Subgroups — #26186987

Order 12032.

726 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253967	C₂	1	1	yes	maximal, characteristic
2	#20252101	C₂	1	2	yes	maximal, complemented
3	#20242219	C₂	1	8	yes	maximal, complemented
4	#20242147	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241840	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094030	—	2	2	no	—
8	#15689673	C₂²	1	2	yes	—
9	#15689528	C₄	1	1	yes	characteristic
10	#15689500	C₄	1	1	yes	characteristic
11	#15689470	C₄	1	2	yes	—
12	#15689359	C₄	1	4	yes	—
13	C₄×D₃₇₆	C₂²	1	4	yes	—
14	#15689202	—	2	2	no	—
15	#15689202	C₂²	1	4	yes	—
16	C₈×D₁₈₈	—	2	2	no	—
17	C₈×D₁₈₈	C₂²	1	4	yes	—
18	#15689147	C₂²	1	2	yes	—
19	#15689089	—	2	2	no	—
20	#15688876	C₂²	1	4	yes	—
21	#15688852	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#322	C₂²	1	4	yes	—
26	C₂×#10625350	—	2	2	no	—
27	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
28	#13094030	—	4	8	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625324	—	2	4	no	—
31	C₂×C₈×D₄₇	—	2	4	no	—
32	#13094024	C₂×C₄	1	2	yes	—
33	#13094021	C₂×C₄	1	2	yes	—
34	#13094012	—	2	2	no	—
35	#13093998	—	2	2	no	—
36	#13093998	C₂×C₄	1	4	yes	—
37	C₄×D₁₈₈	—	2	2	no	—
38	C₄×D₁₈₈	C₂³	1	4	yes	—
39	#13093958	C₂×C₄	1	4	yes	—
40	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
42	#13093946	C₂×C₄	1	8	yes	—
43	(C₂×D₄₇)⋊C₈	—	2	4	no	—
44	(C₂×D₄₇)⋊C₈	—	2	4	no	—
45	C₂²×C₈×C₄₇	—	2	1	no	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₄₇	—	2	1	no	—
51	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	D₄	1	2	yes	—
54	C₄×C₈×C₄₇	C₂³	1	2	yes	—
55	C₄×C₈×C₄₇	C₂³	1	2	yes	—
56	C₄×C₈×C₄₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₇	—	2	4	no	—
61	#10625350	—	4	4	no	—
62	#10625350	—	4	4	no	—
63	C₂×D₁₈₈	—	2	2	no	—
64	C₂×D₁₈₈	—	2	2	no	—
65	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
66	C₂×C₄×D₄₇	—	2	4	no	—
67	C₂×Dic₉₄	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	4	2	no	—
72	C₂×C₈×C₄₇	D₈	1	4	yes	complemented
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₇	#41	1	1	yes	characteristic
80	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
81	(C₂×D₄₇)⋊C₄	—	2	4	no	—
82	(C₂×D₄₇)⋊C₄	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
85	#10625329	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	D₃₇₆	—	4	4	no	—
88	#10625324	—	4	8	no	—
89	C₈×D₄₇	—	4	8	no	—
90	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
91	C₂²×D₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₄₇	#135	1	2	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
102	C₂²⋊D₄₇	—	4	8	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	D₁₈₈	—	2	8	no	—
105	D₁₈₈	—	4	4	no	—
106	D₁₈₈	—	4	4	no	—
107	C₄×D₄₇	—	4	4	no	—
108	C₄×D₄₇	—	4	4	no	—
109	Dic₉₄	—	4	4	no	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	—	2	2	no	—
113	C₈×C₄₇	—	4	2	no	—
114	C₈×C₄₇	C₂×D₈	1	4	yes	—
115	C₄₇⋊C₈	—	4	4	no	—
116	#17975	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	1	no	—
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	C₄×D₈	1	2	yes	—
120	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×D₄₇	—	4	4	no	—
122	C₂×D₄₇	—	4	4	no	—
123	C₂×D₄₇	—	4	4	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	C₄×D₈	1	2	yes	—
127	C₄×C₄₇	#444	1	2	yes	—
128	C₄×C₄₇	#568	1	2	yes	—
129	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
131	Dic₄₇	—	4	4	no	—
132	#3169	—	47	1	no	—
133	#2931	—	47	2	no	—
134	#1581	—	47	8	no	—
135	#1567	—	47	1	no	—
136	#1492	—	47	2	no	—
137	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
138	C₂×C₄₇	—	2	2	no	—
139	C₂×C₄₇	#2932	1	1	yes	characteristic
140	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
142	D₄₇	—	8	4	no	—
143	C₂×#131	—	94	2	no	—
144	#532	—	47	2	no	—
145	#500	—	47	1	no	—
146	#493	—	47	1	no	—
147	#486	—	47	2	no	—
148	#466	—	47	4	no	—
149	C₄×D₈	—	47	4	no	—
150	#436	—	47	4	no	—
151	#436	—	94	2	no	—
152	C₈×D₄	—	47	4	no	—
153	C₈×D₄	—	94	2	no	—
154	#423	—	47	2	no	—
155	#408	—	94	2	no	—
156	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
159	#335	—	47	4	no	—
160	D₄⋊C₈	—	47	8	no	—
161	#322	C₂×D₄₇	1	4	yes	—
162	C₄₇	#17975	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	94	2	no	—
164	C₂×D₈	—	47	4	no	—
165	#131	—	188	8	no	—
166	C₂×M₁₆	—	94	2	no	—
167	C₂×M₁₆	—	94	4	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	94	4	no	—
170	C₂²×C₈	C₄×D₄₇	1	2	yes	—
171	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
172	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
173	C₄⋊Q₈	—	47	2	no	—
174	#127	—	47	2	no	—
175	#124	—	94	2	no	—
176	#121	—	47	4	no	—
177	#121	—	94	2	no	—
178	C₄×D₄	—	47	4	no	—
179	C₄×D₄	—	94	2	no	—
180	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
181	#107	—	47	4	no	—
182	C₄⋊C₈	—	47	4	no	—
183	C₄⋊C₈	—	94	2	no	—
184	#102	—	47	8	no	—
185	C₂²⋊C₈	—	94	4	no	—
186	C₂²⋊C₈	—	94	4	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
189	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
190	C₄×C₈	D₁₈₈	1	2	yes	—
191	C₄×C₈	C₂²×D₄₇	1	2	yes	—
192	C₄×C₈	C₂²×D₄₇	1	2	yes	—
193	C₄×C₈	C₂²×D₄₇	1	2	yes	—
194	#41	—	188	4	no	—
195	#41	—	188	4	no	—
196	C₂×Q₈	—	94	2	no	—
197	C₂×D₄	—	47	4	no	—
198	C₂×D₄	—	94	4	no	—
199	C₂×D₄	—	94	2	no	—
200	C₂×D₄	—	94	2	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂²×C₄	#10625350	1	1	yes	characteristic
205	D₈	—	188	4	no	—
206	M₁₆	—	188	4	no	—
207	M₁₆	—	188	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	94	4	no	—
213	C₂×C₈	—	188	8	no	—
214	C₂×C₈	D₃₇₆	1	4	yes	complemented
215	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
217	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
218	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
219	C₄⋊C₄	—	47	4	no	—
220	C₄⋊C₄	—	94	2	no	—
221	C₂²⋊C₄	—	94	4	no	—
222	C₂²⋊C₄	—	94	4	no	—
223	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
225	C₄²	#10625350	1	1	yes	characteristic
226	C₄²	C₂³×D₄₇	1	1	yes	characteristic
227	C₂³	—	94	4	no	—
228	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
229	Q₈	—	188	4	no	—
230	D₄	—	94	8	no	—
231	D₄	—	188	8	no	—
232	D₄	—	188	4	no	—
233	D₄	—	188	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	94	4	no	—
237	C₂×C₄	—	188	4	no	—
238	C₂×C₄	—	188	4	no	—
239	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
240	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
241	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
243	C₂×C₄	#13094046	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
246	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	188	4	no	—
252	C₈	C₂×D₃₇₆	1	4	yes	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	188	4	no	—
256	C₂²	—	188	4	no	—
257	C₂²	—	188	4	no	—
258	C₂²	C₄×D₃₇₆	1	2	yes	—
259	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	188	4	no	—
263	C₄	C₄×D₃₇₆	1	2	yes	—
264	C₄	#15689246	1	2	yes	—
265	C₄	#15689889	1	2	yes	—
266	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
267	C₄	C₂×#13094046	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	376	4	no	—
270	C₂	#20252107	1	1	yes	characteristic
271	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic
272	C₂	C₂×#15689246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#20253967	C₂
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#17975
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #20252101	C₂	semidirect
3. #20242219	C₂	semidirect
4. #20242147	C₂	semidirect
5. #20241840	C₂	semidirect
6. C₂×C₄₇×#322	C₂	semidirect
72. C₂×C₈×C₄₇	D₈	semidirect
137. C₂×#322	D₄₇	semidirect
162. C₄₇	#17975	semidirect
214. C₂×C₈	D₃₇₆	semidirect

Maximal subgroups

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

subgroup	index
#20253967	2
#20252101	2
#20242219	2
#20242147	2
#20241840	2
C₂×C₄₇×#322	2
#17975	47