Subgroups — C₂×#20242026 #26184320

Order 12032.

954 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	2	yes	maximal
2	#20242026	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688865	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	4	no	—
6	#15689180	C₂²	1	8	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×#13093935	C₂²	1	4	yes	—
10	#15688865	C₂²	1	16	yes	complemented
11	C₄₇×#533	—	2	4	no	—
12	C₄₇×#533	C₄	1	4	yes	—
13	C₄₇×#533	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
15	C₂×#125×C₄₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
17	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
18	C₂×#117×C₄₇	C₄	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
23	#13093935	—	2	4	no	—
24	#13093935	C₂³	1	8	yes	—
25	C₂²×Q₈×C₄₇	—	2	2	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	D₄	1	2	yes	—
28	C₄⋊Q₈×C₄₇	—	2	4	no	—
29	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
30	#125×C₄₇	—	2	4	no	—
31	#125×C₄₇	C₂×C₄	1	4	yes	—
32	#125×C₄₇	C₂³	1	4	yes	—
33	C₂²⋊Q₈×C₄₇	—	2	8	no	—
34	C₂²⋊Q₈×C₄₇	—	2	8	no	—
35	C₄×Q₈×C₄₇	—	2	8	no	—
36	C₄×Q₈×C₄₇	—	2	8	no	—
37	#117×C₄₇	—	2	4	no	—
38	#117×C₄₇	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
50	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₇	—	2	8	no	—
52	C₂×Q₈×C₄₇	—	4	8	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
64	C₄⋊C₄×C₄₇	—	2	8	no	—
65	C₄⋊C₄×C₄₇	—	2	8	no	—
66	C₄⋊C₄×C₄₇	—	2	4	no	—
67	C₄⋊C₄×C₄₇	—	2	8	no	—
68	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
69	C₂²⋊C₄×C₄₇	—	2	8	no	—
70	C₄²×C₄₇	—	2	8	no	—
71	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
72	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₇	C₂×D₄	1	4	yes	—
76	C₄²×C₄₇	C₂×D₄	1	2	yes	—
77	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₄₇⋊C₈	—	2	8	no	—
79	C₂×C₄₇⋊C₈	—	4	8	no	—
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₄₇	—	4	16	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	—	4	4	no	—
93	C₂×C₄×C₄₇	—	4	8	no	—
94	C₂×C₄×C₄₇	#101	1	4	yes	—
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
101	C₄₇⋊C₈	—	4	8	no	—
102	C₂×#1537	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	4	no	—
105	C₂²×C₄₇	—	4	4	no	—
106	C₂²×C₄₇	C₂×#101	1	2	yes	—
107	C₂²×C₄₇	#421	1	4	yes	—
108	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₄₇	—	2	1	no	—
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	—	2	2	no	—
112	C₄×C₄₇	—	2	2	no	—
113	C₄×C₄₇	—	2	4	no	—
114	C₄×C₄₇	—	4	8	no	—
115	C₄×C₄₇	C₂×#101	1	4	yes	—
116	C₂×#533	D₄₇	1	1	yes	characteristic
117	C₂×#432	—	47	2	no	—
118	#1537	—	47	8	no	—
119	C₂×#330	—	47	4	no	—
120	C₂×C₄₇	—	4	2	no	—
121	C₂×C₄₇	#1537	1	4	yes	complemented
122	C₂×C₄₇	C₂²×#101	1	1	yes	characteristic
123	C₂×C₄₇	C₂×#421	1	2	yes	—
124	#533	—	2	4	no	—
125	#533	Dic₄₇	1	4	yes	—
126	#533	C₂×D₄₇	1	4	yes	—
127	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
128	C₂×#125	C₂×D₄₇	1	2	yes	—
129	C₂×C₂²⋊Q₈	—	2	2	no	—
130	C₂×C₄×Q₈	—	2	2	no	—
131	C₂×#117	Dic₄₇	1	2	yes	—
132	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
133	#432	—	47	8	no	—
134	#432	—	94	4	no	—
135	C₂×C₄⋊C₈	—	94	2	no	—
136	C₂×C₂²⋊C₈	—	94	2	no	—
137	C₂×#97	—	47	4	no	—
138	#330	—	47	16	no	—
139	C₄₇	C₂×#1537	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	—	2	2	no	—
141	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
142	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
143	C₂²×C₈	—	94	4	no	—
144	C₄⋊Q₈	—	2	4	no	—
145	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
146	#125	—	2	4	no	—
147	#125	C₂×Dic₄₇	1	4	yes	—
148	#125	C₂²×D₄₇	1	4	yes	—
149	C₂²⋊Q₈	—	2	8	no	—
150	C₂²⋊Q₈	—	2	8	no	—
151	C₄×Q₈	—	2	8	no	—
152	C₄×Q₈	—	2	8	no	—
153	#117	—	2	4	no	—
154	#117	C₂×Dic₄₇	1	8	yes	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
158	C₂×C₂²⋊C₄	—	2	2	no	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
162	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
166	C₄⋊C₈	—	94	8	no	—
167	C₂²⋊C₈	—	94	8	no	—
168	#97	—	47	8	no	—
169	#97	—	94	4	no	—
170	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	—	4	8	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
177	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
179	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
184	C₂×C₈	—	94	8	no	—
185	C₂×C₈	—	188	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	8	no	—
190	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
191	C₂²⋊C₄	—	2	8	no	—
192	C₄²	—	2	8	no	—
193	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
194	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
195	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
196	C₄²	C₂²×Dic₄₇	1	2	yes	—
197	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
198	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
199	C₄²	C₂³×D₄₇	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
203	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
204	Q₈	—	4	16	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	#13093943	1	4	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
217	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
218	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
219	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
221	C₈	—	188	8	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×#13093943	1	2	yes	—
226	C₂²	#15689140	1	4	yes	—
227	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	8	no	—
234	C₄	C₂×#13093943	1	4	yes	—
235	C₂	—	4	2	no	—
236	C₂	#20242026	1	4	yes	direct product factor
237	C₂	C₂²×#13093943	1	1	yes	characteristic
238	C₂	C₂×#15689140	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#1537
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20242026	C₂	direct (N × H)
3. C₂×#15688865	C₂	semidirect
10. #15688865	C₂²	semidirect
121. C₂×C₄₇	#1537	semidirect
139. C₄₇	C₂×#1537	semidirect
236. C₂	#20242026	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×#15689180	2
#20242026	2
C₂×#15688865	2
C₂×C₄₇×#533	2
C₂×#1537	47

Subgroups — C2×#20242026 #26184320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242026 #26184320