Subgroups — C₂×#20243525 #26216503

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20243525	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688887	C₂	1	4	yes	maximal
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15689147	—	2	2	no	—
9	#15689147	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
11	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
15	#15688887	C₂²	1	16	yes	—
16	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#423	—	2	2	no	—
19	C₄₇×#423	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	C₂³	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	4	no	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
31	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
32	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	—	2	2	no	—
36	C₂×M₁₆×C₄₇	—	2	8	no	—
37	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₄⋊C₈×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×C₄×C₄₇	#41	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₄₇	—	2	8	no	—
62	M₁₆×C₄₇	—	4	4	no	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	4	4	no	—
65	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂×D₄	1	4	yes	—
71	#10625329	—	2	8	no	—
72	#10625329	—	4	4	no	—
73	C₂×C₄₇⋊C₈	—	2	8	no	—
74	C₂×C₄₇⋊C₈	—	2	8	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	4	4	no	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
80	C₂³×C₄₇	—	2	4	no	—
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₄₇	#125	1	4	yes	—
84	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	#108	1	8	yes	—
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₄₇	#117	1	4	yes	—
94	C₂×C₄×C₄₇	#117	1	4	yes	—
95	C₂×C₄×C₄₇	#124	1	2	yes	—
96	C₂×C₄×C₄₇	#124	1	2	yes	—
97	C₂×C₄×C₄₇	#124	1	2	yes	—
98	C₂×C₄×C₄₇	#124	1	2	yes	—
99	C₂×C₄×C₄₇	#125	1	4	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
104	C₈×C₄₇	—	4	4	no	—
105	C₄₇⋊C₈	—	2	8	no	—
106	C₄₇⋊C₈	—	4	4	no	—
107	C₂×#1837	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	8	no	—
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	#382	1	2	yes	—
112	C₂²×C₄₇	#382	1	2	yes	—
113	C₂²×C₄₇	C₂×#108	1	4	yes	—
114	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×#125	1	2	yes	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	#382	1	2	yes	—
118	C₄×C₄₇	#382	1	2	yes	—
119	C₄×C₄₇	C₂×#108	1	8	yes	—
120	C₄×C₄₇	C₂×#117	1	2	yes	—
121	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
123	C₂×#423	—	47	1	no	—
124	C₂×#423	D₄₇	1	1	yes	characteristic
125	C₂²×C₄×C₈	—	47	1	no	—
126	#1837	—	47	8	no	—
127	C₂×#341	—	47	4	no	—
128	C₂×C₄₇	—	2	2	no	—
129	C₂×C₄₇	—	2	2	no	—
130	C₂×C₄₇	#1837	1	4	yes	complemented
131	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
132	C₂×C₄₇	C₂²×#108	1	2	yes	—
133	C₂²×M₁₆	—	47	2	no	—
134	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
135	C₂³×C₈	—	47	2	no	—
136	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
137	#423	—	2	2	no	—
138	#423	—	47	4	no	—
139	#423	—	94	2	no	—
140	#423	C₂×D₄₇	1	4	yes	—
141	C₂×C₄⋊C₈	—	2	2	no	—
142	C₂×C₄⋊C₈	—	94	2	no	—
143	C₂×C₄×C₈	—	47	2	no	—
144	C₂×C₄×C₈	—	47	4	no	—
145	C₂×C₄×C₈	—	47	2	no	—
146	C₂×C₄×C₈	—	47	4	no	—
147	#341	—	47	16	no	—
148	C₄₇	C₂×#1837	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	Dic₉₄	1	2	yes	—
150	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	8	no	—
153	C₂×M₁₆	—	47	4	no	—
154	C₂×M₁₆	—	94	2	no	—
155	C₂×M₁₆	—	94	8	no	—
156	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	47	4	no	—
159	C₂²×C₈	—	47	4	no	—
160	C₂²×C₈	—	47	4	no	—
161	C₂²×C₈	—	94	4	no	—
162	C₂²×C₈	—	94	4	no	—
163	C₂²×C₈	—	94	2	no	—
164	C₂×C₄²	Dic₉₄	1	2	yes	—
165	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₄×D₄₇	1	2	yes	—
167	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
170	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
171	C₄⋊C₈	—	2	8	no	—
172	C₄⋊C₈	—	94	8	no	—
173	C₄×C₈	—	47	8	no	—
174	C₄×C₈	—	47	8	no	—
175	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
178	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
179	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
180	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
181	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
184	C₂²×C₄	#10625352	1	4	yes	—
185	C₂²×C₄	#10625354	1	4	yes	—
186	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
189	M₁₆	—	2	8	no	—
190	M₁₆	—	4	4	no	—
191	M₁₆	—	94	8	no	—
192	M₁₆	—	188	4	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	47	8	no	—
196	C₂×C₈	—	94	8	no	—
197	C₂×C₈	—	94	8	no	—
198	C₂×C₈	—	94	4	no	—
199	C₂×C₈	—	94	4	no	—
200	C₂×C₈	—	94	4	no	—
201	C₂×C₈	—	188	4	no	—
202	C₄²	Dic₄₇⋊C₄	1	2	yes	—
203	C₄²	Dic₄₇⋊C₄	1	4	yes	—
204	C₄²	Dic₄₇⋊C₄	1	2	yes	—
205	C₄²	C₂×C₄×D₄₇	1	2	yes	—
206	C₄²	C₂×C₄×D₄₇	1	2	yes	—
207	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
208	C₂³	—	2	4	no	—
209	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
210	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
211	C₂³	#13094015	1	4	yes	—
212	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	#13093959	1	8	yes	—
217	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
218	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
221	C₂×C₄	#13093982	1	4	yes	—
222	C₂×C₄	#13093982	1	4	yes	—
223	C₂×C₄	#13094011	1	2	yes	—
224	C₂×C₄	#13094011	1	2	yes	—
225	C₂×C₄	#13094013	1	2	yes	—
226	C₂×C₄	#13094013	1	2	yes	—
227	C₂×C₄	#13094015	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₄	C₂×#10625352	1	2	yes	—
231	C₂×C₄	C₂×#10625354	1	2	yes	—
232	C₈	—	4	4	no	—
233	C₈	—	94	8	no	—
234	C₈	—	188	4	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	#15688988	1	2	yes	—
239	C₂²	#15688988	1	2	yes	—
240	C₂²	C₂×#13093959	1	4	yes	—
241	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
242	C₂²	C₂×#13094015	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	#15688988	1	2	yes	—
245	C₄	#15688988	1	2	yes	—
246	C₄	C₂×#13093959	1	8	yes	—
247	C₄	C₂×#13093982	1	2	yes	—
248	C₄	C₂×#13094011	1	1	yes	characteristic
249	C₄	C₂×#13094013	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	2	2	no	—
252	C₂	#20243525	1	4	yes	direct product factor
253	C₂	C₂×#15688988	1	1	yes	characteristic
254	C₂	C₂²×#13093959	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	C₂×#1837
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20243525	C₂	direct (N × H)
130. C₂×C₄₇	#1837	semidirect
148. C₄₇	C₂×#1837	semidirect
252. C₂	#20243525	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂²×C₄×C₄₇⋊C₈	2
#20243525	2
C₂×#15688887	2
C₂×C₄₇×#423	2
C₂×#1837	47

Subgroups — C2×#20243525 #26216503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243525 #26216503