Subgroups — #26226916

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252026	C₂	1	1	yes	maximal, characteristic
2	#20251952	C₂	1	1	yes	maximal, characteristic
3	#20244164	C₂	1	4	yes	maximal, complemented
4	#20243595	C₂	1	4	yes	maximal
5	#20243563	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243560	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243529	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243526	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₇×#433	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	2	yes	—
11	#15689184	—	2	4	no	—
12	C₂×#13093960	C₂²	1	4	yes	—
13	C₂×#13093960	C₂²	1	4	yes	—
14	#15689150	—	2	4	no	—
15	#15689147	C₂²	1	1	yes	characteristic
16	#15689147	C₂²	1	1	yes	characteristic
17	#15689147	C₂²	1	2	yes	—
18	#15689147	C₂²	1	1	yes	characteristic
19	#15689147	C₂²	1	1	yes	characteristic
20	#15689087	C₂²	1	2	yes	—
21	C₄×#10625329	C₂²	1	2	yes	—
22	#15688888	C₂²	1	4	yes	—
23	#15688888	C₂²	1	4	yes	—
24	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#433	C₄	1	8	yes	—
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	—	4	4	no	—
32	C₂×#10625329	C₂³	1	2	yes	—
33	C₂×#10625329	C₂³	1	2	yes	—
34	C₂×#10625329	C₂³	1	2	yes	—
35	C₂×#10625329	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	—	2	2	no	—
37	#13093960	—	4	4	no	—
38	#13093960	—	4	4	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
40	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
44	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
45	#13093935	—	2	2	no	—
46	C₄×C₄₇⋊C₈	—	2	2	no	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	#117×C₄₇	C₂×C₄	1	4	yes	—
51	#117×C₄₇	D₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
58	#97×C₄₇	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
76	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₂×D₄	1	2	yes	—
80	C₄²×C₄₇	C₂×D₄	1	2	yes	—
81	#10625329	—	2	4	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	4	4	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₄₇	—	2	4	no	—
109	C₈×C₄₇	—	2	4	no	—
110	C₄₇⋊C₈	—	4	2	no	—
111	C₄₇⋊C₈	—	4	2	no	—
112	C₄₇⋊C₈	—	4	2	no	—
113	C₄₇⋊C₈	—	4	2	no	—
114	#21995	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	8	2	no	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	#544	1	2	yes	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	4	no	—
123	C₄×C₄₇	—	2	4	no	—
124	C₄×C₄₇	#520	1	2	yes	—
125	C₄×C₄₇	#535	1	1	yes	characteristic
126	C₄×C₄₇	#536	1	1	yes	characteristic
127	#2921	—	47	1	no	—
128	C₂×#433	D₄₇	1	1	yes	characteristic
129	#2905	—	47	1	no	—
130	#1950	—	47	4	no	—
131	#1855	—	47	4	no	—
132	#1849	—	47	1	no	—
133	#1848	—	47	1	no	—
134	#1838	—	47	1	no	—
135	#1837	—	47	1	no	—
136	C₂×C₄₇	—	2	2	no	—
137	C₂×C₄₇	—	4	2	no	—
138	C₂×C₄₇	#2317	1	1	yes	characteristic
139	C₂×C₄₇	#2916	1	2	yes	—
140	C₂²×M₁₆	—	47	2	no	—
141	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
142	#433	—	94	4	no	—
143	#433	Dic₄₇	1	8	yes	—
144	C₂×#108	—	47	4	no	—
145	C₂×#108	—	47	4	no	—
146	#424	—	94	4	no	—
147	#423	—	47	1	no	—
148	#423	—	47	1	no	—
149	#423	—	47	1	no	—
150	#423	—	47	1	no	—
151	#423	—	47	2	no	—
152	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
153	#407	—	47	2	no	—
154	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
155	C₄×M₁₆	—	47	2	no	—
156	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
158	#341	—	47	4	no	—
159	#341	—	47	4	no	—
160	C₄₇	#21995	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×M₁₆	—	47	2	no	—
163	C₂×M₁₆	—	47	2	no	—
164	C₂×M₁₆	—	47	2	no	—
165	C₂×M₁₆	—	47	2	no	—
166	C₂×M₁₆	—	94	2	no	—
167	C₂×M₁₆	—	188	4	no	—
168	C₂²×C₈	—	94	2	no	—
169	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
170	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
171	#117	D₁₈₈	1	4	yes	—
172	#117	C₂×Dic₄₇	1	4	yes	—
173	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
174	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
177	#108	—	188	4	no	—
178	#108	—	188	4	no	—
179	C₄⋊C₈	—	94	4	no	—
180	C₄⋊C₈	—	94	2	no	—
181	C₄⋊C₈	—	94	2	no	—
182	C₄⋊C₈	—	94	2	no	—
183	C₄⋊C₈	—	94	2	no	—
184	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
185	C₂²⋊C₈	—	94	4	no	—
186	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
187	#97	—	94	2	no	—
188	#97	C₂×Dic₄₇	1	4	yes	—
189	C₄×C₈	—	94	2	no	—
190	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
191	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
196	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
197	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
200	M₁₆	—	94	4	no	—
201	M₁₆	—	94	4	no	—
202	M₁₆	—	94	4	no	—
203	M₁₆	—	94	4	no	—
204	M₁₆	—	188	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	—	94	2	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	—	188	4	no	—
214	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
215	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
216	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
217	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
218	C₄²	C₂×D₁₈₈	1	2	yes	—
219	C₄²	C₂×D₁₈₈	1	2	yes	—
220	C₄²	C₂²×Dic₄₇	1	2	yes	—
221	C₄²	C₂²×Dic₄₇	1	2	yes	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
225	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
233	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
235	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
237	C₂×C₄	#13094070	1	2	yes	—
238	C₂×C₄	#13094071	1	2	yes	—
239	C₈	—	2	4	no	—
240	C₈	—	2	4	no	—
241	C₈	—	188	2	no	—
242	C₈	—	188	2	no	—
243	C₈	—	188	2	no	—
244	C₈	—	188	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
250	C₂²	#15689751	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	#15689606	1	2	yes	—
255	C₄	#15689692	1	1	yes	characteristic
256	C₄	#15689699	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	#20246011	1	1	yes	characteristic
260	C₂	#20251999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#433	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#433	D₄₇
hypocenter	C₄₇	#21995
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #20244164	C₂	semidirect
5. #20243563	C₂	semidirect
6. #20243560	C₂	semidirect
7. #20243529	C₂	semidirect
8. #20243526	C₂	semidirect
160. C₄₇	#21995	semidirect

Maximal subgroups

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

subgroup	index
#20252026	2
#20251952	2
#20244164	2
#20243595	2
#20243563	2
#20243560	2
#20243529	2
#20243526	2
C₂×C₄₇×#433	2
#21995	47