Subgroups — C₂×#20242114 #26184815

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#20242114	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688872	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	4	no	—
6	#15689180	C₂²	1	8	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×#13093935	—	2	2	no	—
10	#15688872	C₂²	1	16	yes	complemented
11	C₂³×Q₈×C₄₇	C₄	1	2	yes	—
12	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₄₇	C₄	1	4	yes	—
15	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
16	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
24	#13093935	—	2	8	no	—
25	C₂²×Q₈×C₄₇	—	2	4	no	—
26	C₂²×Q₈×C₄₇	—	2	2	no	—
27	C₂²×Q₈×C₄₇	—	4	8	no	—
28	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
29	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₄₇	—	2	2	no	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	D₄	1	2	yes	—
33	C₄⋊Q₈×C₄₇	—	2	4	no	—
34	C₄⋊Q₈×C₄₇	—	2	8	no	—
35	C₄⋊Q₈×C₄₇	—	2	8	no	—
36	C₄⋊Q₈×C₄₇	—	2	4	no	—
37	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
38	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
39	C₄⋊Q₈×C₄₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	4	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	—	2	4	no	—
53	C₂×Q₈×C₄₇	—	4	16	no	—
54	C₂×Q₈×C₄₇	—	4	8	no	—
55	C₂×Q₈×C₄₇	—	4	16	no	—
56	C₂×Q₈×C₄₇	—	4	8	no	—
57	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	8	no	—
60	C₂²×C₄×C₄₇	—	2	8	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	4	8	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₄⋊C₄×C₄₇	—	2	16	no	—
72	C₄⋊C₄×C₄₇	—	2	8	no	—
73	C₄⋊C₄×C₄₇	—	2	8	no	—
74	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	4	yes	—
79	C₄²×C₄₇	C₂×D₄	1	2	yes	—
80	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₄₇⋊C₈	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	4	8	no	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	#100	1	4	yes	—
85	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₄₇	—	4	8	no	—
88	Q₈×C₄₇	—	4	8	no	—
89	Q₈×C₄₇	—	4	16	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	4	16	no	—
99	C₂×C₄×C₄₇	—	4	8	no	—
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	#137	1	8	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
107	C₄₇⋊C₈	—	4	8	no	—
108	C₂×#1558	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	8	no	—
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	C₂×#100	1	4	yes	—
112	C₂²×C₄₇	C₂×#100	1	2	yes	—
113	C₂²×C₄₇	#419	1	4	yes	—
114	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₄₇	—	2	1	no	—
116	C₄×C₄₇	—	2	2	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	4	8	no	—
120	C₄×C₄₇	#419	1	4	yes	—
121	C₄×C₄₇	C₂×#137	1	4	yes	—
122	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
123	C₂×#432	—	47	2	no	—
124	#1558	—	47	8	no	—
125	C₂×#333	—	47	4	no	—
126	C₂×C₄₇	—	2	4	no	—
127	C₂×C₄₇	#1558	1	4	yes	complemented
128	C₂×C₄₇	C₂²×#100	1	1	yes	characteristic
129	C₂×C₄₇	C₂×#419	1	2	yes	—
130	C₂³×Q₈	Dic₄₇	1	2	yes	—
131	C₂×C₄⋊Q₈	—	2	2	no	—
132	C₂×C₄⋊Q₈	—	2	4	no	—
133	C₂×C₄⋊Q₈	Dic₄₇	1	4	yes	—
134	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
135	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
136	C₂×C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
137	C₂²×C₄⋊C₄	—	2	2	no	—
138	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
139	#432	—	47	8	no	—
140	#432	—	94	4	no	—
141	C₂×C₄⋊C₈	—	47	4	no	—
142	C₂×C₂²⋊C₈	—	94	2	no	—
143	C₂×#97	—	94	2	no	—
144	#333	—	47	16	no	—
145	C₄₇	C₂×#1558	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	2	4	no	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	4	8	no	—
149	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
150	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
153	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
154	C₂²×C₈	—	94	4	no	—
155	C₄⋊Q₈	—	2	4	no	—
156	C₄⋊Q₈	—	2	8	no	—
157	C₄⋊Q₈	—	2	8	no	—
158	C₄⋊Q₈	—	2	4	no	—
159	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
160	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
161	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	8	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	8	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
167	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
172	C₄⋊C₈	—	47	8	no	—
173	C₄⋊C₈	—	94	4	no	—
174	C₂²⋊C₈	—	94	8	no	—
175	#97	—	94	8	no	—
176	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
177	C₂×Q₈	—	2	8	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	4	16	no	—
180	C₂×Q₈	—	4	8	no	—
181	C₂×Q₈	—	4	16	no	—
182	C₂×Q₈	—	4	8	no	—
183	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
190	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
191	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
192	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
197	C₂×C₈	—	94	8	no	—
198	C₂×C₈	—	188	8	no	—
199	C₄⋊C₄	—	2	16	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
203	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
204	C₄²	C₂²×Dic₄₇	1	2	yes	—
205	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
206	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
207	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
208	C₄²	C₂³×D₄₇	1	1	yes	characteristic
209	C₂³	—	2	4	no	—
210	C₂³	#13093941	1	4	yes	—
211	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
212	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
213	Q₈	—	4	8	no	—
214	Q₈	—	4	8	no	—
215	Q₈	—	4	16	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	16	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
230	C₂×C₄	#13094055	1	8	yes	—
231	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
233	C₈	—	188	8	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	C₂×#13093941	1	4	yes	—
237	C₂²	C₂×#13093941	1	2	yes	—
238	C₂²	#15689132	1	4	yes	—
239	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	4	8	no	—
245	C₄	#15689132	1	4	yes	—
246	C₄	C₂×#13094055	1	4	yes	—
247	C₂	—	2	4	no	—
248	C₂	#20242114	1	4	yes	direct product factor
249	C₂	C₂²×#13093941	1	1	yes	characteristic
250	C₂	C₂×#15689132	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#1558
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20242114	C₂	direct (N × H)
3. C₂×#15688872	C₂	semidirect
10. #15688872	C₂²	semidirect
127. C₂×C₄₇	#1558	semidirect
145. C₄₇	C₂×#1558	semidirect
248. C₂	#20242114	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
#20242114	2
C₂×#15688872	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#1558	47

Subgroups — C2×#20242114 #26184815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242114 #26184815