Subgroups — #26220730

Order 12032.

1198 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20244325	C₂	1	8	yes	maximal, complemented
4	#20243085	C₂	1	4	yes	maximal
5	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15689147	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
12	#15688878	C₂²	1	16	yes	—
13	C₄₇×#533	—	2	4	no	—
14	C₄₇×#533	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
16	C₂×#125×C₄₇	C₄	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₄₇	C₄	1	4	yes	—
18	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
19	C₂×#117×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	2	8	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
29	C₄×C₄₇⋊C₈	—	2	8	no	—
30	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₇	—	2	8	no	—
34	#125×C₄₇	—	2	8	no	—
35	C₂²⋊Q₈×C₄₇	—	2	16	no	—
36	C₄×Q₈×C₄₇	—	2	8	no	—
37	C₄×Q₈×C₄₇	—	2	8	no	—
38	#117×C₄₇	—	2	4	no	—
39	#117×C₄₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	4	yes	—
47	C₂×C₄²×C₄₇	D₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
50	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₇	—	2	8	no	—
52	C₂×Q₈×C₄₇	—	2	16	no	—
53	C₂²×C₄×C₄₇	—	2	8	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₄₇	#41	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	—	2	8	no	—
65	C₄⋊C₄×C₄₇	—	2	16	no	—
66	C₄⋊C₄×C₄₇	—	2	8	no	—
67	C₂²⋊C₄×C₄₇	—	2	8	no	—
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	4	no	—
70	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
71	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
72	C₄²×C₄₇	C₂×D₄	1	4	yes	—
73	C₄²×C₄₇	C₂×D₄	1	4	yes	—
74	#10625329	—	4	8	no	—
75	C₂×C₄₇⋊C₈	—	2	16	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	8	no	—
80	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
84	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₄₇	—	4	16	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	16	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	2	16	no	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
93	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
94	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
95	C₂×C₄×C₄₇	#124	1	4	yes	—
96	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₄₇	#131	1	8	yes	—
98	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₄₇⋊C₈	—	2	8	no	—
103	C₄₇⋊C₈	—	4	4	no	—
104	#21300	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	—	2	8	no	—
107	C₂²×C₄₇	#391	1	4	yes	—
108	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
110	C₂²×C₄₇	C₂×#131	1	4	yes	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	—	4	8	no	—
113	C₄×C₄₇	#391	1	4	yes	—
114	C₄×C₄₇	#408	1	8	yes	—
115	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
116	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
117	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₂×#533	D₄₇	1	1	yes	characteristic
119	C₂×#423	—	47	1	no	—
120	C₂²×C₄×C₈	—	47	1	no	—
121	#1984	—	47	8	no	—
122	#1729	—	47	4	no	—
123	C₂×C₄₇	—	2	4	no	—
124	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
125	C₂×C₄₇	C₂×#408	1	2	yes	—
126	C₂×C₄₇	#2960	1	4	yes	—
127	C₂²×M₁₆	—	47	2	no	—
128	C₂³×C₈	—	47	2	no	—
129	#533	—	2	4	no	—
130	#533	—	2	4	no	—
131	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
132	C₂×#125	Dic₄₇	1	2	yes	—
133	C₂×C₂²⋊Q₈	Dic₄₇	1	4	yes	—
134	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
135	C₂×#117	C₂×D₄₇	1	2	yes	—
136	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
137	#423	—	94	4	no	—
138	C₂×C₄⋊C₈	—	47	4	no	—
139	C₂×C₄×C₈	—	47	4	no	—
140	C₂×C₄×C₈	—	94	4	no	—
141	#336	—	47	16	no	—
142	C₄₇	#21300	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
144	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
145	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
146	C₂×M₁₆	—	94	8	no	—
147	C₂×M₁₆	—	94	4	no	—
148	C₂²×C₈	—	47	8	no	—
149	C₂²×C₈	—	47	4	no	—
150	C₂²×C₈	—	47	4	no	—
151	C₂²×C₈	—	94	4	no	—
152	C₂²×C₈	—	94	4	no	—
153	C₄⋊Q₈	—	2	8	no	—
154	#125	—	2	8	no	—
155	C₂²⋊Q₈	—	2	16	no	—
156	C₄×Q₈	—	2	8	no	—
157	C₄×Q₈	—	2	8	no	—
158	#117	—	2	4	no	—
159	#117	—	2	4	no	—
160	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
161	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
162	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
164	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
167	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
168	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
170	C₄⋊C₈	—	94	8	no	—
171	C₄×C₈	—	94	8	no	—
172	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	2	16	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
177	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
178	C₂²×C₄	#10625354	1	4	yes	—
179	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
180	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
185	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
186	M₁₆	—	188	8	no	—
187	C₂×C₈	—	47	8	no	—
188	C₂×C₈	—	94	16	no	—
189	C₂×C₈	—	94	4	no	—
190	C₂×C₈	—	94	4	no	—
191	C₂×C₈	—	94	4	no	—
192	C₂×C₈	—	94	8	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	16	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
200	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
201	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
202	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
203	C₂³	—	2	4	no	—
204	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
205	C₂³	Q₈×Dic₄₇	1	4	yes	—
206	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
207	Q₈	—	4	16	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
213	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
215	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
216	C₂×C₄	#13094000	1	8	yes	—
217	C₂×C₄	#13094013	1	4	yes	—
218	C₂×C₄	#13094022	1	4	yes	—
219	C₂×C₄	#13094031	1	8	yes	—
220	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
223	C₂×C₄	C₂×#10625354	1	2	yes	—
224	C₈	—	94	8	no	—
225	C₈	—	188	4	no	—
226	C₂²	—	2	8	no	—
227	C₂²	—	2	4	no	—
228	C₂²	#15689030	1	4	yes	—
229	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₂²	C₂×Q₈×Dic₄₇	1	2	yes	—
231	C₂²	C₂×#13094031	1	4	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	4	8	no	—
234	C₄	#15689030	1	4	yes	—
235	C₄	#15689090	1	8	yes	—
236	C₄	C₂×#13094000	1	2	yes	—
237	C₄	C₂×#13094013	1	1	yes	characteristic
238	C₄	C₂×#13094022	1	1	yes	characteristic
239	C₂	—	2	4	no	—
240	C₂	C₂×#15689030	1	1	yes	characteristic
241	C₂	C₂×#15689090	1	2	yes	—
242	C₂	#20252323	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#533	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#21300
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #20244325	C₂	semidirect
25. C₂²×C₄₇⋊C₈	Q₈	semidirect
142. C₄₇	#21300	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂²×C₄×C₄₇⋊C₈	2
#20244325	2
#20243085	2
C₂×C₄₇×#533	2
#21300	47