Subgroups — #26379103

Order 12032.

2238 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246569	C₂	1	4	yes	maximal, complemented
2	#20246478	C₂	1	4	yes	maximal, complemented
3	#20246257	C₂	1	8	yes	maximal
4	#20246137	C₂	1	8	yes	maximal, complemented
5	#20245974	C₂	1	4	yes	maximal
6	#20245786	C₂	1	2	yes	maximal
7	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₂²	1	8	yes	—
9	#15689030	C₂²	1	16	yes	—
10	#15689030	C₂²	1	16	yes	—
11	#15689022	C₂²	1	16	yes	—
12	#15689009	C₂²	1	32	yes	—
13	#15688991	C₂²	1	4	yes	—
14	#15688989	C₂²	1	4	yes	—
15	#15688986	C₂²	1	4	yes	—
16	#15688986	C₂²	1	8	yes	—
17	#15688983	C₂²	1	4	yes	—
18	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
21	#15688969	C₂²	1	4	yes	—
22	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₄₇	C₄	1	16	yes	complemented
24	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
26	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	#13093983	—	4	16	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
32	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	8	yes	—
36	C₄₇⋊#95	C₂³	1	8	yes	—
37	C₄₇⋊#95	C₂³	1	8	yes	—
38	C₄₇⋊#95	C₂³	1	8	yes	—
39	C₄₇⋊#95	C₂³	1	32	yes	—
40	C₂²×Q₈×C₄₇	—	4	8	no	—
41	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
42	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
43	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
47	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
48	C₄⋊Q₈×C₄₇	C₂×C₄	1	32	yes	—
49	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₄₇	—	4	32	no	—
59	C₂×Q₈×C₄₇	—	4	4	no	—
60	C₂×Q₈×C₄₇	—	4	8	no	—
61	C₂×Q₈×C₄₇	—	4	4	no	—
62	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
63	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	16	no	—
66	C₂²×C₄×C₄₇	—	2	8	no	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
75	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
76	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
77	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²⋊Dic₄₇	—	2	16	no	—
82	C₄⋊Dic₄₇	—	2	16	no	—
83	C₄⋊Dic₄₇	—	4	16	no	—
84	C₄×Dic₄₇	—	2	16	no	—
85	C₄×Dic₄₇	—	4	16	no	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
88	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
89	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
90	Q₈×C₄₇	—	4	16	no	—
91	Q₈×C₄₇	—	4	16	no	—
92	C₂×C₄×C₄₇	—	2	32	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	16	no	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₄₇	2₋¹⁺⁴	1	8	yes	—
106	C₂×Dic₄₇	—	2	16	no	—
107	C₂×Dic₄₇	—	2	16	no	—
108	C₂×Dic₄₇	—	2	16	no	—
109	#34365	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	2	8	no	—
112	C₂²×C₄₇	#518	1	4	yes	—
113	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
115	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	4	yes	—
116	C₄×C₄₇	—	2	16	no	—
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	4	no	—
120	C₄×C₄₇	#519	1	8	yes	—
121	Dic₄₇	—	4	16	no	—
122	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
123	#2381	—	47	4	no	—
124	#2369	—	47	4	no	—
125	#2344	—	47	8	no	—
126	#2331	—	47	8	no	—
127	#2310	—	47	4	no	—
128	#2283	—	47	2	no	—
129	C₂×C₄₇	—	2	4	no	—
130	C₂×C₄₇	C₂×#518	1	1	yes	characteristic
131	C₂×C₄₇	C₂×#519	1	2	yes	—
132	C₂×C₄₇	#3544	1	4	yes	—
133	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
134	C₂×C₄⋊Q₈	Dic₄₇	1	16	yes	complemented
135	C₂×C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
136	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
137	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
138	C₂×#117	—	47	8	no	—
139	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
140	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
141	#391	—	47	16	no	—
142	#391	—	47	16	no	—
143	#389	—	47	16	no	—
144	#387	—	47	32	no	—
145	#383	—	47	4	no	—
146	#382	—	47	4	no	—
147	#381	—	47	4	no	—
148	#381	—	47	8	no	—
149	#380	—	47	4	no	—
150	C₄×C₄⋊C₄	—	47	16	no	—
151	C₄×C₄⋊C₄	—	47	4	no	—
152	C₄×C₂²⋊C₄	—	47	4	no	—
153	#376	—	47	4	no	—
154	C₄₇	#34365	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	4	8	no	—
156	C₂²×Q₈	C₂×Dic₄₇	1	16	yes	—
157	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
158	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
159	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
160	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
161	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₄⋊Q₈	C₂×Dic₄₇	1	32	yes	—
163	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
164	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
165	#117	—	188	16	no	—
166	C₂×C₄⋊C₄	—	47	8	no	—
167	C₂×C₄⋊C₄	—	47	16	no	—
168	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
169	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
170	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
171	C₂×C₂²⋊C₄	—	47	8	no	—
172	C₂×C₄²	—	47	8	no	—
173	C₂×C₄²	—	47	16	no	—
174	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
175	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
176	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
178	#95	—	47	8	no	—
179	#95	—	47	8	no	—
180	#95	—	47	8	no	—
181	#95	—	47	8	no	—
182	#95	—	47	32	no	—
183	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
184	C₂×Q₈	—	4	32	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	8	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
189	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	47	16	no	—
194	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
195	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
196	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
197	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
198	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
201	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
202	C₄⋊C₄	—	94	16	no	—
203	C₄⋊C₄	—	188	16	no	—
204	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
205	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
206	C₂²⋊C₄	—	94	16	no	—
207	C₄²	—	94	16	no	—
208	C₄²	—	188	16	no	—
209	C₄²	C₂²×Dic₄₇	1	4	yes	—
210	C₄²	C₂²×Dic₄₇	1	8	yes	—
211	C₄²	C₂²×Dic₄₇	1	4	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂³×Dic₄₇	1	2	yes	—
214	C₂³	#13094069	1	4	yes	—
215	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
216	Q₈	—	4	16	no	—
217	Q₈	—	4	16	no	—
218	C₂×C₄	—	2	32	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	94	16	no	—
227	C₂×C₄	—	94	16	no	—
228	C₂×C₄	—	94	16	no	—
229	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
230	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
231	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
232	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
233	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
234	C₂×C₄	#13094072	1	8	yes	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	8	no	—
237	C₂²	#15689598	1	4	yes	—
238	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
239	C₂²	C₂×#13094069	1	2	yes	—
240	C₂²	C₂×#13094072	1	4	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	16	no	—
244	C₄	—	2	4	no	—
245	C₄	—	188	16	no	—
246	C₄	#15689601	1	8	yes	—
247	C₂	—	2	4	no	—
248	C₂	C₂×#15689598	1	1	yes	characteristic
249	C₂	C₂×#15689601	1	2	yes	—
250	C₂	#20257732	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#34365
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20246569	C₂	semidirect
2. #20246478	C₂	semidirect
4. #20246137	C₂	semidirect
23. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
134. C₂×C₄⋊Q₈	Dic₄₇	semidirect
154. C₄₇	#34365	semidirect

Maximal subgroups

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

subgroup	index
#20246569	2
#20246478	2
#20246257	2
#20246137	2
#20245974	2
#20245786	2
C₂²×C₄⋊Q₈×C₄₇	2
#34365	47