Subgroups — #26109938

Order 12032.

422 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241222	C₂	1	4	yes	maximal
4	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
22	C₄×C₄₇⋊C₈	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	2	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	4	no	—
26	C₄×C₄₇⋊C₈	C₈	1	4	yes	complemented
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₇	—	2	4	no	—
36	C₄⋊C₈×C₄₇	—	4	2	no	—
37	C₄⋊C₈×C₄₇	—	4	2	no	—
38	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₄₇	—	4	2	no	—
46	C₂×C₈×C₄₇	—	4	2	no	—
47	C₂×C₈×C₄₇	—	4	2	no	—
48	C₄²×C₄₇	—	2	2	no	—
49	C₄²×C₄₇	—	2	2	no	—
50	C₄²×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	C₂×C₈	1	2	yes	—
52	C₄²×C₄₇	M₁₆	1	2	yes	—
53	C₄²×C₄₇	D₈	1	2	yes	—
54	C₄²×C₄₇	SD₁₆	1	2	yes	—
55	C₄²×C₄₇	SD₁₆	1	2	yes	—
56	C₄²×C₄₇	Q₁₆	1	2	yes	—
57	C₄²×C₄₇	#41	1	2	yes	—
58	C₄²×C₄₇	#41	1	2	yes	—
59	C₂×C₄₇⋊C₈	—	2	2	no	—
60	C₂×C₄₇⋊C₈	—	2	2	no	—
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
67	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
68	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	—	2	1	no	—
74	C₂×C₄×C₄₇	—	2	1	no	—
75	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
76	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
77	C₂×C₄×C₄₇	#106	1	2	yes	—
78	C₂×C₄×C₄₇	#106	1	2	yes	—
79	C₂×C₄×C₄₇	#107	1	2	yes	—
80	C₂×C₄×C₄₇	#107	1	2	yes	—
81	C₂×C₄×C₄₇	#108	1	2	yes	—
82	C₂×C₄×C₄₇	#108	1	2	yes	—
83	C₂×C₄×C₄₇	#117	1	2	yes	—
84	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#125	1	2	yes	—
87	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₈×C₄₇	—	8	2	no	—
95	C₄₇⋊C₈	—	2	4	no	—
96	C₄₇⋊C₈	—	2	4	no	—
97	C₄₇⋊C₈	—	8	2	no	—
98	#8784	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	#382	1	1	yes	characteristic
100	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₄₇	#423	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	#334	1	4	yes	—
110	C₄×C₄₇	#335	1	4	yes	—
111	C₄×C₄₇	#431	1	2	yes	—
112	C₄×C₄₇	#432	1	2	yes	—
113	C₄×C₄₇	#486	1	1	yes	characteristic
114	C₄×C₄₇	#486	1	1	yes	characteristic
115	C₄×C₄₇	#487	1	1	yes	characteristic
116	C₄×C₄₇	#487	1	1	yes	characteristic
117	#1842	—	47	1	no	—
118	#1842	D₄₇	1	1	yes	characteristic
119	C₄²×C₈	—	47	1	no	—
120	#1277	—	47	4	no	—
121	C₂×C₄₇	C₂×#334	1	1	yes	characteristic
122	C₂×C₄₇	C₂×#335	1	1	yes	characteristic
123	C₂×C₄₇	#1566	1	1	yes	characteristic
124	C₂×C₄₇	#1567	1	1	yes	characteristic
125	C₂×C₄₇	#1831	1	1	yes	characteristic
126	C₂×C₄₇	#1834	1	1	yes	characteristic
127	C₂×C₄₇	#1837	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	—	2	1	no	—
130	C₂×C₄⋊C₈	—	47	2	no	—
131	C₂×C₄⋊C₈	—	94	1	no	—
132	C₂×C₄⋊C₈	—	94	1	no	—
133	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
134	C₂×C₄×C₈	—	47	1	no	—
135	C₂×C₄×C₈	—	47	2	no	—
136	C₂×C₄×C₈	—	47	1	no	—
137	C₂×C₄×C₈	—	47	2	no	—
138	C₄³	C₂×D₄₇	1	1	yes	characteristic
139	C₄₇	#8784	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	47	2	no	—
142	C₂²×C₈	—	47	2	no	—
143	C₂²×C₈	—	94	2	no	—
144	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
145	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	4	2	no	—
153	C₄⋊C₈	—	4	2	no	—
154	C₄⋊C₈	—	94	4	no	—
155	C₄⋊C₈	—	188	2	no	—
156	C₄⋊C₈	—	188	2	no	—
157	C₄×C₈	—	47	4	no	—
158	C₄×C₈	—	94	4	no	—
159	C₄×C₈	—	94	2	no	—
160	C₄×C₈	—	94	2	no	—
161	C₄×C₈	—	94	2	no	—
162	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	47	4	no	—
173	C₂×C₈	—	47	4	no	—
174	C₂×C₈	—	94	2	no	—
175	C₂×C₈	—	94	2	no	—
176	C₂×C₈	—	94	2	no	—
177	C₂×C₈	—	94	2	no	—
178	C₂×C₈	—	188	2	no	—
179	C₂×C₈	—	188	2	no	—
180	C₂×C₈	—	188	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	C₈×D₄₇	1	2	yes	—
185	C₄²	#10625324	1	2	yes	—
186	C₄²	#10625334	1	2	yes	—
187	C₄²	#10625335	1	2	yes	—
188	C₄²	#10625336	1	2	yes	—
189	C₄²	#10625337	1	2	yes	—
190	C₄²	#10625352	1	2	yes	—
191	C₄²	#10625354	1	2	yes	—
192	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
200	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
201	C₂×C₄	#13093955	1	2	yes	—
202	C₂×C₄	#13093955	1	2	yes	—
203	C₂×C₄	#13093957	1	2	yes	—
204	C₂×C₄	#13093957	1	2	yes	—
205	C₂×C₄	#13093959	1	2	yes	—
206	C₂×C₄	#13093959	1	2	yes	—
207	C₂×C₄	#13093982	1	2	yes	—
208	C₂×C₄	#13094011	1	1	yes	characteristic
209	C₂×C₄	#13094013	1	1	yes	characteristic
210	C₂×C₄	#13094015	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
212	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
213	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
214	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
215	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
216	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
217	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
218	C₈	—	8	2	no	—
219	C₈	—	94	4	no	—
220	C₈	—	94	4	no	—
221	C₈	—	376	2	no	—
222	C₂²	#15688988	1	1	yes	characteristic
223	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
224	C₂²	#15689146	1	1	yes	characteristic
225	C₂²	C₂×#13093955	1	1	yes	characteristic
226	C₂²	C₂×#13093957	1	1	yes	characteristic
227	C₂²	C₂×#13093959	1	1	yes	characteristic
228	C₂²	C₂×#13093959	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	#15688873	1	4	yes	—
233	C₄	#15688875	1	4	yes	—
234	C₄	#15689175	1	2	yes	—
235	C₄	#15689179	1	2	yes	—
236	C₄	#15689469	1	1	yes	characteristic
237	C₄	#15689471	1	1	yes	characteristic
238	C₄	#15689473	1	1	yes	characteristic
239	C₄	#15689475	1	1	yes	characteristic
240	C₂	C₂×#15688873	1	1	yes	characteristic
241	C₂	C₂×#15688875	1	1	yes	characteristic
242	C₂	#20242143	1	1	yes	characteristic
243	C₂	#20242146	1	1	yes	characteristic
244	C₂	#20243504	1	1	yes	characteristic
245	C₂	#20243515	1	1	yes	characteristic
246	C₂	#20243525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#8784
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
26. C₄×C₄₇⋊C₈	C₈	semidirect
139. C₄₇	#8784	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₄²×C₄₇⋊C₈	2
#20241222	2
C₄₇×#1842	2
#8784	47