Subgroups — C₄×#15688867 #26107218

Order 12032.

708 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	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂×#15688867	C₂	1	2	yes	maximal
3	#20241271	C₂	1	2	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
7	#15688878	—	2	2	no	—
8	#15688867	C₄	1	8	yes	direct product factor
9	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₄₇	—	2	2	no	—
11	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
12	C₄₇×#390	—	2	2	no	—
13	C₄₇×#390	C₄	1	2	yes	—
14	C₄₇×#384	C₂²	1	1	yes	characteristic
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
19	C₄×C₄₇⋊C₈	—	4	4	no	—
20	C₂²×D₄×C₄₇	—	2	2	no	—
21	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₄₇	—	2	2	no	—
23	#127×C₄₇	—	2	2	no	—
24	#127×C₄₇	C₂×C₄	1	4	yes	—
25	C₄×D₄×C₄₇	—	4	4	no	—
26	C₄×D₄×C₄₇	—	4	4	no	—
27	C₄×D₄×C₄₇	—	4	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
31	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₂⁴×C₄₇	—	2	2	no	—
38	C₂×D₄×C₄₇	—	2	2	no	—
39	C₂×D₄×C₄₇	—	2	8	no	—
40	C₂×D₄×C₄₇	—	4	2	no	—
41	C₂×D₄×C₄₇	—	4	2	no	—
42	C₂×D₄×C₄₇	—	4	2	no	—
43	C₂×D₄×C₄₇	—	4	2	no	—
44	C₂×D₄×C₄₇	—	4	4	no	—
45	C₂×D₄×C₄₇	C₄²	1	4	yes	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	4	4	no	—
48	C₂²×C₄×C₄₇	—	4	8	no	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₄⋊C₄×C₄₇	—	4	2	no	—
55	C₄⋊C₄×C₄₇	—	4	2	no	—
56	C₄⋊C₄×C₄₇	—	4	2	no	—
57	C₂²⋊C₄×C₄₇	—	4	4	no	—
58	C₂²⋊C₄×C₄₇	—	4	8	no	—
59	C₄²×C₄₇	—	2	2	no	—
60	C₄²×C₄₇	—	2	4	no	—
61	C₄²×C₄₇	—	4	1	no	—
62	C₄²×C₄₇	D₈	1	4	yes	—
63	C₄²×C₄₇	SD₁₆	1	4	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
65	C₄²×C₄₇	#41	1	2	yes	—
66	C₂×C₄₇⋊C₈	—	2	8	no	—
67	C₂×C₄₇⋊C₈	—	4	4	no	—
68	C₂×C₄₇⋊C₈	—	4	4	no	—
69	C₂³×C₄₇	—	2	4	no	—
70	C₂³×C₄₇	—	4	2	no	—
71	C₂³×C₄₇	—	4	2	no	—
72	C₂³×C₄₇	—	4	4	no	—
73	C₂³×C₄₇	—	4	4	no	—
74	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	D₄×C₄₇	—	4	4	no	—
76	D₄×C₄₇	—	4	4	no	—
77	D₄×C₄₇	—	4	8	no	—
78	D₄×C₄₇	—	4	8	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	8	8	no	—
88	C₂×C₄×C₄₇	#100	1	2	yes	—
89	C₂×C₄×C₄₇	#102	1	4	yes	—
90	C₂×C₄×C₄₇	#102	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₄₇	#135	1	4	yes	—
97	C₄₇⋊C₈	—	4	8	no	—
98	C₄×#331	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	4	4	no	—
100	C₂²×C₄₇	—	4	8	no	—
101	C₂²×C₄₇	—	8	2	no	—
102	C₂²×C₄₇	—	8	4	no	—
103	C₂²×C₄₇	—	8	2	no	—
104	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
106	C₂²×C₄₇	#413	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×#102	1	2	yes	—
108	C₂²×C₄₇	#416	1	2	yes	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	4	1	no	—
113	C₄×C₄₇	—	8	2	no	—
114	C₄×C₄₇	#331	1	4	yes	complemented
115	C₄×C₄₇	C₄×D₈	1	4	yes	—
116	C₄×C₄₇	C₄×SD₁₆	1	4	yes	—
117	C₄×#127	D₄₇	1	1	yes	characteristic
118	C₄×C₄⋊C₈	—	47	2	no	—
119	C₂×#331	—	47	2	no	—
120	#1296	—	47	2	no	—
121	C₂×C₄₇	—	8	4	no	—
122	C₂×C₄₇	C₂×#331	1	1	yes	characteristic
123	C₂×C₄₇	#1540	1	2	yes	—
124	C₂×C₄₇	#1543	1	1	yes	characteristic
125	C₂×C₄₇	C₄×#100	1	1	yes	characteristic
126	C₂×C₄₇	C₄×#102	1	2	yes	—
127	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
128	C₂×C₄×D₄	—	2	2	no	—
129	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
130	C₂×C₄⋊C₈	—	47	4	no	—
131	C₂×C₄×C₈	—	94	2	no	—
132	#390	—	2	2	no	—
133	#390	Dic₄₇	1	2	yes	—
134	#384	C₂×D₄₇	1	1	yes	characteristic
135	C₄³	C₂×D₄₇	1	1	yes	characteristic
136	#336	—	94	2	no	—
137	#331	—	47	8	no	—
138	C₄₇	C₄×#331	1	1	yes	characteristic, complemented, Sylow
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
141	C₂³×C₄	—	2	2	no	—
142	C₂²×C₈	—	94	4	no	—
143	#127	—	2	2	no	—
144	#127	C₄×D₄₇	1	4	yes	—
145	C₄×D₄	—	4	4	no	—
146	C₄×D₄	—	4	4	no	—
147	C₄×D₄	—	4	4	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
150	C₂×C₂²⋊C₄	—	2	4	no	—
151	C₂×C₂²⋊C₄	—	2	2	no	—
152	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
156	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
157	C₄⋊C₈	—	47	8	no	—
158	C₄⋊C₈	—	94	4	no	—
159	C₄×C₈	—	188	4	no	—
160	C₂⁴	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	8	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	C₄×Dic₄₇	1	4	yes	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	—	4	8	no	—
172	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
174	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
177	C₂×C₈	—	94	8	no	—
178	C₂×C₈	—	188	4	no	—
179	C₂×C₈	—	188	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₂²⋊C₄	—	4	8	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	1	no	—
188	C₄²	#10625334	1	4	yes	—
189	C₄²	#10625335	1	4	yes	—
190	C₄²	C₂×C₄×D₄₇	1	2	yes	—
191	C₄²	#10625350	1	2	yes	—
192	C₂³	—	2	4	no	—
193	C₂³	—	4	2	no	—
194	C₂³	—	4	2	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	4	4	no	—
197	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	8	no	—
201	D₄	—	4	8	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	8	no	—
211	C₂×C₄	#13093941	1	2	yes	—
212	C₂×C₄	#13093945	1	4	yes	—
213	C₂×C₄	#13093945	1	4	yes	—
214	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
215	C₂×C₄	#13093983	1	1	yes	characteristic
216	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
217	C₂×C₄	C₂×#10625334	1	2	yes	—
218	C₂×C₄	C₂×#10625335	1	2	yes	—
219	C₂×C₄	#13094044	1	4	yes	—
220	C₈	—	188	8	no	—
221	C₂²	—	4	8	no	—
222	C₂²	—	4	4	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	4	no	—
225	C₂²	—	8	2	no	—
226	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
227	C₂²	C₂×#13093941	1	1	yes	characteristic
228	C₂²	#15689107	1	1	yes	characteristic
229	C₂²	C₂×#13093945	1	2	yes	—
230	C₂²	#15689119	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	8	2	no	—
236	C₄	#15688867	1	4	yes	direct product factor
237	C₄	C₄×#10625334	1	4	yes	—
238	C₄	C₄×#10625335	1	4	yes	—
239	C₂	—	8	4	no	—
240	C₂	C₂×#15688867	1	1	yes	characteristic
241	C₂	#20242037	1	2	yes	—
242	C₂	#20242050	1	1	yes	characteristic
243	C₂	C₄×#13093941	1	1	yes	characteristic
244	C₂	C₄×#13093945	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	C₄×#331
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. #20241271	C₂	semidirect
8. #15688867	C₄	direct (N × H)
18. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
114. C₄×C₄₇	#331	semidirect
138. C₄₇	C₄×#331	semidirect
236. C₄	#15688867	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688867	2
#20241271	2
C₄×#127×C₄₇	2
C₄×#331	47

Subgroups — C4×#15688867 #26107218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688867 #26107218