Subgroups — C₂×#20245154 #26369278

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	2	yes	maximal
2	#20245154	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688944	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	—	2	2	no	—
6	C₂×#13093964	—	2	4	no	—
7	C₂×#13093964	C₂²	1	4	yes	—
8	C₂×#13093964	C₂²	1	8	yes	—
9	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
10	#15688944	C₂²	1	16	yes	complemented
11	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
14	C₄₇×#405	C₄	1	4	yes	—
15	C₄₇×#405	C₂²	1	4	yes	—
16	C₂×#97×C₄₇	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
18	#13093964	—	2	8	no	—
19	#13093964	—	2	4	no	—
20	#13093964	C₂³	1	8	yes	—
21	C₂×C₄₇⋊C₁₆	—	2	8	no	—
22	C₂×C₄₇⋊C₁₆	—	2	4	no	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₄₇	C₈	1	4	yes	—
25	C₂×M₁₆×C₄₇	C₈	1	4	yes	—
26	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₇	D₄	1	2	yes	—
31	C₂²×C₈×C₄₇	D₄	1	2	yes	—
32	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
33	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
36	#117×C₄₇	—	2	2	no	—
37	#117×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	#97×C₄₇	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₄₇	C₂³	1	4	yes	—
44	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	M₁₆×C₄₇	C₂×C₈	1	16	yes	—
55	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₂²⋊C₄×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
75	C₄₇⋊C₁₆	—	2	8	no	—
76	C₂³×C₄₇	—	4	2	no	—
77	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
78	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
79	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
85	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₈×C₄₇	C₂²×C₈	1	8	yes	—
100	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×#2153	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	4	no	—
105	C₂²×C₄₇	—	4	2	no	—
106	C₂²×C₄₇	—	4	2	no	—
107	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
109	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
111	C₄×C₄₇	—	4	4	no	—
112	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
115	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
117	C₂²×M₃₂	—	47	2	no	—
118	C₂×#405	D₄₇	1	1	yes	characteristic
119	#2153	—	47	8	no	—
120	C₂×#364	—	47	4	no	—
121	C₂×C₄₇	—	2	1	no	—
122	C₂×C₄₇	—	2	1	no	—
123	C₂×C₄₇	—	2	4	no	—
124	C₂×C₄₇	#2153	1	2	yes	complemented
125	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
127	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
128	C₂×#117	Dic₄₇	1	1	yes	characteristic
129	C₂×M₃₂	—	47	4	no	—
130	C₂×M₃₂	—	47	8	no	—
131	C₂×M₃₂	—	94	4	no	—
132	C₂×M₃₂	—	94	2	no	—
133	C₂²×C₁₆	—	94	2	no	—
134	#405	Dic₄₇	1	4	yes	—
135	#405	C₂×D₄₇	1	4	yes	—
136	C₂×#97	Dic₄₇	1	2	yes	—
137	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
138	#364	—	47	16	no	—
139	C₄₇	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
141	C₂×M₁₆	C₄₇⋊C₈	1	4	yes	—
142	C₂×M₁₆	C₄₇⋊C₈	1	4	yes	—
143	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
144	C₂²×C₈	Dic₉₄	1	2	yes	—
145	C₂²×C₈	Dic₉₄	1	2	yes	—
146	C₂²×C₈	D₁₈₈	1	2	yes	—
147	C₂²×C₈	D₁₈₈	1	2	yes	—
148	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
149	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
151	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
152	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
153	#117	—	2	2	no	—
154	#117	C₂×Dic₄₇	1	4	yes	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₂²⋊C₄	—	2	1	no	—
157	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
158	M₃₂	—	47	8	no	—
159	M₃₂	—	94	8	no	—
160	M₃₂	—	94	4	no	—
161	C₂×C₁₆	—	94	8	no	—
162	C₂×C₁₆	—	94	4	no	—
163	#97	C₂×Dic₄₇	1	8	yes	—
164	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
165	C₄×C₈	C₂²×D₄₇	1	4	yes	—
166	C₂⁴	#10625329	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
169	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
170	C₂²×C₄	#10625329	1	1	yes	characteristic
171	C₂²×C₄	#10625329	1	2	yes	—
172	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
173	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
176	M₁₆	C₂×C₄₇⋊C₈	1	16	yes	—
177	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
178	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
179	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
180	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
181	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
182	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
183	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
184	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
185	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
186	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
187	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
188	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
189	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
191	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
192	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	C₂²×Dic₄₇	1	4	yes	—
197	C₁₆	—	94	8	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
200	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
201	C₂³	C₂×#10625329	1	1	yes	characteristic
202	C₂³	C₂×#10625329	1	2	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
206	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
207	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
208	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
215	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
216	C₂×C₄	C₂×#10625329	1	2	yes	—
217	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
218	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
219	C₈	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
220	C₈	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
221	C₈	C₂²×C₄₇⋊C₈	1	8	yes	—
222	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
223	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
229	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	4	yes	—
230	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
231	C₂²	C₂²×#10625329	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
234	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	4	yes	—
235	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
236	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
237	C₄	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	2	4	no	—
241	C₂	#20245154	1	2	yes	direct product factor
242	C₂	C₂²×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂⁴×C₄₇	M₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	C₂×#2153
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20245154	C₂	direct (N × H)
3. C₂×#15688944	C₂	semidirect
10. #15688944	C₂²	semidirect
124. C₂×C₄₇	#2153	semidirect
139. C₄₇	C₂×#2153	semidirect
241. C₂	#20245154	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×#13093964	2
#20245154	2
C₂×#15688944	2
C₂×C₄₇×#405	2
C₂×#2153	47

Subgroups — C2×#20245154 #26369278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245154 #26369278