Subgroups — #26182056

Order 12032.

688 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252355	C₂	1	2	yes	maximal
2	#20241691	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241688	C₂	1	2	yes	maximal, complemented
4	#20241685	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241683	C₂	1	2	yes	maximal, complemented
6	#20241681	C₂	1	2	yes	maximal, complemented
7	#20241678	C₂	1	2	yes	maximal, complemented
8	#20241677	C₂	1	2	yes	maximal, complemented
9	C₄₇×#2922	C₂	1	1	yes	maximal, characteristic
10	#15689254	C₄	1	4	yes	—
11	#15689201	C₄	1	4	yes	—
12	#15689195	C₄	1	4	yes	—
13	#15689195	C₄	1	4	yes	—
14	#15689176	C₂²	1	2	yes	—
15	#15689176	C₂²	1	2	yes	—
16	#15689176	C₂²	1	2	yes	—
17	C₄×#10625329	C₂²	1	2	yes	—
18	C₄×#10625329	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	#15688845	C₂²	1	2	yes	—
21	#15688845	C₂²	1	2	yes	—
22	#15688845	C₂²	1	2	yes	—
23	#15688845	C₂²	1	2	yes	—
24	#15688844	C₂²	1	2	yes	—
25	#15688844	C₂²	1	2	yes	—
26	#15688844	C₂²	1	1	yes	characteristic
27	#15688844	C₂²	1	1	yes	characteristic
28	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#433	C₄	1	4	yes	—
32	C₄₇×#431	C₂²	1	2	yes	—
33	C₄₇×#431	C₂²	1	2	yes	—
34	C₄₇×#424	C₄	1	2	yes	—
35	C₄₇×#408	C₄	1	2	yes	—
36	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×#10625329	C₂×C₄	1	4	yes	—
39	C₂×#10625329	C₂×C₄	1	4	yes	—
40	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
42	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
44	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
45	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
46	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
47	#13093935	C₂×C₄	1	4	yes	—
48	#13093935	C₂×C₄	1	4	yes	—
49	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
50	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
51	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
52	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
53	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
54	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
61	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
62	#117×C₄₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
66	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
67	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
69	#97×C₄₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	2	yes	—
74	#41×C₄₇	—	4	2	no	—
75	C₂×Q₈×C₄₇	C₄²	1	1	yes	characteristic
76	C₂×D₄×C₄₇	C₄²	1	1	yes	characteristic
77	C₂×D₄×C₄₇	C₄²	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
83	M₁₆×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
89	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
90	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
91	C₂²⋊C₄×C₄₇	C₄²	1	1	yes	characteristic
92	C₂²⋊C₄×C₄₇	C₄²	1	1	yes	characteristic
93	C₂²⋊C₄×C₄₇	C₄²	1	2	yes	—
94	C₂²⋊C₄×C₄₇	C₄²	1	2	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
101	#10625329	—	4	4	no	—
102	#10625329	—	4	4	no	—
103	C₂×C₄₇⋊C₈	—	4	4	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
110	Q₈×C₄₇	—	4	1	no	—
111	D₄×C₄₇	—	4	1	no	—
112	D₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₈×C₄₇	—	2	4	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	—	2	2	no	—
128	C₄₇⋊C₈	—	2	4	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	C₄₇⋊C₈	—	2	4	no	—
131	C₄₇⋊C₈	—	2	4	no	—
132	#17342	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	#568	1	4	yes	—
145	C₄×C₄₇	#568	1	2	yes	—
146	#2965	—	47	2	no	—
147	#2922	D₄₇	1	1	yes	characteristic
148	#1455	—	47	1	no	—
149	#1454	—	47	2	no	—
150	#1453	—	47	2	no	—
151	#1453	—	47	1	no	—
152	#1452	—	47	2	no	—
153	#1451	—	47	2	no	—
154	#1451	—	47	2	no	—
155	C₂×C₄₇	—	4	1	no	—
156	C₂×C₄₇	—	4	2	no	—
157	C₂×C₄₇	#2873	1	1	yes	characteristic
158	C₂×C₄₇	#2875	1	2	yes	—
159	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
160	#446	—	47	4	no	—
161	#436	—	47	4	no	—
162	#435	—	47	4	no	—
163	#435	—	47	4	no	—
164	#433	Dic₄₇	1	4	yes	—
165	#431	—	47	2	no	—
166	#431	—	47	2	no	—
167	#431	—	47	2	no	—
168	#431	C₂×D₄₇	1	2	yes	—
169	#431	C₂×D₄₇	1	2	yes	—
170	#424	Dic₄₇	1	2	yes	—
171	#408	Dic₄₇	1	2	yes	—
172	C₄×M₁₆	—	47	2	no	—
173	C₄×M₁₆	—	47	2	no	—
174	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
175	C₂×C₄×C₈	—	47	2	no	—
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	#322	—	47	2	no	—
178	#322	—	47	1	no	—
179	#322	—	47	2	no	—
180	#322	—	47	1	no	—
181	#322	—	47	2	no	—
182	#322	—	47	2	no	—
183	#322	—	47	2	no	—
184	#322	—	47	2	no	—
185	C₈²	—	47	1	no	—
186	C₈²	—	47	1	no	—
187	C₄₇	#17342	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
189	C₂×M₁₆	—	47	4	no	—
190	C₂×M₁₆	—	47	4	no	—
191	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
192	C₂²×C₈	—	47	4	no	—
193	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
194	C₄×Q₈	C₄×D₄₇	1	2	yes	—
195	C₄×D₄	C₄×D₄₇	1	2	yes	—
196	C₄×D₄	C₄×D₄₇	1	2	yes	—
197	C₄×D₄	C₄×D₄₇	1	2	yes	—
198	#117	C₂×Dic₄₇	1	1	yes	characteristic
199	#117	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₈	—	47	4	no	—
203	C₄⋊C₈	—	47	4	no	—
204	C₄⋊C₈	—	47	4	no	—
205	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
206	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
207	C₂²⋊C₈	—	47	4	no	—
208	C₂²⋊C₈	—	47	4	no	—
209	C₂²⋊C₈	—	47	4	no	—
210	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
211	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
212	#97	—	47	4	no	—
213	#97	—	47	4	no	—
214	#97	C₂×Dic₄₇	1	2	yes	—
215	C₄×C₈	—	47	4	no	—
216	C₄×C₈	—	47	2	no	—
217	C₄×C₈	—	47	2	no	—
218	C₄×C₈	—	47	2	no	—
219	C₄×C₈	—	47	2	no	—
220	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
221	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₄₇	1	2	yes	—
224	#41	—	4	2	no	—
225	C₂×Q₈	C₄×Dic₄₇	1	1	yes	characteristic
226	C₂×D₄	C₄×Dic₄₇	1	1	yes	characteristic
227	C₂×D₄	C₄×Dic₄₇	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
231	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
232	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
233	M₁₆	—	4	2	no	—
234	M₁₆	—	188	4	no	—
235	M₁₆	—	188	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	47	4	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	188	4	no	—
242	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
243	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
244	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
245	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
246	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
247	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
248	C₂²⋊C₄	C₄×Dic₄₇	1	1	yes	characteristic
249	C₂²⋊C₄	C₄×Dic₄₇	1	1	yes	characteristic
250	C₂²⋊C₄	C₄×Dic₄₇	1	2	yes	—
251	C₂²⋊C₄	C₄×Dic₄₇	1	2	yes	—
252	C₄²	C₂×C₄×D₄₇	1	2	yes	—
253	C₄²	C₂×C₄×D₄₇	1	2	yes	—
254	C₄²	C₂×C₄×D₄₇	1	2	yes	—
255	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
256	C₄²	C₂²×Dic₄₇	1	2	yes	—
257	C₄²	C₂³×D₄₇	1	1	yes	characteristic
258	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
259	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
260	Q₈	—	4	1	no	—
261	D₄	—	4	1	no	—
262	D₄	—	4	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
269	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
272	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
274	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	94	4	no	—
279	C₈	—	94	4	no	—
280	C₈	—	94	4	no	—
281	C₈	—	94	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	#15689888	1	4	yes	—
294	C₄	#15689890	1	2	yes	—
295	C₂	—	4	1	no	—
296	C₂	—	4	2	no	—
297	C₂	#20251807	1	1	yes	characteristic
298	C₂	#20251814	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2922	C₂
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#2922	D₄₇
hypocenter	C₄₇	#17342
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #2922

Complements

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

N	H ≅ G/N	type
2. #20241691	C₂	semidirect
3. #20241688	C₂	semidirect
4. #20241685	C₂	semidirect
5. #20241683	C₂	semidirect
6. #20241681	C₂	semidirect
7. #20241678	C₂	semidirect
8. #20241677	C₂	semidirect
187. C₄₇	#17342	semidirect

Maximal subgroups

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

subgroup	index
#20252355	2
#20241691	2
#20241688	2
#20241685	2
#20241683	2
#20241681	2
#20241678	2
#20241677	2
C₄₇×#2922	2
#17342	47