4	no	—
155	C₂²×C₈	—	94	4	no	—
156	C₂²×C₈	—	94	4	no	—
157	C₂²×C₈	—	94	4	no	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	Dic₉₄	1	4	yes	—
160	C₂×C₄²	C₄×D₄₇	1	8	yes	—
161	C₂×C₄²	D₁₈₈	1	4	yes	—
162	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
163	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
164	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
166	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
167	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
168	C₄⋊C₈	—	47	32	no	—
169	#97	—	94	8	no	—
170	C₄×C₈	—	94	8	no	—
171	#95	C₂×Dic₄₇	1	16	yes	—
172	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
178	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
179	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
181	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
182	C₂²×C₄	#10625350	1	4	yes	—
183	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
184	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
185	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
186	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
187	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
188	C₂×C₈	—	47	8	no	—
189	C₂×C₈	—	47	8	no	—
190	C₂×C₈	—	94	16	no	—
191	C₂×C₈	—	94	16	no	—
192	C₂×C₈	—	94	4	no	—
193	C₂×C₈	—	94	4	no	—
194	C₂×C₈	—	94	4	no	—
195	C₂×C₈	—	94	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	C₄×Dic₄₇	1	8	yes	—
199	C₄²	C₄⋊Dic₄₇	1	8	yes	—
200	C₄²	C₂×Dic₉₄	1	4	yes	—
201	C₄²	C₂×C₄×D₄₇	1	8	yes	—
202	C₄²	C₂×D₁₈₈	1	4	yes	—
203	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
204	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
205	C₂³	#13093983	1	2	yes	—
206	C₂³	#13094031	1	8	yes	—
207	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	4	8	no	—
215	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
216	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
219	C₂×C₄	#13093983	1	2	yes	—
220	C₂×C₄	C₄×D₁₈₈	1	8	yes	—
221	C₂×C₄	C₄×Dic₉₄	1	8	yes	—
222	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
223	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
225	C₂×C₄	C₂×#10625350	1	2	yes	—
226	C₈	—	94	8	no	—
227	C₈	—	94	8	no	—
228	C₂²	C₄×C₄⋊Dic₄₇	1	4	yes	—
229	C₂²	#15689079	1	8	yes	—
230	C₂²	#15689150	1	8	yes	—
231	C₂²	#15689184	1	8	yes	—
232	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
233	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
234	C₂²	C₂×#13093983	1	1	yes	characteristic
235	C₂²	C₂×#13094031	1	4	yes	—
236	C₄	—	2	8	no	—
237	C₄	—	2	8	no	—
238	C₄	—	4	4	no	—
239	C₄	C₄×C₄⋊Dic₄₇	1	4	yes	—
240	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
241	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
242	C₂	#20243245	1	8	yes	direct product factor
243	C₂	C₂×C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
244	C₂	C₂×#15689079	1	2	yes	—
245	C₂	C₂×#15689150	1	2	yes	—
246	C₂	C₂×#15689184	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#376	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	C₂×#1768
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
4. #20243245	C₂	direct (N × H)
9. C₂×C₄⋊(C₄₇⋊C₈)	C₄	semidirect
30. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
125. C₂×C₄₇	#1768	semidirect
145. C₄₇	C₂×#1768	semidirect
242. C₂	#20243245	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
C₂²×#13093935	2
C₂²×C₄×C₄₇⋊C₈	2
#20243245	2
C₂×#15688878	2
C₂×C₄₇×#376	2
C₂×#1768	47

Subgroups — C2×#20243245 #26214154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243245 #26214154