Subgroups — C₂×#20246478 #26375919

Order 12032.

2726 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246478	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689030	C₂	1	4	yes	maximal
3	C₂×#15689022	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₄×Dic₄₇	C₂	1	4	yes	maximal
5	C₂×C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
11	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
12	#15689030	C₂²	1	32	yes	—
13	#15689022	C₂²	1	32	yes	—
14	C₄⋊C₄×Dic₄₇	C₂²	1	32	yes	—
15	C₄×C₄⋊Dic₄₇	C₂²	1	16	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
17	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
18	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Q₈×C₄₇	C₄	1	16	yes	complemented
20	C₂×C₄⋊Q₈×C₄₇	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×Dic₄₇	C₂³	1	4	yes	—
24	C₂³×Dic₄₇	C₂³	1	4	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	8	no	—
26	C₂×C₄⋊Dic₄₇	—	4	8	no	—
27	C₂×C₄⋊Dic₄₇	Q₈	1	16	yes	complemented
28	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
31	C₂×C₄×Dic₄₇	—	2	16	no	—
32	C₂×C₄×Dic₄₇	—	4	8	no	—
33	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	16	yes	—
36	C₄₇⋊#95	C₂³	1	32	yes	—
37	C₂²×Q₈×C₄₇	—	4	4	no	—
38	C₂²×Q₈×C₄₇	—	4	4	no	—
39	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
40	C₂²×Q₈×C₄₇	C₂³	1	8	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₄₇	C₂×C₄	1	64	yes	complemented
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
49	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₄₇	—	4	32	no	—
51	C₂×Q₈×C₄₇	—	4	8	no	—
52	C₂×Q₈×C₄₇	—	4	8	no	—
53	C₂×Q₈×C₄₇	C₂²×C₄	1	32	yes	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
62	C₂²×C₄×C₄₇	#41	1	8	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
66	C₂²×Dic₄₇	—	2	8	no	—
67	C₂²×Dic₄₇	—	2	8	no	—
68	C₂²×Dic₄₇	—	2	8	no	—
69	C₂²×Dic₄₇	—	2	8	no	—
70	C₂²×Dic₄₇	—	2	8	no	—
71	C₂²×Dic₄₇	C₂×Q₈	1	16	yes	—
72	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
73	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
74	C₄⋊C₄×C₄₇	C₂²×C₄	1	64	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
76	C₄⋊Dic₄₇	—	2	16	no	—
77	C₄⋊Dic₄₇	—	4	16	no	—
78	C₄⋊Dic₄₇	C₂×Q₈	1	32	yes	complemented
79	C₄×Dic₄₇	—	2	32	no	—
80	C₄×Dic₄₇	—	4	16	no	—
81	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
82	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
83	C₂³×C₄₇	C₂×#41	1	4	yes	—
84	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
85	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
86	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
87	Q₈×C₄₇	—	4	32	no	—
88	C₂×C₄×C₄₇	—	2	32	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	C₄×Q₈	1	32	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
98	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
99	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
100	C₂×Dic₄₇	—	2	32	no	—
101	C₂×Dic₄₇	—	2	8	no	—
102	C₂×Dic₄₇	—	2	8	no	—
103	C₂×Dic₄₇	—	2	8	no	—
104	C₂×Dic₄₇	—	2	8	no	—
105	C₂×Dic₄₇	—	2	8	no	—
106	C₂×Dic₄₇	—	4	16	no	—
107	C₂×Dic₄₇	C₂²×Q₈	1	16	yes	—
108	C₂×#2369	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	C₂×C₄×Q₈	1	8	yes	—
110	C₂²×C₄₇	#518	1	4	yes	—
111	C₂²×C₄₇	#554	1	8	yes	—
112	C₂²×C₄₇	#555	1	4	yes	—
113	C₂²×C₄₇	Q₈²	1	4	yes	—
114	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
116	C₂²×C₄₇	C₂²×#41	1	2	yes	—
117	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
119	C₄×C₄₇	—	2	16	no	—
120	C₄×C₄₇	—	2	4	no	—
121	C₄×C₄₇	C₂×C₄×Q₈	1	16	yes	—
122	Dic₄₇	—	2	16	no	—
123	Dic₄₇	—	4	8	no	—
124	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
125	#2369	—	47	16	no	—
126	C₂×#391	—	47	4	no	—
127	C₂×#389	—	47	4	no	—
128	C₂×C₄×C₄⋊C₄	—	47	2	no	—
129	C₂×C₄×C₄⋊C₄	—	47	4	no	—
130	C₂×C₄₇	#2369	1	8	yes	complemented
131	C₂×C₄₇	C₂²×C₄×Q₈	1	2	yes	—
132	C₂×C₄₇	C₂×#518	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#554	1	2	yes	—
134	C₂×C₄₇	C₂×#555	1	1	yes	characteristic
135	C₂×C₄₇	C₂×Q₈²	1	1	yes	characteristic
136	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
137	C₂×C₄⋊Q₈	Dic₄₇	1	16	yes	complemented
138	C₂×C₄⋊Q₈	C₂×D₄₇	1	8	yes	—
139	C₂²×C₄⋊C₄	—	47	2	no	—
140	C₂²×C₄⋊C₄	—	47	2	no	—
141	C₂²×C₄⋊C₄	—	47	4	no	—
142	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
143	C₂²×C₄²	—	47	4	no	—
144	C₂²×C₄²	—	47	4	no	—
145	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
146	#391	—	47	32	no	—
147	#389	—	47	32	no	—
148	C₄×C₄⋊C₄	—	47	32	no	—
149	C₄×C₄⋊C₄	—	47	16	no	—
150	C₂×#95	—	47	8	no	—
151	C₂×#95	—	47	4	no	—
152	C₄₇	C₂×#2369	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	4	4	no	—
154	C₂²×Q₈	—	4	4	no	—
155	C₂²×Q₈	C₂×Dic₄₇	1	16	yes	—
156	C₂²×Q₈	C₂²×D₄₇	1	8	yes	—
157	C₂³×C₄	—	47	4	no	—
158	C₂³×C₄	—	47	4	no	—
159	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
160	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
161	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
162	C₄⋊Q₈	C₂×Dic₄₇	1	64	yes	complemented
163	C₂×C₄⋊C₄	—	47	16	no	—
164	C₂×C₄⋊C₄	—	47	8	no	—
165	C₂×C₄⋊C₄	—	47	8	no	—
166	C₂×C₄⋊C₄	—	47	16	no	—
167	C₂×C₄⋊C₄	—	94	8	no	—
168	C₂×C₄⋊C₄	—	188	8	no	—
169	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
170	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
171	C₂×C₄²	—	47	16	no	—
172	C₂×C₄²	—	47	16	no	—
173	C₂×C₄²	—	94	16	no	—
174	C₂×C₄²	—	188	8	no	—
175	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
176	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
177	#95	—	47	32	no	—
178	#95	—	47	16	no	—
179	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
180	C₂×Q₈	—	4	32	no	—
181	C₂×Q₈	—	4	8	no	—
182	C₂×Q₈	—	4	8	no	—
183	C₂×Q₈	C₂²×Dic₄₇	1	32	yes	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	47	16	no	—
189	C₂²×C₄	—	47	8	no	—
190	C₂²×C₄	—	47	8	no	—
191	C₂²×C₄	—	94	8	no	—
192	C₂²×C₄	—	94	8	no	—
193	C₂²×C₄	—	94	8	no	—
194	C₂²×C₄	—	94	8	no	—
195	C₂²×C₄	—	94	8	no	—
196	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
197	C₂²×C₄	#10625354	1	8	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
200	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
201	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
203	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
204	C₄⋊C₄	—	47	32	no	—
205	C₄⋊C₄	—	94	16	no	—
206	C₄⋊C₄	—	188	16	no	—
207	C₄⋊C₄	C₂²×Dic₄₇	1	64	yes	—
208	C₄²	—	94	32	no	—
209	C₄²	—	188	16	no	—
210	C₄²	C₂²×Dic₄₇	1	16	yes	—
211	C₂³	C₂³×Dic₄₇	1	2	yes	—
212	C₂³	C₂×Q₈×D₄₇	1	4	yes	—
213	C₂³	C₂×#10625354	1	4	yes	—
214	C₂³	#13094069	1	2	yes	—
215	C₂³	#13094069	1	2	yes	—
216	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
217	Q₈	—	4	32	no	—
218	C₂×C₄	—	2	32	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	47	16	no	—
225	C₂×C₄	—	94	32	no	—
226	C₂×C₄	—	94	8	no	—
227	C₂×C₄	—	94	8	no	—
228	C₂×C₄	—	94	8	no	—
229	C₂×C₄	—	94	8	no	—
230	C₂×C₄	—	94	8	no	—
231	C₂×C₄	—	188	16	no	—
232	C₂×C₄	Q₈×Dic₄₇	1	32	yes	—
233	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
234	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
235	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
236	C₂×C₄	C₂×Q₈×D₄₇	1	8	yes	—
237	C₂×C₄	C₂×#10625354	1	8	yes	—
238	C₂²	C₂×Q₈×Dic₄₇	1	8	yes	—
239	C₂²	#15689598	1	4	yes	—
240	C₂²	#15689834	1	8	yes	—
241	C₂²	#15689839	1	4	yes	—
242	C₂²	#15689853	1	4	yes	—
243	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
244	C₂²	C₂²×Q₈×D₄₇	1	2	yes	—
245	C₂²	C₂²×#10625354	1	2	yes	—
246	C₂²	C₂×#13094069	1	1	yes	characteristic
247	C₂²	C₂×#13094069	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	2	16	no	—
250	C₄	—	94	16	no	—
251	C₄	—	188	8	no	—
252	C₄	C₂×Q₈×Dic₄₇	1	16	yes	—
253	C₂	#20246478	1	8	yes	direct product factor
254	C₂	C₂²×Q₈×Dic₄₇	1	2	yes	—
255	C₂	C₂×#15689598	1	1	yes	characteristic
256	C₂	C₂×#15689834	1	2	yes	—
257	C₂	C₂×#15689839	1	1	yes	characteristic
258	C₂	C₂×#15689853	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#2369
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20246478	C₂	direct (N × H)
19. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
27. C₂×C₄⋊Dic₄₇	Q₈	semidirect
44. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
78. C₄⋊Dic₄₇	C₂×Q₈	semidirect
130. C₂×C₄₇	#2369	semidirect
137. C₂×C₄⋊Q₈	Dic₄₇	semidirect
152. C₄₇	C₂×#2369	semidirect
162. C₄⋊Q₈	C₂×Dic₄₇	semidirect
253. C₂	#20246478	direct (N × H)

Maximal subgroups

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

subgroup	index
#20246478	2
C₂×#15689030	2
C₂×#15689022	2
C₂×C₄⋊C₄×Dic₄₇	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#2369	47

Subgroups — C2×#20246478 #26375919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246478 #26375919