Subgroups — C₄×#15689890 #27079695

Order 12032.

3250 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	2	yes	maximal
2	#20251816	C₂	1	8	yes	maximal, complemented
3	C₄×#13094031	C₂	1	8	yes	maximal, complemented
4	#20251807	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×#10625329	C₂	1	6	yes	maximal, complemented
6	C₂×C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689890	C₄	1	32	yes	direct product factor
8	C₂×#13094031	C₂²	1	16	yes	—
9	C₂²×#10625329	C₂²	1	12	yes	—
10	#15689079	C₂²	1	48	yes	complemented
11	C₄×#10625329	C₂²	1	4	yes	—
12	C₄×#10625329	C₂²	1	12	yes	—
13	C₄×#10625329	C₂²	1	12	yes	complemented
14	C₄×#10625329	C₂²	1	12	yes	complemented
15	C₂×#13093935	C₂²	1	12	yes	complemented
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	12	yes	complemented
17	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×#41×C₄₇	C₄	1	8	yes	—
19	C₄×#41×C₄₇	C₂²	1	4	yes	—
20	C₄×#41×C₄₇	C₂²	1	4	yes	—
21	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
22	C₂×C₄×D₄×C₄₇	C₄	1	6	yes	—
23	C₂×#117×C₄₇	C₂²	1	3	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
25	#13094031	C₂×C₄	1	128	yes	complemented
26	C₂×#10625329	C₂×C₄	1	96	yes	complemented
27	C₂×#10625329	C₂³	1	8	yes	—
28	C₂×#10625329	C₂³	1	24	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	24	yes	—
30	#13093935	C₂³	1	8	yes	—
31	#13093935	C₂³	1	24	yes	—
32	C₄×C₄₇⋊C₈	C₂³	1	8	yes	—
33	C₄×C₄₇⋊C₈	C₂³	1	24	yes	—
34	C₂×#41×C₄₇	C₂×C₄	1	16	yes	—
35	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×#41×C₄₇	C₂³	1	4	yes	—
37	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₄₇	C₂×C₄	1	3	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
41	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
42	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
43	C₄×D₄×C₄₇	C₂×C₄	1	24	yes	—
44	C₄×D₄×C₄₇	C₂×C₄	1	24	yes	—
45	#117×C₄₇	C₂×C₄	1	12	yes	—
46	#117×C₄₇	C₂³	1	12	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
48	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	6	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	12	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	3	yes	—
56	#41×C₄₇	C₄²	1	32	yes	—
57	#41×C₄₇	C₂²×C₄	1	32	yes	—
58	C₂×Q₈×C₄₇	C₄²	1	8	yes	—
59	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₄₇	C₄²	1	24	yes	—
61	C₂×D₄×C₄₇	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₄₇	—	4	12	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
72	C₄⋊C₄×C₄₇	C₂²×C₄	1	24	yes	—
73	C₂²⋊C₄×C₄₇	C₂²×C₄	1	24	yes	—
74	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	6	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
78	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₄²×C₄₇	C₂⁴	1	6	yes	—
80	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
81	#10625329	C₂²×C₄	1	32	yes	—
82	#10625329	C₂²×C₄	1	96	yes	complemented
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	96	yes	complemented
84	C₂×C₄₇⋊C₈	C₂⁴	1	16	yes	—
85	C₂³×C₄₇	—	4	6	no	—
86	C₂³×C₄₇	C₂×C₄²	1	12	yes	—
87	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
88	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	Q₈×C₄₇	C₂×C₄²	1	16	yes	—
90	D₄×C₄₇	C₂×C₄²	1	48	yes	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	24	no	—
93	C₂×C₄×C₄₇	—	4	12	no	—
94	C₂×C₄×C₄₇	—	4	6	no	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
96	C₂×C₄×C₄₇	C₂×C₄²	1	24	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	24	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
104	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
105	C₄₇⋊C₈	C₂³×C₄	1	32	yes	—
106	C₄×#568	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	2	2	no	—
108	C₂²×C₄₇	—	2	12	no	—
109	C₂²×C₄₇	—	4	12	no	—
110	C₂²×C₄₇	C₂²×C₄²	1	2	yes	—
111	C₂²×C₄₇	C₂²×C₄²	1	12	yes	—
112	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
113	C₄×C₄₇	—	2	2	no	—
114	C₄×C₄₇	—	2	12	no	—
115	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
116	C₄×C₄₇	C₂²×C₄²	1	12	yes	—
117	C₄×C₄₇	#568	1	4	yes	complemented
118	C₄×C₄₇	C₂⁴×C₄	1	2	yes	—
119	C₂×#568	—	47	2	no	—
120	C₂×C₄×#41	D₄₇	1	1	yes	characteristic
121	#2875	—	47	8	no	—
122	C₄×#131	—	47	8	no	—
123	#2873	—	47	6	no	—
124	C₂×C₄×M₁₆	—	47	6	no	—
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	—	2	12	no	—
127	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
128	C₂×C₄₇	C₂×#568	1	2	yes	—
129	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
130	#568	—	47	32	no	—
131	C₂×#131	—	47	16	no	—
132	C₂²×M₁₆	—	47	12	no	—
133	C₄×#41	Dic₄₇	1	8	yes	—
134	C₄×#41	C₂×D₄₇	1	4	yes	—
135	C₄×#41	C₂×D₄₇	1	4	yes	—
136	C₂×C₄×Q₈	Dic₄₇	1	2	yes	—
137	C₂×C₄×D₄	Dic₄₇	1	6	yes	—
138	C₂×#117	C₂×D₄₇	1	3	yes	—
139	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
140	#405	—	47	48	no	—
141	C₄×M₁₆	—	47	4	no	—
142	C₄×M₁₆	—	47	12	no	—
143	C₄×M₁₆	—	47	12	no	—
144	C₄×M₁₆	—	47	12	no	—
145	C₂×#97	—	47	12	no	—
146	C₂×C₄×C₈	—	47	12	no	—
147	C₄₇	C₄×#568	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	C₄×D₄₇	1	16	yes	—
149	C₂×#41	C₂×Dic₄₇	1	4	yes	—
150	C₂×#41	C₂²×D₄₇	1	4	yes	—
151	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
152	C₂²×D₄	C₂×Dic₄₇	1	3	yes	—
153	C₂³×C₄	C₂×Dic₄₇	1	6	yes	—
154	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
155	#131	—	47	128	no	—
156	C₂×M₁₆	—	47	96	no	—
157	C₂×M₁₆	—	47	8	no	—
158	C₂×M₁₆	—	47	24	no	—
159	C₂²×C₈	—	47	24	no	—
160	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
161	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
162	C₄×D₄	C₂×Dic₄₇	1	24	yes	—
163	C₄×D₄	C₂×Dic₄₇	1	24	yes	—
164	#117	C₂×Dic₄₇	1	12	yes	—
165	#117	C₂²×D₄₇	1	12	yes	—
166	C₂×C₄⋊C₄	C₂×Dic₄₇	1	6	yes	—
167	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	6	yes	—
168	C₂×C₄²	C₂×Dic₄₇	1	6	yes	—
169	C₂×C₄²	C₂×Dic₄₇	1	12	yes	—
170	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
171	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
172	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	#97	—	47	8	no	—
175	#97	—	47	24	no	—
176	C₄×C₈	—	47	24	no	—
177	C₄×C₈	—	47	8	no	—
178	C₂⁴	C₂²×Dic₄₇	1	3	yes	—
179	#41	C₄×Dic₄₇	1	32	yes	—
180	#41	C₂×C₄×D₄₇	1	32	yes	—
181	C₂×Q₈	C₄×Dic₄₇	1	8	yes	—
182	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
183	C₂×D₄	C₄×Dic₄₇	1	24	yes	—
184	C₂×D₄	C₂²×Dic₄₇	1	12	yes	—
185	C₂²×C₄	—	4	12	no	—
186	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
187	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
189	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
190	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
191	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
192	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
193	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
195	M₁₆	—	47	32	no	—
196	M₁₆	—	47	96	no	—
197	C₂×C₈	—	47	96	no	—
198	C₂×C₈	—	47	16	no	—
199	C₄⋊C₄	C₂²×Dic₄₇	1	24	yes	—
200	C₂²⋊C₄	C₂²×Dic₄₇	1	24	yes	—
201	C₄²	C₂²×Dic₄₇	1	2	yes	—
202	C₄²	C₂²×Dic₄₇	1	6	yes	—
203	C₄²	C₂²×Dic₄₇	1	12	yes	—
204	C₄²	C₂²×Dic₄₇	1	12	yes	—
205	C₄²	C₂³×D₄₇	1	6	yes	—
206	C₄²	C₂³×D₄₇	1	1	yes	characteristic
207	C₄²	C₂³×D₄₇	1	1	yes	characteristic
208	C₂³	—	4	6	no	—
209	C₂³	C₂×C₄×Dic₄₇	1	12	yes	—
210	C₂³	C₂³×Dic₄₇	1	6	yes	—
211	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
212	Q₈	C₂×C₄×Dic₄₇	1	16	yes	—
213	D₄	C₂×C₄×Dic₄₇	1	48	yes	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	24	no	—
216	C₂×C₄	—	4	12	no	—
217	C₂×C₄	—	4	6	no	—
218	C₂×C₄	C₂×C₄×Dic₄₇	1	12	yes	—
219	C₂×C₄	C₂×C₄×Dic₄₇	1	24	yes	—
220	C₂×C₄	C₂²×C₄×D₄₇	1	24	yes	—
221	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
222	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
223	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
224	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
226	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
227	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
228	C₈	—	47	32	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	12	no	—
231	C₂²	—	4	12	no	—
232	C₂²	C₂²×C₄×Dic₄₇	1	2	yes	—
233	C₂²	C₂²×C₄×Dic₄₇	1	12	yes	—
234	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	12	no	—
237	C₄	C₂²×C₄×Dic₄₇	1	2	yes	—
238	C₄	C₂²×C₄×Dic₄₇	1	12	yes	—
239	C₄	#15689890	1	4	yes	direct product factor
240	C₄	C₂³×C₄×D₄₇	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	—	2	12	no	—
243	C₂	C₂³×C₄×Dic₄₇	1	1	yes	characteristic
244	C₂	C₂×#15689890	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂³×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄×#41×C₄₇	C₂
socle	C₂²×C₄₇	C₂⁴×C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×C₄×#41	D₄₇
hypocenter	C₄₇	C₄×#568
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #20251816	C₂	semidirect
3. C₄×#13094031	C₂	semidirect
4. #20251807	C₂	semidirect
5. C₂×C₄×#10625329	C₂	semidirect
7. #15689890	C₄	direct (N × H)
10. #15689079	C₂²	semidirect
13. C₄×#10625329	C₂²	semidirect
14. C₄×#10625329	C₂²	semidirect
15. C₂×#13093935	C₂²	semidirect
16. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
25. #13094031	C₂×C₄	semidirect
26. C₂×#10625329	C₂×C₄	semidirect
82. #10625329	C₂²×C₄	semidirect
83. C₂×C₄₇⋊C₈	C₂²×C₄	semidirect
117. C₄×C₄₇	#568	semidirect
147. C₄₇	C₄×#568	semidirect
239. C₄	#15689890	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689890	2
#20251816	2
C₄×#13094031	2
#20251807	2
C₂×C₄×#10625329	2
C₂×C₄×#41×C₄₇	2
C₄×#568	47

Subgroups — C4×#15689890 #27079695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689890 #27079695