Subgroups — #27079915

Order 12032.

2666 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689890	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13094031	C₂	1	1	yes	maximal, characteristic
3	#20251956	C₂	1	16	yes	maximal, complemented
4	#20251952	C₂	1	6	yes	maximal, complemented
5	C₂×#15689150	C₂	1	6	yes	maximal, complemented
6	C₄₇×#3428	C₂	1	1	yes	maximal, characteristic
7	#15689890	—	2	8	no	—
8	C₂×#13094031	—	2	8	no	—
9	C₂×#13094031	C₂²	1	8	yes	—
10	C₂×#13094031	C₂²	1	8	yes	—
11	C₂²×#10625329	C₂²	1	6	yes	—
12	C₂²×#10625329	C₂²	1	3	yes	—
13	C₂³×C₄₇⋊C₈	C₂²	1	3	yes	—
14	#15689150	C₂²	1	4	yes	—
15	#15689150	C₂²	1	12	yes	—
16	#15689150	C₂²	1	24	yes	complemented
17	#15689150	C₂²	1	24	yes	complemented
18	#15689147	C₂²	1	24	yes	complemented
19	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	24	yes	complemented
20	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#520	C₄	1	8	yes	—
22	C₄₇×#519	C₄	1	2	yes	—
23	C₄₇×#518	C₄	1	6	yes	—
24	C₄×#41×C₄₇	C₂²	1	8	yes	—
25	C₂×#117×C₄₇	C₂²	1	6	yes	—
26	#13094031	—	2	32	no	—
27	#13094031	—	2	32	no	—
28	C₂×#10625329	—	2	12	no	—
29	C₂×#10625329	—	2	24	no	—
30	C₂×#10625329	C₂³	1	4	yes	—
31	C₂×#10625329	C₂³	1	12	yes	—
32	C₂×#10625329	C₂³	1	12	yes	—
33	C₂²×C₄₇⋊C₈	—	2	12	no	—
34	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
38	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂³	1	48	yes	—
40	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×#41×C₄₇	D₄	1	8	yes	—
42	C₂×#41×C₄₇	Q₈	1	8	yes	—
43	C₂×#41×C₄₇	C₂³	1	4	yes	—
44	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₄₇	C₂×C₄	1	3	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
47	C₄×Q₈×C₄₇	C₂×C₄	1	16	yes	—
48	C₄×D₄×C₄₇	C₂×C₄	1	48	yes	—
49	#117×C₄₇	C₂×C₄	1	12	yes	—
50	#117×C₄₇	C₂×C₄	1	12	yes	—
51	#117×C₄₇	C₂³	1	4	yes	—
52	#117×C₄₇	C₂³	1	12	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	12	yes	—
54	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	12	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	3	yes	—
57	#41×C₄₇	C₄⋊C₄	1	32	yes	—
58	#41×C₄₇	C₂×D₄	1	16	yes	—
59	#41×C₄₇	C₂×Q₈	1	16	yes	—
60	C₂×Q₈×C₄₇	C₄⋊C₄	1	8	yes	—
61	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₄₇	C₄⋊C₄	1	24	yes	—
63	C₂×D₄×C₄₇	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	12	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
72	C₄⋊C₄×C₄₇	C₂²×C₄	1	24	yes	—
73	C₂²⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂²⋊C₄×C₄₇	C₂²×C₄	1	24	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	24	yes	—
76	C₄²×C₄₇	C₂⁴	1	8	yes	—
77	#10625329	—	2	8	no	—
78	#10625329	—	2	24	no	—
79	#10625329	—	2	24	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	6	no	—
82	C₂×C₄₇⋊C₈	—	2	24	no	—
83	C₂×C₄₇⋊C₈	—	2	24	no	—
84	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
85	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
86	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
87	C₂³×C₄₇	—	4	6	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
89	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
91	Q₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
92	D₄×C₄₇	C₂×C₄⋊C₄	1	48	yes	—
93	C₂×C₄×C₄₇	—	4	24	no	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
101	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
103	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
104	C₄₇⋊C₈	—	2	2	no	—
105	C₄₇⋊C₈	—	2	6	no	—
106	C₄₇⋊C₈	—	2	8	no	—
107	#60100	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	12	no	—
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	4	12	no	—
111	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
112	C₂²×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
113	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
114	C₄×C₄₇	—	2	16	no	—
115	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
116	C₄×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
117	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
118	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
119	C₂×#568	—	47	1	no	—
120	C₂²×#131	—	47	1	no	—
121	#3428	D₄₇	1	1	yes	characteristic
122	#2906	—	47	16	no	—
123	#2905	—	47	6	no	—
124	C₂×#424	—	47	6	no	—
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	—	2	12	no	—
127	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄₇	#3573	1	2	yes	—
129	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
130	#568	—	94	8	no	—
131	C₂×#131	—	47	8	no	—
132	C₂×#131	—	47	8	no	—
133	C₂×#131	—	94	8	no	—
134	C₂²×M₁₆	—	47	6	no	—
135	C₂²×M₁₆	—	47	3	no	—
136	C₂³×C₈	—	47	3	no	—
137	#520	Dic₄₇	1	8	yes	—
138	#519	Dic₄₇	1	2	yes	—
139	#518	Dic₄₇	1	6	yes	—
140	C₄×#41	C₂×D₄₇	1	8	yes	—
141	C₂×#117	C₂×D₄₇	1	6	yes	—
142	#424	—	47	4	no	—
143	#424	—	47	12	no	—
144	#424	—	47	24	no	—
145	#424	—	47	24	no	—
146	#423	—	47	24	no	—
147	C₂×C₄⋊C₈	—	47	24	no	—
148	C₄₇	#60100	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	Dic₉₄	1	8	yes	—
150	C₂×#41	D₁₈₈	1	8	yes	—
151	C₂×#41	C₂×Dic₄₇	1	4	yes	—
152	C₂×#41	C₂²×D₄₇	1	4	yes	—
153	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
154	C₂²×D₄	C₂×Dic₄₇	1	3	yes	—
155	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
156	#131	—	94	32	no	—
157	#131	—	94	32	no	—
158	C₂×M₁₆	—	47	4	no	—
159	C₂×M₁₆	—	47	12	no	—
160	C₂×M₁₆	—	47	12	no	—
161	C₂×M₁₆	—	94	12	no	—
162	C₂×M₁₆	—	94	24	no	—
163	C₂²×C₈	—	47	1	no	—
164	C₂²×C₈	—	47	3	no	—
165	C₂²×C₈	—	47	12	no	—
166	C₂²×C₈	—	47	12	no	—
167	C₂²×C₈	—	94	12	no	—
168	C₄×Q₈	C₂×Dic₄₇	1	16	yes	—
169	C₄×D₄	C₂×Dic₄₇	1	48	yes	—
170	#117	C₂×Dic₄₇	1	12	yes	—
171	#117	C₂×Dic₄₇	1	12	yes	—
172	#117	C₂²×D₄₇	1	4	yes	—
173	#117	C₂²×D₄₇	1	12	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₄₇	1	12	yes	—
175	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	12	yes	—
176	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
177	C₄⋊C₈	—	47	16	no	—
178	C₄⋊C₈	—	47	48	no	—
179	C₂⁴	C₂²×Dic₄₇	1	3	yes	—
180	#41	C₄⋊Dic₄₇	1	32	yes	—
181	#41	C₂×Dic₉₄	1	16	yes	—
182	#41	C₂×D₁₈₈	1	16	yes	—
183	C₂×Q₈	C₄⋊Dic₄₇	1	8	yes	—
184	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
185	C₂×D₄	C₄⋊Dic₄₇	1	24	yes	—
186	C₂×D₄	C₂²×Dic₄₇	1	12	yes	—
187	C₂²×C₄	C₂×Dic₉₄	1	12	yes	—
188	C₂²×C₄	C₂×D₁₈₈	1	12	yes	—
189	C₂²×C₄	C₂²×Dic₄₇	1	24	yes	—
190	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
191	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
192	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
193	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
194	M₁₆	—	94	8	no	—
195	M₁₆	—	94	24	no	—
196	M₁₆	—	94	24	no	—
197	C₂×C₈	—	47	2	no	—
198	C₂×C₈	—	47	8	no	—
199	C₂×C₈	—	47	6	no	—
200	C₂×C₈	—	94	2	no	—
201	C₂×C₈	—	94	6	no	—
202	C₂×C₈	—	94	24	no	—
203	C₂×C₈	—	94	24	no	—
204	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
205	C₄⋊C₄	C₂²×Dic₄₇	1	24	yes	—
206	C₂²⋊C₄	C₂²×Dic₄₇	1	8	yes	—
207	C₂²⋊C₄	C₂²×Dic₄₇	1	24	yes	—
208	C₄²	C₂²×Dic₄₇	1	24	yes	—
209	C₄²	C₂³×D₄₇	1	8	yes	—
210	C₂³	—	4	6	no	—
211	C₂³	C₂×C₄⋊Dic₄₇	1	12	yes	—
212	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
213	C₂³	C₂³×Dic₄₇	1	6	yes	—
214	Q₈	C₂×C₄⋊Dic₄₇	1	16	yes	—
215	D₄	C₂×C₄⋊Dic₄₇	1	48	yes	—
216	C₂×C₄	—	4	24	no	—
217	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₄₇	1	24	yes	—
219	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
220	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
221	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
223	C₂×C₄	C₂²×D₁₈₈	1	12	yes	—
224	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
225	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
226	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
227	C₈	—	94	2	no	—
228	C₈	—	94	6	no	—
229	C₈	—	94	8	no	—
230	C₂²	—	2	12	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	12	no	—
233	C₂²	C₂²×C₄⋊Dic₄₇	1	2	yes	—
234	C₂²	C₂²×C₄⋊Dic₄₇	1	12	yes	—
235	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
236	C₄	—	2	16	no	—
237	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
238	C₄	C₂²×C₄⋊Dic₄₇	1	12	yes	—
239	C₄	C₂³×D₁₈₈	1	1	yes	characteristic
240	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	12	no	—
243	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
244	C₂	#20258100	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₄₇×#3428	C₂
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#3428	D₄₇
hypocenter	C₄₇	#60100
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #3428

Complements

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

N	H ≅ G/N	type
3. #20251956	C₂	semidirect
4. #20251952	C₂	semidirect
5. C₂×#15689150	C₂	semidirect
16. #15689150	C₂²	semidirect
17. #15689150	C₂²	semidirect
18. #15689147	C₂²	semidirect
19. C₂×C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
148. C₄₇	#60100	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689890	2
C₂²×#13094031	2
#20251956	2
#20251952	2
C₂×#15689150	2
C₄₇×#3428	2
#60100	47