Subgroups — #26216513

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
3	#20243528	C₂	1	4	yes	maximal, complemented
4	#20243525	C₂	1	4	yes	maximal, complemented
5	#20243114	C₂	1	2	yes	maximal
6	C₂×#15688887	C₂	1	2	yes	maximal
7	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	#15689150	—	2	4	no	—
12	#15689147	C₂²	1	4	yes	—
13	#15689079	C₄	1	8	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15688887	C₂²	1	4	yes	—
17	#15688887	C₂²	1	8	yes	—
18	#15688887	C₂²	1	4	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#424	—	2	4	no	—
22	C₄₇×#423	C₂²	1	4	yes	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	C₂×C₄	1	4	yes	—
29	C₂×#10625329	C₂³	1	4	yes	—
30	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
38	#13093935	C₂×C₄	1	8	yes	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	4	no	—
43	C₂×M₁₆×C₄₇	—	2	4	no	—
44	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	#117×C₄₇	C₂×C₄	1	4	yes	—
47	#117×C₄₇	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	C₄⋊C₈×C₄₇	—	2	8	no	—
52	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	#41	1	2	yes	—
58	C₂²×C₄×C₄₇	#41	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	M₁₆×C₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	4	4	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	4	4	no	—
67	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
68	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₄₇	C₂×D₄	1	4	yes	—
71	#10625329	—	2	8	no	—
72	#10625329	—	2	4	no	—
73	#10625329	—	2	4	no	—
74	#10625329	—	4	4	no	—
75	C₂×C₄₇⋊C₈	—	2	8	no	—
76	C₂×C₄₇⋊C₈	—	2	1	no	—
77	C₂×C₄₇⋊C₈	—	2	1	no	—
78	C₂×C₄₇⋊C₈	—	2	1	no	—
79	C₂×C₄₇⋊C₈	—	2	1	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	4	4	no	—
84	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂³×C₄₇	—	4	2	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	#125	1	2	yes	—
90	C₂³×C₄₇	#125	1	2	yes	—
91	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	4	4	no	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	#117	1	4	yes	—
96	C₂×C₄×C₄₇	#117	1	4	yes	—
97	C₂×C₄×C₄₇	#124	1	4	yes	—
98	C₂×C₄×C₄₇	#124	1	4	yes	—
99	C₂×C₄×C₄₇	#125	1	2	yes	—
100	C₂×C₄×C₄₇	#125	1	2	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
107	C₈×C₄₇	—	4	4	no	—
108	C₄₇⋊C₈	—	2	2	no	—
109	C₄₇⋊C₈	—	2	2	no	—
110	C₄₇⋊C₈	—	2	4	no	—
111	C₄₇⋊C₈	—	4	4	no	—
112	#20768	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	—	4	2	no	—
116	C₂²×C₄₇	—	4	2	no	—
117	C₂²×C₄₇	#382	1	4	yes	—
118	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#125	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#125	1	1	yes	characteristic
121	C₄×C₄₇	—	2	8	no	—
122	C₄×C₄₇	#382	1	4	yes	—
123	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
124	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
125	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
126	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
127	#2905	—	47	1	no	—
128	#2905	D₄₇	1	1	yes	characteristic
129	C₂×#405	—	47	1	no	—
130	#1838	—	47	4	no	—
131	#1837	—	47	4	no	—
132	#1738	—	47	2	no	—
133	C₂×#341	—	47	2	no	—
134	C₂×C₄₇	—	2	1	no	—
135	C₂×C₄₇	—	2	1	no	—
136	C₂×C₄₇	—	2	4	no	—
137	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
138	C₂×C₄₇	#2916	1	2	yes	—
139	C₂²×M₁₆	—	47	1	no	—
140	C₂²×M₁₆	—	47	2	no	—
141	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
142	C₂³×C₈	—	47	1	no	—
143	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
144	#424	—	2	4	no	—
145	#424	—	94	4	no	—
146	#423	—	47	4	no	—
147	#423	C₂×D₄₇	1	4	yes	—
148	#405	—	47	8	no	—
149	C₂×#97	—	47	2	no	—
150	C₂×C₄×C₈	—	47	2	no	—
151	#341	—	47	4	no	—
152	#341	—	47	8	no	—
153	#341	—	47	4	no	—
154	C₄₇	#20768	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	47	4	no	—
160	C₂×M₁₆	—	47	4	no	—
161	C₂×M₁₆	—	94	2	no	—
162	C₂×M₁₆	—	94	2	no	—
163	C₂×M₁₆	—	94	2	no	—
164	C₂×M₁₆	—	94	4	no	—
165	C₂×M₁₆	—	94	4	no	—
166	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	47	4	no	—
169	C₂²×C₈	—	47	1	no	—
170	C₂²×C₈	—	47	1	no	—
171	C₂²×C₈	—	47	2	no	—
172	C₂²×C₈	—	94	2	no	—
173	C₂²×C₈	—	94	2	no	—
174	C₂²×C₈	—	94	2	no	—
175	#117	C₄×D₄₇	1	4	yes	—
176	#117	C₂²⋊D₄₇	1	4	yes	—
177	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
178	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
179	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
180	C₄⋊C₈	—	2	8	no	—
181	C₄⋊C₈	—	94	8	no	—
182	#97	—	47	8	no	—
183	C₄×C₈	—	47	8	no	—
184	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
186	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
188	C₂²×C₄	#10625352	1	2	yes	—
189	C₂²×C₄	#10625352	1	2	yes	—
190	C₂²×C₄	#10625354	1	2	yes	—
191	C₂²×C₄	#10625354	1	2	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
193	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
194	M₁₆	—	2	4	no	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	94	8	no	—
198	M₁₆	—	94	4	no	—
199	M₁₆	—	94	4	no	—
200	M₁₆	—	188	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	47	2	no	—
204	C₂×C₈	—	47	4	no	—
205	C₂×C₈	—	47	2	no	—
206	C₂×C₈	—	94	8	no	—
207	C₂×C₈	—	94	1	no	—
208	C₂×C₈	—	94	1	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	—	94	1	no	—
211	C₂×C₈	—	94	1	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	188	4	no	—
215	C₄⋊C₄	Dic₄₇⋊C₄	1	8	yes	—
216	C₂²⋊C₄	Dic₄₇⋊C₄	1	8	yes	—
217	C₄²	C₂×C₄×D₄₇	1	4	yes	—
218	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
221	C₂³	#13094015	1	2	yes	—
222	C₂³	#13094015	1	2	yes	—
223	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
226	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
227	C₂×C₄	#13093982	1	4	yes	—
228	C₂×C₄	#13093982	1	4	yes	—
229	C₂×C₄	#13094011	1	4	yes	—
230	C₂×C₄	#13094013	1	4	yes	—
231	C₂×C₄	#13094015	1	2	yes	—
232	C₂×C₄	#13094015	1	2	yes	—
233	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
236	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
238	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
239	C₈	—	4	4	no	—
240	C₈	—	94	2	no	—
241	C₈	—	94	2	no	—
242	C₈	—	94	4	no	—
243	C₈	—	188	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#15688988	1	4	yes	—
249	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
250	C₂²	C₂×#13094015	1	1	yes	characteristic
251	C₂²	C₂×#13094015	1	1	yes	characteristic
252	C₄	—	2	8	no	—
253	C₄	#15688988	1	4	yes	—
254	C₄	C₂×#13093982	1	1	yes	characteristic
255	C₄	C₂×#13093982	1	1	yes	characteristic
256	C₄	C₂×#13094011	1	1	yes	characteristic
257	C₄	C₂×#13094013	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	C₂×#15688988	1	1	yes	characteristic
262	C₂	#20251997	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2905	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2905	D₄₇
hypocenter	C₄₇	#20768
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2905

Complements

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

N	H ≅ G/N	type
3. #20243528	C₂	semidirect
4. #20243525	C₂	semidirect
154. C₄₇	#20768	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
C₂×#15689079	2
#20243528	2
#20243525	2
#20243114	2
C₂×#15688887	2
C₄₇×#2905	2
#20768	47