Subgroups — C₂×#20243562 #26216737

Order 12032.

1006 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093959	C₂	1	4	yes	maximal
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
4	#20243562	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#13093959	—	2	16	no	—
9	C₂×#13093959	C₂²	1	16	yes	—
10	#15689147	—	2	2	no	—
11	#15689147	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
13	C₂×#13093935	C₄	1	4	yes	—
14	C₂×#13093935	C₄	1	4	yes	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#423	—	2	2	no	—
19	C₄₇×#423	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	C₂³	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	D₄	1	8	yes	—
28	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	#13093959	—	2	32	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
32	#13093935	C₂×C₄	1	16	yes	—
33	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	—	2	2	no	—
36	C₂×M₁₆×C₄₇	—	2	8	no	—
37	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	C₄⋊C₈×C₄₇	—	2	8	no	—
46	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	—	4	4	no	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₄₇	—	2	8	no	—
60	M₁₆×C₄₇	—	4	4	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	—	4	4	no	—
63	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
66	C₄²×C₄₇	C₂×D₄	1	4	yes	—
67	#10625329	—	2	8	no	—
68	#10625329	—	4	4	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	4	4	no	—
71	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	—
72	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	—
73	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
74	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
75	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
76	C₂³×C₄₇	—	2	4	no	—
77	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
80	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#127	1	2	yes	—
90	C₂×C₄×C₄₇	#127	1	2	yes	—
91	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
93	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
98	C₈×C₄₇	—	4	4	no	—
99	C₄₇⋊C₈	—	4	4	no	—
100	C₄₇⋊C₈	C₂×C₄⋊C₄	1	16	yes	—
101	C₂×#1849	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	—	2	8	no	—
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	#384	1	2	yes	—
106	C₂²×C₄₇	#384	1	2	yes	—
107	C₂²×C₄₇	#430	1	4	yes	—
108	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
110	C₄×C₄₇	—	2	8	no	—
111	C₄×C₄₇	#384	1	2	yes	—
112	C₄×C₄₇	#384	1	2	yes	—
113	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
114	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
115	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×#108	—	47	4	no	—
117	C₂×#423	—	47	1	no	—
118	C₂×#423	D₄₇	1	1	yes	characteristic
119	C₂²×#97	—	47	1	no	—
120	#1849	—	47	8	no	—
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	—	2	2	no	—
123	C₂×C₄₇	#1849	1	4	yes	complemented
124	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
125	C₂×C₄₇	C₂×#430	1	2	yes	—
126	C₂²×M₁₆	—	47	2	no	—
127	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
128	C₂³×C₈	—	47	2	no	—
129	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
130	C₂×#108	—	47	16	no	—
131	C₂×#108	—	94	16	no	—
132	#423	—	2	2	no	—
133	#423	—	47	4	no	—
134	#423	—	94	2	no	—
135	#423	C₂×D₄₇	1	4	yes	—
136	C₂×C₄⋊C₈	—	2	2	no	—
137	C₂×C₄⋊C₈	—	94	2	no	—
138	C₂×#97	—	47	4	no	—
139	C₂×#97	—	47	4	no	—
140	C₂×#97	—	47	4	no	—
141	C₄₇	C₂×#1849	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	Dic₉₄	1	2	yes	—
143	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	—	47	4	no	—
147	C₂×M₁₆	—	94	2	no	—
148	C₂×M₁₆	—	94	8	no	—
149	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	47	4	no	—
152	C₂²×C₈	—	47	4	no	—
153	C₂²×C₈	—	47	8	no	—
154	C₂²×C₈	—	47	8	no	—
155	C₂²×C₈	—	47	4	no	—
156	C₂²×C₈	—	94	2	no	—
157	C₂×C₄²	Dic₉₄	1	2	yes	—
158	C₂×C₄²	C₄×D₄₇	1	2	yes	—
159	C₂×C₄²	C₄×D₄₇	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
161	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
162	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
163	#108	—	94	32	no	—
164	C₄⋊C₈	—	2	8	no	—
165	C₄⋊C₈	—	94	8	no	—
166	#97	—	47	16	no	—
167	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
170	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
171	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
172	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
175	C₂²×C₄	D₄×D₄₇	1	4	yes	—
176	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
180	M₁₆	—	2	8	no	—
181	M₁₆	—	4	4	no	—
182	M₁₆	—	94	8	no	—
183	M₁₆	—	188	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	47	16	no	—
187	C₂×C₈	—	47	16	no	—
188	C₂×C₈	—	47	8	no	—
189	C₂×C₈	—	47	8	no	—
190	C₂×C₈	—	47	8	no	—
191	C₂×C₈	—	94	4	no	—
192	C₂×C₈	—	188	4	no	—
193	C₄²	Dic₄₇⋊C₄	1	4	yes	—
194	C₄²	Dic₄₇⋊C₄	1	4	yes	—
195	C₄²	C₂×C₄×D₄₇	1	4	yes	—
196	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
199	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
200	C₂³	#13094023	1	4	yes	—
201	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
206	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
207	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
208	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
210	C₂×C₄	#13094020	1	2	yes	—
211	C₂×C₄	#13094020	1	2	yes	—
212	C₂×C₄	#13094022	1	2	yes	—
213	C₂×C₄	#13094022	1	2	yes	—
214	C₂×C₄	#13094023	1	4	yes	—
215	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
217	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
219	C₈	—	4	4	no	—
220	C₈	—	47	16	no	—
221	C₈	—	188	4	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	8	no	—
225	C₂²	#15688992	1	2	yes	—
226	C₂²	#15688992	1	2	yes	—
227	C₂²	#15689171	1	4	yes	—
228	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
229	C₂²	C₂×#13094023	1	2	yes	—
230	C₄	—	2	8	no	—
231	C₄	#15688992	1	2	yes	—
232	C₄	#15688992	1	2	yes	—
233	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
234	C₄	C₂×#13094020	1	1	yes	characteristic
235	C₄	C₂×#13094022	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	2	2	no	—
238	C₂	#20243562	1	4	yes	direct product factor
239	C₂	C₂×#15688992	1	1	yes	characteristic
240	C₂	C₂×#15689171	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	C₂×#1849
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20243562	C₂	direct (N × H)
123. C₂×C₄₇	#1849	semidirect
141. C₄₇	C₂×#1849	semidirect
238. C₂	#20243562	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13093959	2
C₂×#15689147	2
C₂²×#13093935	2
#20243562	2
C₂×C₄₇×#423	2
C₂×#1849	47

Subgroups — C2×#20243562 #26216737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243562 #26216737