Subgroups — C₂×#20242206 #26185670

Order 12032.

718 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	C₂×#15689247	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20242206	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688853	C₂	1	2	yes	maximal
5	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₈₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
9	#15689247	—	2	4	no	—
10	#15689247	C₂²	1	8	yes	—
11	C₄×Dic₁₈₈	C₄	1	4	yes	—
12	C₄×Dic₁₈₈	C₂²	1	4	yes	—
13	C₂×#13093958	C₄	1	1	yes	characteristic
14	C₂×Dic₄₇⋊C₈	—	2	2	no	—
15	C₂×#13093949	C₄	1	2	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	—	2	2	no	—
19	#15688853	C₂²	1	8	yes	—
20	#15688844	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#334	C₂²	1	4	yes	—
24	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₈₈	C₈	1	8	yes	complemented
26	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₄×Dic₉₄	—	2	2	no	—
30	C₄×Dic₉₄	—	2	2	no	—
31	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
32	C₄×Dic₉₄	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	#13093958	C₂×C₄	1	4	yes	—
37	Dic₄₇⋊C₈	—	2	8	no	—
38	#13093949	C₂×C₄	1	8	yes	—
39	C₄×C₄₇⋊C₈	—	2	1	no	—
40	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
41	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
43	C₈×Dic₄₇	—	2	8	no	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₇	—	2	2	no	—
49	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
50	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	2	no	—
56	C₂²×Dic₄₇	—	2	2	no	—
57	C₂×Dic₉₄	—	2	2	no	—
58	C₂×Dic₉₄	—	2	4	no	—
59	C₂×Dic₉₄	C₂×C₈	1	8	yes	—
60	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	—	4	4	no	—
64	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₄₇	#41	1	1	yes	characteristic
70	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₄⋊Dic₄₇	—	2	2	no	—
72	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
73	Dic₄₇⋊C₄	—	2	4	no	—
74	Dic₄₇⋊C₄	—	2	4	no	—
75	C₄×Dic₄₇	—	2	4	no	—
76	C₄×Dic₄₇	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
81	Dic₁₈₈	C₂×C₈	1	16	yes	complemented
82	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₄₇	#131	1	2	yes	—
89	C₂×C₄×C₄₇	#137	1	4	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
93	C₂×Dic₄₇	—	2	4	no	—
94	C₂×Dic₄₇	—	2	8	no	—
95	C₂×Dic₄₇	—	2	4	no	—
96	C₂×Dic₄₇	—	4	4	no	—
97	Dic₉₄	—	2	4	no	—
98	Dic₉₄	—	2	8	no	—
99	Dic₉₄	C₂²×C₈	1	8	yes	—
100	C₈×C₄₇	—	2	2	no	—
101	C₈×C₄₇	—	4	4	no	—
102	C₈×C₄₇	C₂²×C₈	1	4	yes	—
103	C₄₇⋊C₈	—	2	4	no	—
104	C₄₇⋊C₈	—	4	2	no	—
105	C₂×#1579	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	C₈×D₄	1	2	yes	—
107	C₂²×C₄₇	#441	1	2	yes	—
108	C₂²×C₄₇	#444	1	2	yes	—
109	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	C₈×D₄	1	2	yes	—
113	C₄×C₄₇	#441	1	2	yes	—
114	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
116	C₄×C₄₇	C₂×#137	1	2	yes	—
117	Dic₄₇	—	2	8	no	—
118	Dic₄₇	—	4	4	no	—
119	C₂×C₈×Q₈	—	47	2	no	—
120	C₂×C₄×Q₁₆	—	47	1	no	—
121	#1579	—	47	8	no	—
122	C₂×#334	D₄₇	1	1	yes	characteristic
123	C₂×Q₈⋊C₈	—	47	2	no	—
124	C₂×#322	—	47	1	no	—
125	C₂×C₄₇	#1579	1	4	yes	complemented
126	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
127	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
128	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
129	C₂²×Q₁₆	—	47	1	no	—
130	C₂×C₄×Q₈	—	47	2	no	—
131	C₈×Q₈	—	47	8	no	—
132	C₈×Q₈	—	94	4	no	—
133	C₄×Q₁₆	—	47	4	no	—
134	C₄×Q₁₆	—	47	4	no	—
135	C₂×#107	—	47	1	no	—
136	C₂×C₄⋊C₈	—	94	2	no	—
137	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
138	C₂×#103	—	47	2	no	—
139	C₂×C₄×C₈	—	47	1	no	—
140	C₂×C₄×C₈	—	47	1	no	—
141	C₂×C₄×C₈	—	94	2	no	—
142	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
143	#334	C₂×D₄₇	1	4	yes	—
144	Q₈⋊C₈	—	47	8	no	—
145	#322	—	47	4	no	—
146	C₄₇	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	47	2	no	—
148	C₂×Q₁₆	—	47	8	no	—
149	C₂×Q₁₆	—	47	4	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	47	2	no	—
152	C₂²×C₈	—	94	1	no	—
153	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
154	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
155	C₄×Q₈	—	47	4	no	—
156	C₄×Q₈	—	47	4	no	—
157	C₄×Q₈	—	94	2	no	—
158	C₄×Q₈	—	94	2	no	—
159	C₂×C₄⋊C₄	—	47	2	no	—
160	C₂×C₄⋊C₄	—	94	2	no	—
161	C₂×C₄²	—	94	2	no	—
162	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
163	#107	—	47	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	94	8	no	—
166	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
167	#103	—	47	8	no	—
168	C₄×C₈	—	47	2	no	—
169	C₄×C₈	—	47	2	no	—
170	C₄×C₈	—	47	2	no	—
171	C₄×C₈	—	94	8	no	—
172	C₄×C₈	—	94	1	no	—
173	C₄×C₈	C₄×D₄₇	1	2	yes	—
174	C₄×C₈	C₂²×D₄₇	1	2	yes	—
175	C₂×Q₈	—	47	8	no	—
176	C₂×Q₈	—	47	4	no	—
177	C₂×Q₈	—	94	2	no	—
178	C₂×Q₈	—	94	4	no	—
179	C₂²×C₄	—	94	2	no	—
180	C₂²×C₄	—	94	2	no	—
181	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
183	C₂²×C₄	#10625354	1	1	yes	characteristic
184	Q₁₆	—	47	16	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	47	4	no	—
189	C₂×C₈	—	94	4	no	—
190	C₂×C₈	—	94	2	no	—
191	C₂×C₈	—	188	2	no	—
192	C₂×C₈	C₈×D₄₇	1	4	yes	—
193	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
195	C₄⋊C₄	—	47	4	no	—
196	C₄⋊C₄	—	94	4	no	—
197	C₄⋊C₄	—	94	2	no	—
198	C₄⋊C₄	—	94	4	no	—
199	C₄²	—	94	4	no	—
200	C₄²	—	94	4	no	—
201	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₄²	D₄×D₄₇	1	1	yes	characteristic
203	C₄²	#10625354	1	1	yes	characteristic
204	C₄²	C₂³×D₄₇	1	1	yes	characteristic
205	C₂³	#13093986	1	1	yes	characteristic
206	Q₈	—	47	8	no	—
207	Q₈	—	94	4	no	—
208	Q₈	—	94	8	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	94	4	no	—
211	C₂×C₄	—	94	8	no	—
212	C₂×C₄	—	94	4	no	—
213	C₂×C₄	—	188	4	no	—
214	C₂×C₄	#13093986	1	1	yes	characteristic
215	C₂×C₄	#13093986	1	1	yes	characteristic
216	C₂×C₄	#13093986	1	1	yes	characteristic
217	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
218	C₂×C₄	#13094032	1	2	yes	—
219	C₂×C₄	#13094053	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₈	—	94	4	no	—
226	C₈	—	188	2	no	—
227	C₈	C₂×C₈×D₄₇	1	4	yes	—
228	C₂²	#15689188	1	2	yes	—
229	C₂²	#15689229	1	2	yes	—
230	C₂²	#15689244	1	2	yes	—
231	C₂²	C₂×#13093986	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	94	8	no	—
235	C₄	—	188	4	no	—
236	C₄	#15689188	1	2	yes	—
237	C₄	#15689229	1	2	yes	—
238	C₄	C₂²×C₈×D₄₇	1	1	yes	characteristic
239	C₄	C₂×#13094032	1	1	yes	characteristic
240	C₄	C₂×#13094053	1	2	yes	—
241	C₂	#20242206	1	4	yes	direct product factor
242	C₂	C₂×#15689188	1	1	yes	characteristic
243	C₂	C₂×#15689229	1	1	yes	characteristic
244	C₂	C₂×#15689244	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	C₂×#1579
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #20242206	C₂	direct (N × H)
25. C₂×Dic₁₈₈	C₈	semidirect
81. Dic₁₈₈	C₂×C₈	semidirect
125. C₂×C₄₇	#1579	semidirect
146. C₄₇	C₂×#1579	semidirect
241. C₂	#20242206	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×#15689247	2
C₂×C₄×Dic₁₈₈	2
#20242206	2
C₂×#15688853	2
C₂×#15688844	2
C₂×C₄₇×#334	2
C₂×#1579	47

Subgroups — C2×#20242206 #26185670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242206 #26185670