Subgroups — C₄×#15688926 #26131490

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688926	C₂	1	2	yes	maximal
3	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#20241479	C₂	1	2	yes	maximal
5	C₄×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₉₄	—	2	1	no	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₄×Dic₁₈₈	—	2	2	no	—
10	C₄×Dic₁₈₈	C₄	1	4	yes	complemented
11	C₂×#13093958	C₂²	1	1	yes	characteristic
12	C₂×#13093949	—	2	1	no	—
13	#15688926	C₄	1	8	yes	direct product factor
14	C₄×C₄₇⋊C₁₆	—	2	2	no	—
15	C₂×#107×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₄₇×#340	—	2	1	no	—
19	C₂²×Dic₉₄	—	2	1	no	—
20	C₂×Dic₁₈₈	—	2	1	no	—
21	C₂×Dic₁₈₈	—	2	2	no	—
22	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
24	C₄×Dic₉₄	—	2	4	no	—
25	C₄×Dic₉₄	—	4	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×Dic₄₇⋊C₄	—	4	1	no	—
28	C₂×C₄×Dic₄₇	—	4	1	no	—
29	C₂×C₄₇⋊C₁₆	—	2	2	no	—
30	C₂×C₄₇⋊C₁₆	—	2	2	no	—
31	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
32	#13093958	—	2	1	no	—
33	#13093958	C₂×C₄	1	2	yes	—
34	#13093949	—	2	4	no	—
35	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	—	4	1	no	—
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	#107×C₄₇	—	2	2	no	—
41	#107×C₄₇	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	#95×C₄₇	—	4	1	no	—
45	C₂²×C₄×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	4	1	no	—
50	C₂²×Dic₄₇	—	4	1	no	—
51	C₂×Dic₉₄	—	2	2	no	—
52	C₂×Dic₉₄	—	2	4	no	—
53	C₂×Dic₉₄	—	4	1	no	—
54	C₂×Dic₉₄	—	4	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	#41	1	2	yes	—
61	C₄⋊C₄×C₄₇	—	2	4	no	—
62	C₄⋊C₄×C₄₇	—	4	2	no	—
63	C₄²×C₄₇	—	8	2	no	—
64	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₄₇	SD₁₆	1	2	yes	—
66	C₄⋊Dic₄₇	—	2	2	no	—
67	C₄⋊Dic₄₇	—	4	1	no	—
68	Dic₄₇⋊C₄	—	4	4	no	—
69	C₄×Dic₄₇	—	4	4	no	—
70	Dic₁₈₈	—	2	2	no	—
71	Dic₁₈₈	—	2	4	no	—
72	Dic₁₈₈	C₄²	1	4	yes	complemented
73	C₄₇⋊C₁₆	—	2	4	no	—
74	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	—	4	4	no	—
77	C₂×C₄×C₄₇	—	8	2	no	—
78	C₂×C₄×C₄₇	—	8	2	no	—
79	C₂×C₄×C₄₇	#102	1	2	yes	—
80	C₂×C₄×C₄₇	#102	1	2	yes	—
81	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
82	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#135	1	2	yes	—
87	C₂×Dic₄₇	—	4	2	no	—
88	C₂×Dic₄₇	—	4	4	no	—
89	C₂×Dic₄₇	—	4	2	no	—
90	C₂×Dic₄₇	—	8	2	no	—
91	Dic₉₄	—	2	4	no	—
92	Dic₉₄	—	4	2	no	—
93	Dic₉₄	—	4	4	no	—
94	C₈×C₄₇	—	2	2	no	—
95	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₈×C₄₇	#117	1	1	yes	characteristic
97	C₈×C₄₇	C₄×D₄	1	2	yes	—
98	C₄×#358	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
100	C₂²×C₄₇	C₄×D₈	1	2	yes	—
101	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²×C₄₇	#508	1	2	yes	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	8	4	no	—
106	C₄×C₄₇	#358	1	4	yes	complemented
107	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₄₇	#416	1	1	yes	characteristic
109	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
110	Dic₄₇	—	4	4	no	—
111	Dic₄₇	—	8	2	no	—
112	C₂×C₄×Q₁₆	—	47	1	no	—
113	C₂×#358	—	47	2	no	—
114	C₂×C₄×C₁₆	—	47	1	no	—
115	C₄×#107	D₄₇	1	1	yes	characteristic
116	#1380	—	47	2	no	—
117	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
119	C₂×C₄₇	#2138	1	1	yes	characteristic
120	C₂×C₄₇	C₄×SD₃₂	1	2	yes	—
121	C₂×C₄₇	C₄×Q₃₂	1	2	yes	—
122	C₂²×Q₁₆	—	47	1	no	—
123	C₂×C₄×Q₈	—	94	1	no	—
124	C₂²×C₁₆	—	47	2	no	—
125	C₄×Q₁₆	—	47	4	no	—
126	C₄×Q₁₆	—	94	2	no	—
127	C₂×#107	—	47	1	no	—
128	C₂×#107	C₂×D₄₇	1	2	yes	—
129	C₂×#103	—	94	1	no	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	C₄×C₄⋊C₄	—	2	1	no	—
132	#358	—	47	8	no	—
133	C₄×C₁₆	—	94	2	no	—
134	#340	—	2	1	no	—
135	C₄₇	C₄×#358	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	94	1	no	—
137	C₂×Q₁₆	—	47	4	no	—
138	C₂×Q₁₆	—	47	2	no	—
139	C₂×Q₁₆	—	94	1	no	—
140	C₂×Q₁₆	—	94	2	no	—
141	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
143	C₄×Q₈	—	94	4	no	—
144	C₄×Q₈	—	188	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	94	1	no	—
147	C₂×C₄⋊C₄	—	188	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	—	188	1	no	—
150	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
151	C₂×C₁₆	—	47	4	no	—
152	C₂×C₁₆	—	94	2	no	—
153	C₂×C₁₆	—	94	2	no	—
154	#107	—	2	2	no	—
155	#107	—	47	2	no	—
156	#107	—	94	1	no	—
157	#107	C₄×D₄₇	1	4	yes	—
158	#103	—	94	4	no	—
159	C₄×C₈	C₄×D₄₇	1	2	yes	—
160	C₄×C₈	D₁₈₈	1	2	yes	—
161	#95	—	4	1	no	—
162	C₂×Q₈	—	94	2	no	—
163	C₂×Q₈	—	94	4	no	—
164	C₂×Q₈	—	188	1	no	—
165	C₂×Q₈	—	188	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	188	1	no	—
168	C₂²×C₄	—	188	1	no	—
169	C₂²×C₄	#10625334	1	1	yes	characteristic
170	C₂²×C₄	#10625334	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
172	Q₁₆	—	47	4	no	—
173	Q₁₆	—	94	2	no	—
174	Q₁₆	—	94	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
179	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
180	C₂×C₈	#10625350	1	2	yes	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	94	2	no	—
184	C₄⋊C₄	—	188	4	no	—
185	C₄⋊C₄	—	188	1	no	—
186	C₄²	—	8	2	no	—
187	C₄²	—	188	4	no	—
188	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
189	C₄²	#10625336	1	2	yes	—
190	C₁₆	—	94	4	no	—
191	C₂³	C₂×#10625334	1	1	yes	characteristic
192	Q₈	—	94	4	no	—
193	Q₈	—	188	2	no	—
194	Q₈	—	188	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	376	2	no	—
200	C₂×C₄	—	188	2	no	—
201	C₂×C₄	—	188	2	no	—
202	C₂×C₄	—	188	4	no	—
203	C₂×C₄	#13093944	1	2	yes	—
204	C₂×C₄	#13093944	1	2	yes	—
205	C₂×C₄	#13093968	1	2	yes	—
206	C₂×C₄	#13093970	1	2	yes	—
207	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
209	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
210	C₂×C₄	#13094044	1	2	yes	—
211	C₈	—	2	2	no	—
212	C₈	C₄²×D₄₇	1	1	yes	characteristic
213	C₈	#13093981	1	1	yes	characteristic
214	C₈	C₄×D₁₈₈	1	2	yes	—
215	C₂²	C₂×#13093944	1	1	yes	characteristic
216	C₂²	C₄×#10625334	1	2	yes	—
217	C₂²	C₂×#13093968	1	1	yes	characteristic
218	C₂²	C₂×#13093970	1	1	yes	characteristic
219	C₂²	#15689553	1	2	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	8	4	no	—
222	C₄	—	376	2	no	—
223	C₄	—	188	4	no	—
224	C₄	#15688926	1	4	yes	direct product factor
225	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
226	C₄	#15689117	1	1	yes	characteristic
227	C₄	C₄×#10625336	1	2	yes	—
228	C₂	C₄×#13093944	1	1	yes	characteristic
229	C₂	C₂×#15688926	1	1	yes	characteristic
230	C₂	#20245093	1	1	yes	characteristic
231	C₂	C₄×#13093968	1	2	yes	—
232	C₂	C₄×#13093970	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄×#107×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄×#107	D₄₇
hypocenter	C₄₇	C₄×#358
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₁₈₈	C₄	semidirect
13. #15688926	C₄	direct (N × H)
72. Dic₁₈₈	C₄²	semidirect
106. C₄×C₄₇	#358	semidirect
135. C₄₇	C₄×#358	semidirect
224. C₄	#15688926	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×C₄×Dic₁₈₈	2
C₂×#15688926	2
C₂×C₄×C₄₇⋊C₁₆	2
#20241479	2
C₄×#107×C₄₇	2
C₄×#358	47

Subgroups — C4×#15688926 #26131490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688926 #26131490