Subgroups — C₂×#20245974 #26374838

Order 12032.

5214 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093983	C₂	1	2	yes	maximal
2	#20245974	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689030	C₂	1	8	yes	maximal, complemented
4	C₂×#15688986	C₂	1	4	yes	maximal
5	C₂⁴×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	—	2	16	no	—
7	C₂×#13093983	C₂²	1	16	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
11	#15689030	C₂²	1	64	yes	complemented
12	#15688986	C₂²	1	32	yes	—
13	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
14	C₂³×Q₈×C₄₇	—	2	16	no	—
15	C₂³×Q₈×C₄₇	C₄	1	16	yes	complemented
16	C₂³×Q₈×C₄₇	C₂²	1	4	yes	—
17	C₂³×Q₈×C₄₇	C₂²	1	8	yes	—
18	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×Dic₄₇	C₂³	1	8	yes	—
21	#13093983	—	2	64	no	—
22	C₂×C₄⋊Dic₄₇	—	2	16	no	—
23	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
24	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
25	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
26	C₂×C₄×Dic₄₇	—	2	16	no	—
27	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
28	C₄₇⋊#95	C₂³	1	64	yes	—
29	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Q₈×C₄₇	—	2	128	no	—
31	C₂²×Q₈×C₄₇	—	2	16	no	—
32	C₂²×Q₈×C₄₇	—	2	16	no	—
33	C₂²×Q₈×C₄₇	—	2	64	no	—
34	C₂²×Q₈×C₄₇	C₂×C₄	1	32	yes	—
35	C₂²×Q₈×C₄₇	C₂×C₄	1	64	yes	complemented
36	C₂²×Q₈×C₄₇	C₂³	1	16	yes	—
37	C₂³×C₄×C₄₇	—	2	16	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₇	D₄	1	16	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
45	C₂⁴×C₄₇	—	2	8	no	—
46	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
48	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
49	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₄₇	—	2	128	no	—
51	C₂×Q₈×C₄₇	—	2	256	no	—
52	C₂×Q₈×C₄₇	—	2	32	no	—
53	C₂×Q₈×C₄₇	—	2	32	no	—
54	C₂×Q₈×C₄₇	C₂²×C₄	1	64	yes	—
55	C₂²×C₄×C₄₇	—	2	64	no	—
56	C₂²×C₄×C₄₇	—	2	32	no	—
57	C₂²×C₄×C₄₇	—	2	16	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	64	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	32	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	32	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
68	C₂²×Dic₄₇	—	2	16	no	—
69	C₂²×Dic₄₇	—	2	32	no	—
70	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
71	C₂²⋊Dic₄₇	—	2	32	no	—
72	C₄⋊Dic₄₇	—	2	32	no	—
73	C₄×Dic₄₇	—	2	32	no	—
74	C₂³×C₄₇	—	2	16	no	—
75	C₂³×C₄₇	—	2	32	no	—
76	C₂³×C₄₇	—	2	8	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
78	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
79	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
80	C₂³×C₄₇	C₂²×D₄	1	8	yes	—
81	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
82	C₂³×C₄₇	2₋¹⁺⁴	1	8	yes	—
83	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
84	Q₈×C₄₇	—	2	128	no	—
85	C₂×C₄×C₄₇	—	2	64	no	—
86	C₂×C₄×C₄₇	—	2	32	no	—
87	C₂×C₄×C₄₇	—	2	16	no	—
88	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	64	yes	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	32	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
91	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
92	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
93	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
94	C₂×Dic₄₇	—	2	16	no	—
95	C₂×Dic₄₇	—	2	32	no	—
96	C₂×Dic₄₇	—	4	32	no	—
97	C₂×#2310	—	47	1	no	maximal, Sylow
98	C₂²×C₄₇	—	2	32	no	—
99	C₂²×C₄₇	—	2	8	no	—
100	C₂²×C₄₇	—	2	16	no	—
101	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	8	yes	—
102	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	4	yes	—
103	C₂²×C₄₇	#519	1	8	yes	—
104	C₂²×C₄₇	#536	1	16	yes	—
105	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
106	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
107	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	4	yes	—
108	C₄×C₄₇	—	2	16	no	—
109	C₄×C₄₇	C₂²×C₂²⋊C₄	1	16	yes	—
110	Dic₄₇	—	4	16	no	—
111	C₂⁴×Q₈	D₄₇	1	1	yes	characteristic
112	C₂²×#117	—	47	2	no	—
113	#2310	—	47	16	no	—
114	C₂×#391	—	47	8	no	—
115	C₂×#381	—	47	4	no	—
116	C₂×C₄₇	—	2	8	no	—
117	C₂×C₄₇	#2310	1	8	yes	complemented
118	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄₇	C₂×#519	1	2	yes	—
120	C₂×C₄₇	C₂×#536	1	4	yes	—
121	C₂³×Q₈	—	2	16	no	—
122	C₂³×Q₈	Dic₄₇	1	16	yes	complemented
123	C₂³×Q₈	C₂×D₄₇	1	4	yes	—
124	C₂³×Q₈	C₂×D₄₇	1	8	yes	—
125	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
126	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
127	C₂×#117	—	47	16	no	—
128	C₂×#117	—	94	16	no	—
129	C₂²×C₄⋊C₄	—	47	4	no	—
130	C₂²×C₂²⋊C₄	—	47	4	no	—
131	C₂²×C₄²	—	47	4	no	—
132	#391	—	47	64	no	—
133	#381	—	47	32	no	—
134	C₂×#95	—	47	16	no	—
135	C₄₇	C₂×#2310	1	1	yes	characteristic, complemented, Sylow
136	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
137	C₂²×Q₈	—	2	128	no	—
138	C₂²×Q₈	—	2	16	no	—
139	C₂²×Q₈	—	2	16	no	—
140	C₂²×Q₈	—	2	64	no	—
141	C₂²×Q₈	C₂×Dic₄₇	1	32	yes	—
142	C₂²×Q₈	C₂×Dic₄₇	1	64	yes	complemented
143	C₂²×Q₈	C₂²×D₄₇	1	16	yes	—
144	C₂³×C₄	—	2	16	no	—
145	C₂³×C₄	—	47	8	no	—
146	C₂³×C₄	C₂×Dic₄₇	1	16	yes	—
147	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
148	C₂³×C₄	C₂²⋊D₄₇	1	16	yes	—
149	C₂³×C₄	C₂²×D₄₇	1	8	yes	—
150	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
151	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
152	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
153	#117	—	94	64	no	—
154	C₂×C₄⋊C₄	—	47	16	no	—
155	C₂×C₄⋊C₄	—	94	16	no	—
156	C₂×C₂²⋊C₄	—	47	16	no	—
157	C₂×C₂²⋊C₄	—	94	16	no	—
158	C₂×C₄²	—	47	16	no	—
159	C₂×C₄²	—	94	16	no	—
160	#95	—	47	64	no	—
161	C₂⁴	—	2	8	no	—
162	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
163	C₂⁴	C₂×C₂²⋊D₄₇	1	8	yes	—
164	C₂⁴	C₂³×D₄₇	1	2	yes	—
165	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
166	C₂×Q₈	—	2	128	no	—
167	C₂×Q₈	—	2	256	no	—
168	C₂×Q₈	—	2	32	no	—
169	C₂×Q₈	—	2	32	no	—
170	C₂×Q₈	C₂²×Dic₄₇	1	64	yes	—
171	C₂²×C₄	—	2	64	no	—
172	C₂²×C₄	—	2	32	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	47	16	no	—
175	C₂²×C₄	—	94	16	no	—
176	C₂²×C₄	—	94	32	no	—
177	C₂²×C₄	C₂²⋊Dic₄₇	1	64	yes	—
178	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
179	C₂²×C₄	C₂²×Dic₄₇	1	32	yes	—
180	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₄₇	1	32	yes	—
185	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
186	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
187	C₄⋊C₄	—	94	32	no	—
188	C₂²⋊C₄	—	94	32	no	—
189	C₄²	—	94	32	no	—
190	C₂³	—	2	16	no	—
191	C₂³	—	2	32	no	—
192	C₂³	—	2	8	no	—
193	C₂³	C₂×C₂²⋊Dic₄₇	1	16	yes	—
194	C₂³	C₂³×Dic₄₇	1	4	yes	—
195	C₂³	C₂³×Dic₄₇	1	2	yes	—
196	C₂³	C₂²×C₂²⋊D₄₇	1	8	yes	—
197	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
198	C₂³	#13094072	1	8	yes	—
199	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
200	Q₈	—	2	128	no	—
201	C₂×C₄	—	2	64	no	—
202	C₂×C₄	—	2	32	no	—
203	C₂×C₄	—	2	16	no	—
204	C₂×C₄	—	94	16	no	—
205	C₂×C₄	—	94	32	no	—
206	C₂×C₄	—	188	32	no	—
207	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	64	yes	—
208	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	32	yes	—
209	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
210	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
211	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
213	C₂²	—	2	8	no	—
214	C₂²	—	2	16	no	—
215	C₂²	—	2	32	no	—
216	C₂²	C₂²×C₂²⋊Dic₄₇	1	8	yes	—
217	C₂²	C₂²×C₂²⋊Dic₄₇	1	4	yes	—
218	C₂²	#15689601	1	8	yes	—
219	C₂²	#15689697	1	16	yes	—
220	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
221	C₂²	C₂³×C₂²⋊D₄₇	1	2	yes	—
222	C₂²	C₂×#13094072	1	4	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	188	16	no	—
225	C₄	C₂²×C₂²⋊Dic₄₇	1	16	yes	—
226	C₂	—	2	8	no	—
227	C₂	#20245974	1	8	yes	direct product factor
228	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic
229	C₂	C₂×#15689601	1	2	yes	—
230	C₂	C₂×#15689697	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂⁴×Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁵×C₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂⁴×Q₈	D₄₇
hypocenter	C₄₇	C₂×#2310
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
2. #20245974	C₂	direct (N × H)
3. C₂×#15689030	C₂	semidirect
11. #15689030	C₂²	semidirect
15. C₂³×Q₈×C₄₇	C₄	semidirect
35. C₂²×Q₈×C₄₇	C₂×C₄	semidirect
117. C₂×C₄₇	#2310	semidirect
122. C₂³×Q₈	Dic₄₇	semidirect
135. C₄₇	C₂×#2310	semidirect
142. C₂²×Q₈	C₂×Dic₄₇	semidirect
227. C₂	#20245974	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×#13093983	2
#20245974	2
C₂×#15689030	2
C₂×#15688986	2
C₂⁴×Q₈×C₄₇	2
C₂×#2310	47

Subgroups — C2×#20245974 #26374838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245974 #26374838