Subgroups — C₄²×#10625337 #26214754

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625337	C₂	1	3	yes	maximal
2	C₄²×Dic₉₄	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093957	C₂	1	3	yes	maximal
4	C₄×#13093947	C₂	1	3	yes	maximal
5	C₄×#13093948	C₂	1	3	yes	maximal
6	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄²×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625337	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉₄	C₂²	1	3	yes	—
10	C₄×#10625337	C₄	1	24	yes	direct product factor
11	C₂×#13093957	C₂²	1	3	yes	—
12	C₂×#13093948	C₂²	1	3	yes	—
13	C₂×#13093947	C₂²	1	3	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
16	C₄×Dic₄₇⋊C₄	—	2	3	no	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	3	yes	—
18	C₄²×Dic₄₇	—	2	1	no	—
19	#15688885	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
21	C₄×C₄⋊C₄×C₄₇	—	2	3	no	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
23	C₄³×C₄₇	—	2	1	no	—
24	C₄³×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₄	C₂³	1	1	yes	characteristic
26	C₂×#10625337	C₂×C₄	1	12	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄×Dic₉₄	—	2	6	no	—
30	C₄×Dic₉₄	C₂×C₄	1	12	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	3	yes	—
32	C₂×Dic₄₇⋊C₄	—	2	3	no	—
33	C₂×C₄×Dic₄₇	—	2	3	no	—
34	C₂×C₄×Dic₄₇	—	2	3	no	—
35	#13093957	C₂×C₄	1	12	yes	—
36	#13093948	C₂×C₄	1	12	yes	—
37	#13093947	C₂×C₄	1	12	yes	—
38	C₄×C₄₇⋊C₈	—	2	6	no	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	12	yes	—
40	C₄₇⋊#95	—	2	3	no	—
41	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₄₇	—	2	6	no	—
43	C₄×Q₈×C₄₇	C₂×C₄	1	12	yes	—
44	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
46	C₂×C₄²×C₄₇	—	2	3	no	—
47	C₂×C₄²×C₄₇	—	2	3	no	—
48	C₂×C₄²×C₄₇	D₄	1	3	yes	—
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
51	#95×C₄₇	—	2	3	no	—
52	C₂×Q₈×C₄₇	—	2	3	no	—
53	C₂×Q₈×C₄₇	C₂²×C₄	1	6	yes	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	3	no	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	1	no	—
60	C₂²×Dic₄₇	—	2	3	no	—
61	C₂×Dic₉₄	—	2	3	no	—
62	C₂×Dic₉₄	C₂²×C₄	1	6	yes	—
63	C₄⋊C₄×C₄₇	—	2	12	no	—
64	C₄⋊C₄×C₄₇	—	2	3	no	—
65	C₄⋊C₄×C₄₇	C₂²×C₄	1	6	yes	—
66	C₄²×C₄₇	—	2	12	no	—
67	C₄²×C₄₇	—	2	6	no	—
68	C₄²×C₄₇	—	4	6	no	—
69	C₄²×C₄₇	Q₁₆	1	4	yes	complemented
70	C₄²×C₄₇	C₂²×C₄	1	6	yes	—
71	C₄²×C₄₇	#41	1	6	yes	—
72	#10625337	C₄²	1	16	yes	direct product factor
73	C₄⋊Dic₄₇	—	2	3	no	—
74	C₄⋊Dic₄₇	C₂²×C₄	1	6	yes	—
75	Dic₄₇⋊C₄	—	2	12	no	—
76	C₄×Dic₄₇	—	2	12	no	—
77	C₄×Dic₄₇	—	4	6	no	—
78	C₂×C₄₇⋊C₈	—	2	6	no	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
80	C₂×C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
81	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₄₇	—	2	6	no	—
83	Q₈×C₄₇	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₄₇	—	2	6	no	—
85	C₂×C₄×C₄₇	—	2	6	no	—
86	C₂×C₄×C₄₇	—	2	6	no	—
87	C₂×C₄×C₄₇	—	4	6	no	—
88	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
89	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
90	C₂×C₄×C₄₇	C₂×Q₁₆	1	6	yes	—
91	C₂×C₄×C₄₇	#135	1	6	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
93	C₂×C₄×C₄₇	C₂×#41	1	3	yes	—
94	C₂×Dic₄₇	—	2	6	no	—
95	C₂×Dic₄₇	—	2	6	no	—
96	C₂×Dic₄₇	—	4	6	no	—
97	Dic₉₄	—	2	6	no	—
98	Dic₉₄	C₂×C₄²	1	4	yes	—
99	C₄₇⋊C₈	—	2	6	no	—
100	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
101	C₄²×Q₁₆	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	C₂×C₄×D₄	1	3	yes	—
103	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×#135	1	3	yes	—
105	C₄×C₄₇	—	2	6	no	—
106	C₄×C₄₇	—	2	4	no	—
107	C₄×C₄₇	—	4	6	no	—
108	C₄×C₄₇	C₄×Q₁₆	1	12	yes	complemented
109	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₄₇	C₄×#41	1	3	yes	—
111	Dic₄₇	—	2	4	no	—
112	Dic₄₇	—	4	6	no	—
113	C₂×C₄×Q₁₆	—	47	3	no	—
114	C₄²×Q₈	—	47	1	no	—
115	C₄²×Q₈	D₄₇	1	1	yes	characteristic
116	C₄×#107	—	47	3	no	—
117	C₄×#103	—	47	3	no	—
118	C₄×#103	—	47	3	no	—
119	C₄²×C₈	—	47	1	no	—
120	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
121	C₂×C₄₇	C₂×C₄×Q₁₆	1	3	yes	—
122	C₂×C₄₇	C₄×#135	1	3	yes	—
123	C₂²×Q₁₆	—	47	1	no	—
124	C₂×C₄×Q₈	—	47	3	no	—
125	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
126	C₄×Q₁₆	—	47	24	no	—
127	C₂×#107	—	47	3	no	—
128	C₂×#103	—	47	3	no	—
129	C₂×#103	—	47	3	no	—
130	C₂×C₄×C₈	—	47	3	no	—
131	C₂×C₄×C₈	—	47	3	no	—
132	C₄×C₄⋊C₄	—	2	3	no	—
133	C₄×C₄⋊C₄	—	47	3	no	—
134	C₄×C₄⋊C₄	—	94	3	no	—
135	C₄×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
136	C₄³	—	2	1	no	—
137	C₄³	—	94	1	no	—
138	C₄³	C₂×D₄₇	1	1	yes	characteristic
139	#340	—	47	6	no	—
140	C₄₇	C₄²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	47	1	no	—
142	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
143	C₂×Q₁₆	—	47	12	no	—
144	C₂²×C₈	—	47	1	no	—
145	C₂²×C₈	—	47	3	no	—
146	C₄×Q₈	—	2	6	no	—
147	C₄×Q₈	—	47	12	no	—
148	C₄×Q₈	—	94	6	no	—
149	C₄×Q₈	C₄×D₄₇	1	12	yes	—
150	C₂×C₄⋊C₄	—	2	3	no	—
151	C₂×C₄⋊C₄	—	47	3	no	—
152	C₂×C₄⋊C₄	—	94	3	no	—
153	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
154	C₂×C₄²	—	2	3	no	—
155	C₂×C₄²	—	2	3	no	—
156	C₂×C₄²	—	94	3	no	—
157	C₂×C₄²	—	94	3	no	—
158	C₂×C₄²	C₂²⋊D₄₇	1	3	yes	—
159	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
161	#107	—	47	12	no	—
162	#103	—	47	12	no	—
163	#103	—	47	12	no	—
164	C₄×C₈	—	47	12	no	—
165	C₄×C₈	—	94	6	no	—
166	#95	—	2	3	no	—
167	#95	—	94	3	no	—
168	C₂×Q₈	—	2	3	no	—
169	C₂×Q₈	—	47	6	no	—
170	C₂×Q₈	—	94	3	no	—
171	C₂×Q₈	C₂×C₄×D₄₇	1	6	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	3	no	—
174	C₂²×C₄	—	94	1	no	—
175	C₂²×C₄	—	94	3	no	—
176	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
178	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
179	Q₁₆	—	47	16	no	—
180	C₂×C₈	—	47	6	no	—
181	C₂×C₈	—	47	6	no	—
182	C₂×C₈	—	94	6	no	—
183	C₄⋊C₄	—	2	12	no	—
184	C₄⋊C₄	—	2	3	no	—
185	C₄⋊C₄	—	47	6	no	—
186	C₄⋊C₄	—	94	12	no	—
187	C₄⋊C₄	—	94	3	no	—
188	C₄⋊C₄	C₂×C₄×D₄₇	1	6	yes	—
189	C₄²	—	2	12	no	—
190	C₄²	—	2	6	no	—
191	C₄²	—	4	6	no	—
192	C₄²	—	94	12	no	—
193	C₄²	—	188	6	no	—
194	C₄²	#10625337	1	4	yes	direct product factor
195	C₄²	C₂×C₄×D₄₇	1	6	yes	—
196	C₄²	#10625350	1	6	yes	—
197	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
198	Q₈	—	2	6	no	—
199	Q₈	—	47	4	no	—
200	Q₈	—	94	6	no	—
201	Q₈	C₄²×D₄₇	1	4	yes	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	2	6	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	4	6	no	—
206	C₂×C₄	—	94	6	no	—
207	C₂×C₄	—	94	6	no	—
208	C₂×C₄	—	188	6	no	—
209	C₂×C₄	C₄×C₂²⋊D₄₇	1	6	yes	—
210	C₂×C₄	C₄×C₂²⋊D₄₇	1	6	yes	—
211	C₂×C₄	C₂×#10625337	1	6	yes	—
212	C₂×C₄	#13094044	1	6	yes	—
213	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
214	C₂×C₄	C₂×#10625350	1	3	yes	—
215	C₈	—	47	4	no	—
216	C₈	—	94	6	no	—
217	C₂²	C₂×C₄×C₂²⋊D₄₇	1	3	yes	—
218	C₂²	C₂²×#10625337	1	1	yes	characteristic
219	C₂²	C₂×#13094044	1	3	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	6	no	—
222	C₄	—	4	6	no	—
223	C₄	—	94	4	no	—
224	C₄	—	188	6	no	—
225	C₄	C₄×#10625337	1	12	yes	direct product factor
226	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
227	C₄	C₄×#10625350	1	3	yes	—
228	C₂	C₄²×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂	C₂×C₄×#10625337	1	3	yes	—
230	C₂	C₄×#13094044	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×Q₈	D₄₇
hypocenter	C₄₇	C₄²×Q₁₆
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
10. C₄×#10625337	C₄	direct (N × H)
69. C₄²×C₄₇	Q₁₆	semidirect
72. #10625337	C₄²	direct (N × H)
108. C₄×C₄₇	C₄×Q₁₆	semidirect
140. C₄₇	C₄²×Q₁₆	semidirect
194. C₄²	#10625337	direct (N × H)
225. C₄	C₄×#10625337	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×#10625337	2
C₄²×Dic₉₄	2
C₄×#13093957	2
C₄×#13093947	2
C₄×#13093948	2
C₄²×C₄₇⋊C₈	2
C₄²×Q₈×C₄₇	2
C₄²×Q₁₆	47

Subgroups — C42×#10625337 #26214754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#10625337 #26214754