Subgroups — C₄×#15688938 #26131539

Order 12032.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 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×#15688938	C₂	1	2	yes	maximal
3	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
4	#20241498	C₂	1	2	yes	maximal
5	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₉₄	—	2	1	no	—
8	C₂×#13093964	C₂²	1	2	yes	—
9	C₄×Dic₁₈₈	—	2	1	no	—
10	C₄×Dic₁₈₈	—	2	1	no	—
11	C₄×Dic₁₈₈	C₄	1	2	yes	—
12	C₄×Dic₁₈₈	C₄	1	2	yes	—
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093949	—	2	1	no	—
15	#15688938	C₄	1	8	yes	direct product factor
16	#15688899	—	2	1	no	—
17	C₄×C₄₇⋊C₁₆	—	2	1	no	—
18	C₂×#108×C₄₇	C₂²	1	2	yes	—
19	C₄×M₁₆×C₄₇	—	2	1	no	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#341	—	2	1	no	—
22	C₂²×Dic₉₄	—	2	1	no	—
23	C₂×Dic₁₈₈	—	2	1	no	—
24	C₂×Dic₁₈₈	—	2	2	no	—
25	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
26	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
27	C₄×Dic₉₄	—	2	2	no	—
28	C₄×Dic₉₄	—	2	2	no	—
29	C₄×Dic₉₄	—	4	1	no	—
30	C₄×Dic₉₄	—	4	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×Dic₄₇⋊C₄	—	4	1	no	—
33	C₂×C₄×Dic₄₇	—	4	1	no	—
34	#13093964	—	2	2	no	—
35	#13093964	C₂×C₄	1	4	yes	—
36	C₂×C₄₇⋊C₁₆	—	2	2	no	—
37	#13093958	—	2	1	no	—
38	#13093958	C₂×C₄	1	2	yes	—
39	#13093949	—	2	2	no	—
40	#13093949	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	#108×C₄₇	—	2	2	no	—
46	#108×C₄₇	C₂×C₄	1	4	yes	—
47	#97×C₄₇	—	4	1	no	—
48	C₄×C₈×C₄₇	—	4	1	no	—
49	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	4	1	no	—
57	C₂²×Dic₄₇	—	4	1	no	—
58	C₂×Dic₉₄	—	2	2	no	—
59	C₂×Dic₉₄	—	2	4	no	—
60	C₂×Dic₉₄	—	4	1	no	—
61	C₂×Dic₉₄	—	4	2	no	—
62	M₁₆×C₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	4	2	no	—
64	C₂×C₈×C₄₇	—	2	2	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	#41	1	2	yes	—
72	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₄₇	D₈	1	1	yes	characteristic
75	C₄²×C₄₇	D₈	1	1	yes	characteristic
76	C₄⋊Dic₄₇	—	2	2	no	—
77	C₄⋊Dic₄₇	—	4	1	no	—
78	Dic₄₇⋊C₄	—	4	2	no	—
79	Dic₄₇⋊C₄	—	4	2	no	—
80	C₄×Dic₄₇	—	4	2	no	—
81	C₄×Dic₄₇	—	4	2	no	—
82	Dic₁₈₈	—	2	2	no	—
83	Dic₁₈₈	—	2	4	no	—
84	Dic₁₈₈	C₄²	1	4	yes	—
85	C₄₇⋊C₁₆	—	2	4	no	—
86	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	4	1	no	—
89	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#135	1	2	yes	—
97	C₂×Dic₄₇	—	4	2	no	—
98	C₂×Dic₄₇	—	4	4	no	—
99	C₂×Dic₄₇	—	4	2	no	—
100	C₂×Dic₄₇	—	8	2	no	—
101	Dic₉₄	—	2	4	no	—
102	Dic₉₄	—	4	2	no	—
103	Dic₉₄	—	4	4	no	—
104	C₈×C₄₇	—	2	1	no	—
105	C₈×C₄₇	—	2	1	no	—
106	C₈×C₄₇	—	4	4	no	—
107	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
108	C₈×C₄₇	#117	1	1	yes	characteristic
109	C₈×C₄₇	C₄×D₄	1	2	yes	—
110	C₄×#362	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	1	no	—
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
114	C₂²×C₄₇	C₄×SD₁₆	1	2	yes	—
115	C₄×C₄₇	—	2	1	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	#362	1	2	yes	complemented
119	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₄₇	#416	1	1	yes	characteristic
121	C₄×C₄₇	C₄×D₈	1	2	yes	—
122	Dic₄₇	—	4	4	no	—
123	Dic₄₇	—	8	2	no	—
124	C₂×C₄×Q₁₆	—	47	1	no	—
125	C₂×#362	—	47	2	no	—
126	C₄×M₃₂	—	47	1	no	—
127	C₄×#108	D₄₇	1	1	yes	characteristic
128	#1387	—	47	2	no	—
129	C₂×C₄₇	—	2	2	no	—
130	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
131	C₂×C₄₇	C₂×#362	1	1	yes	characteristic
132	C₂×C₄₇	#2148	1	1	yes	characteristic
133	C₂²×Q₁₆	—	47	1	no	—
134	C₂×C₄×Q₈	—	94	1	no	—
135	C₂×M₃₂	—	47	2	no	—
136	C₄×Q₁₆	—	47	2	no	—
137	C₄×Q₁₆	—	47	2	no	—
138	C₄×Q₁₆	—	94	1	no	—
139	C₄×Q₁₆	—	94	1	no	—
140	C₂×#108	C₂×D₄₇	1	2	yes	—
141	C₂×#107	—	47	1	no	—
142	C₂×#103	—	94	1	no	—
143	C₄×M₁₆	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
145	#362	—	47	8	no	—
146	#346	—	94	1	no	—
147	C₄×C₁₆	—	94	1	no	—
148	#341	—	2	1	no	—
149	C₄₇	C₄×#362	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	—	94	1	no	—
151	C₂×Q₁₆	—	47	4	no	—
152	C₂×Q₁₆	—	47	2	no	—
153	C₂×Q₁₆	—	94	1	no	—
154	C₂×Q₁₆	—	94	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
158	C₄×Q₈	—	94	2	no	—
159	C₄×Q₈	—	94	2	no	—
160	C₄×Q₈	—	188	1	no	—
161	C₄×Q₈	—	188	1	no	—
162	C₂×C₄⋊C₄	—	94	1	no	—
163	C₂×C₄⋊C₄	—	188	1	no	—
164	C₂×C₄²	—	188	1	no	—
165	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
166	M₃₂	—	47	4	no	—
167	M₃₂	—	94	2	no	—
168	C₂×C₁₆	—	94	2	no	—
169	#108	—	2	2	no	—
170	#108	C₄×D₄₇	1	4	yes	—
171	#107	—	47	2	no	—
172	#107	—	94	1	no	—
173	#103	—	94	2	no	—
174	#103	—	94	2	no	—
175	#97	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
178	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
179	C₄×C₈	D₁₈₈	1	1	yes	characteristic
180	C₄×C₈	D₁₈₈	1	1	yes	characteristic
181	C₂×Q₈	—	94	2	no	—
182	C₂×Q₈	—	94	4	no	—
183	C₂×Q₈	—	188	1	no	—
184	C₂×Q₈	—	188	2	no	—
185	C₂²×C₄	—	188	1	no	—
186	C₂²×C₄	—	188	1	no	—
187	C₂²×C₄	#10625336	1	1	yes	characteristic
188	C₂²×C₄	#10625336	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
190	Q₁₆	—	47	4	no	—
191	Q₁₆	—	94	2	no	—
192	Q₁₆	—	94	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
201	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
202	C₂×C₈	#10625350	1	2	yes	—
203	C₄⋊C₄	—	94	2	no	—
204	C₄⋊C₄	—	188	2	no	—
205	C₄⋊C₄	—	188	2	no	—
206	C₄⋊C₄	—	188	1	no	—
207	C₄²	—	188	2	no	—
208	C₄²	—	188	2	no	—
209	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
210	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
211	C₄²	#10625334	1	1	yes	characteristic
212	C₄²	#10625334	1	1	yes	characteristic
213	C₁₆	—	94	4	no	—
214	C₂³	C₂×#10625336	1	1	yes	characteristic
215	Q₈	—	94	4	no	—
216	Q₈	—	188	2	no	—
217	Q₈	—	188	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	376	2	no	—
221	C₂×C₄	—	188	2	no	—
222	C₂×C₄	—	188	2	no	—
223	C₂×C₄	—	188	4	no	—
224	C₂×C₄	#13093944	1	1	yes	characteristic
225	C₂×C₄	#13093944	1	1	yes	characteristic
226	C₂×C₄	#13093944	1	1	yes	characteristic
227	C₂×C₄	#13093944	1	1	yes	characteristic
228	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
230	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
231	C₂×C₄	#13094044	1	2	yes	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	4	4	no	—
235	C₈	C₄²×D₄₇	1	1	yes	characteristic
236	C₈	#13093981	1	1	yes	characteristic
237	C₈	C₄×D₁₈₈	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×#13093944	1	1	yes	characteristic
241	C₂²	C₄×#10625336	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	376	2	no	—
246	C₄	—	188	4	no	—
247	C₄	#15688938	1	2	yes	direct product factor
248	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
249	C₄	#15689117	1	1	yes	characteristic
250	C₄	C₄×#10625334	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×#13093944	1	1	yes	characteristic
253	C₂	C₂×#15688938	1	1	yes	characteristic
254	C₂	#20245135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093944
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	C₄×#362
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
15. #15688938	C₄	direct (N × H)
118. C₄×C₄₇	#362	semidirect
149. C₄₇	C₄×#362	semidirect
247. C₄	#15688938	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×#15688938	2
C₄×#13093964	2
#20241498	2
C₄×#108×C₄₇	2
C₄×#362	47

Subgroups — C4×#15688938 #26131539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688938 #26131539