Subgroups — C₂×#20242985 #26203927

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689521	C₂	1	2	yes	maximal
3	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689474	C₂	1	2	yes	maximal
5	#20242985	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₈₈	—	2	2	no	—
8	C₂×#13094024	C₂²	1	2	yes	—
9	C₂×#13094024	C₂²	1	2	yes	—
10	#15689528	—	2	2	no	—
11	#15689528	C₂²	1	4	yes	—
12	#15689521	—	2	4	no	—
13	#15689521	C₂²	1	8	yes	—
14	#15689503	—	2	2	no	—
15	#15689503	C₂²	1	4	yes	—
16	#15689474	—	2	4	no	—
17	#15689474	C₂²	1	8	yes	—
18	C₂×#13093958	—	2	2	no	—
19	C₂×#13093956	—	2	4	no	—
20	C₂×#13093949	—	2	4	no	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₈²×C₄₇	C₂²	1	4	yes	—
25	C₂²×Dic₉₄	—	2	2	no	—
26	C₂²×Dic₉₄	—	2	2	no	—
27	C₂×Dic₁₈₈	—	2	8	no	—
28	C₂×Dic₁₈₈	—	4	8	no	—
29	#13094024	—	2	2	no	—
30	#13094024	—	2	2	no	—
31	#13094024	—	2	2	no	—
32	#13094024	—	2	2	no	—
33	#13094024	—	2	2	no	—
34	#13094024	—	2	2	no	—
35	#13094024	C₂³	1	4	yes	—
36	#13094024	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	#13093958	—	2	8	no	—
42	#13093956	—	2	16	no	—
43	#13093949	—	2	16	no	—
44	C₂²×C₈×C₄₇	D₄	1	2	yes	—
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₄₇	D₄	1	2	yes	—
49	C₄×C₈×C₄₇	D₄	1	2	yes	—
50	C₄×C₈×C₄₇	D₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	C₄×C₈×C₄₇	C₂³	1	2	yes	—
53	C₄×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	4	2	no	—
58	C₂²×Dic₄₇	—	4	2	no	—
59	C₂×Dic₉₄	—	2	4	no	—
60	C₂×Dic₉₄	—	2	4	no	—
61	C₂×Dic₉₄	—	4	8	no	—
62	C₂×Dic₉₄	—	4	8	no	—
63	C₂×Dic₉₄	—	4	2	no	—
64	C₂×Dic₉₄	—	4	2	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	SD₁₆	1	8	yes	—
67	C₂×C₈×C₄₇	Q₁₆	1	8	yes	complemented
68	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
71	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₄⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	2	4	no	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	2	no	—
80	C₄⋊Dic₄₇	—	4	2	no	—
81	C₄⋊Dic₄₇	—	4	2	no	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	Dic₁₈₈	—	4	16	no	—
84	C₂³×C₄₇	#127	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
89	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
90	C₂×C₄×C₄₇	#135	1	4	yes	—
91	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₄₇	—	4	4	no	—
95	C₂×Dic₄₇	—	4	4	no	—
96	C₂×Dic₄₇	—	8	4	no	—
97	C₂×Dic₄₇	—	8	4	no	—
98	Dic₉₄	—	4	8	no	—
99	Dic₉₄	—	4	8	no	—
100	Dic₉₄	—	8	4	no	—
101	Dic₉₄	—	8	4	no	—
102	C₈×C₄₇	—	2	4	no	—
103	C₈×C₄₇	C₂×SD₁₆	1	8	yes	—
104	C₈×C₄₇	C₂×Q₁₆	1	8	yes	complemented
105	C₂×#1707	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	#492	1	2	yes	—
107	C₂²×C₄₇	#494	1	2	yes	—
108	C₂²×C₄₇	#495	1	2	yes	—
109	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
110	C₄×C₄₇	#492	1	2	yes	—
111	C₄×C₄₇	#494	1	2	yes	—
112	C₄×C₄₇	#495	1	2	yes	—
113	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
114	C₄×C₄₇	C₂²×Q₁₆	1	2	yes	—
115	C₄×C₄₇	C₂×#135	1	2	yes	—
116	Dic₄₇	—	8	4	no	—
117	Dic₄₇	—	8	4	no	—
118	C₂×#500	—	47	1	no	—
119	C₂×#498	—	47	2	no	—
120	C₂×#494	—	47	1	no	—
121	C₂×#487	—	47	2	no	—
122	#1707	—	47	8	no	—
123	C₂×C₈²	D₄₇	1	1	yes	characteristic
124	C₂×C₄₇	#1707	1	4	yes	complemented
125	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#494	1	1	yes	characteristic
127	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
128	C₂²×Q₁₆	—	94	2	no	—
129	C₂×C₄⋊Q₈	—	47	2	no	—
130	C₂×C₄⋊Q₈	—	47	2	no	—
131	#500	—	47	4	no	—
132	#500	—	94	2	no	—
133	#498	—	47	8	no	—
134	#498	—	94	4	no	—
135	#494	—	47	4	no	—
136	#494	—	94	2	no	—
137	#487	—	47	8	no	—
138	#487	—	94	4	no	—
139	C₂×#107	—	94	2	no	—
140	C₂×#106	—	94	4	no	—
141	C₂×#103	—	94	4	no	—
142	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
145	C₈²	C₂×D₄₇	1	4	yes	—
146	C₄₇	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	94	2	no	—
148	C₂²×Q₈	—	94	2	no	—
149	C₂×Q₁₆	—	94	8	no	—
150	C₂×Q₁₆	—	188	8	no	—
151	C₂²×C₈	D₁₈₈	1	2	yes	—
152	C₂²×C₈	D₁₈₈	1	2	yes	—
153	C₂²×C₈	D₁₈₈	1	2	yes	—
154	C₄⋊Q₈	—	47	4	no	—
155	C₄⋊Q₈	—	47	4	no	—
156	C₄⋊Q₈	—	94	2	no	—
157	C₄⋊Q₈	—	94	2	no	—
158	C₄⋊Q₈	—	94	2	no	—
159	C₄⋊Q₈	—	94	2	no	—
160	C₄⋊Q₈	—	94	2	no	—
161	C₄⋊Q₈	—	94	2	no	—
162	C₂×C₄⋊C₄	—	94	2	no	—
163	C₂×C₄⋊C₄	—	94	2	no	—
164	C₂×C₄⋊C₄	—	94	2	no	—
165	C₂×C₄⋊C₄	—	94	2	no	—
166	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
167	#107	—	94	8	no	—
168	#106	—	94	16	no	—
169	#103	—	94	16	no	—
170	C₄×C₈	D₁₈₈	1	2	yes	—
171	C₄×C₈	D₁₈₈	1	2	yes	—
172	C₄×C₈	D₁₈₈	1	2	yes	—
173	C₄×C₈	C₂²×D₄₇	1	2	yes	—
174	C₄×C₈	C₂²×D₄₇	1	2	yes	—
175	C₄×C₈	C₂²×D₄₇	1	2	yes	—
176	C₂×Q₈	—	94	4	no	—
177	C₂×Q₈	—	94	4	no	—
178	C₂×Q₈	—	188	8	no	—
179	C₂×Q₈	—	188	8	no	—
180	C₂×Q₈	—	188	2	no	—
181	C₂×Q₈	—	188	2	no	—
182	C₂²×C₄	—	188	2	no	—
183	C₂²×C₄	—	188	2	no	—
184	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
187	Q₁₆	—	188	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	#10625325	1	8	yes	—
190	C₂×C₈	Dic₁₈₈	1	8	yes	complemented
191	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
192	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
193	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
194	C₄⋊C₄	—	94	4	no	—
195	C₄⋊C₄	—	94	4	no	—
196	C₄⋊C₄	—	94	4	no	—
197	C₄⋊C₄	—	94	4	no	—
198	C₄⋊C₄	—	188	2	no	—
199	C₄⋊C₄	—	188	2	no	—
200	C₄⋊C₄	—	188	2	no	—
201	C₄⋊C₄	—	188	2	no	—
202	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
203	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
204	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
205	C₄²	C₂³×D₄₇	1	1	yes	characteristic
206	C₂³	#13094021	1	1	yes	characteristic
207	Q₈	—	376	4	no	—
208	Q₈	—	376	4	no	—
209	Q₈	—	188	8	no	—
210	Q₈	—	188	8	no	—
211	C₂×C₄	—	376	4	no	—
212	C₂×C₄	—	376	4	no	—
213	C₂×C₄	—	188	4	no	—
214	C₂×C₄	—	188	4	no	—
215	C₂×C₄	#13094021	1	1	yes	characteristic
216	C₂×C₄	#13094021	1	1	yes	characteristic
217	C₂×C₄	#13094021	1	1	yes	characteristic
218	C₂×C₄	C₂×#10625325	1	4	yes	—
219	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
220	C₂×C₄	#13094046	1	4	yes	—
221	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₂×#10625325	1	8	yes	—
226	C₈	C₂×Dic₁₈₈	1	8	yes	complemented
227	C₂²	#15689496	1	2	yes	—
228	C₂²	#15689503	1	2	yes	—
229	C₂²	#15689507	1	2	yes	—
230	C₂²	C₂×#13094021	1	1	yes	characteristic
231	C₄	—	376	4	no	—
232	C₄	—	376	4	no	—
233	C₄	#15689496	1	2	yes	—
234	C₄	#15689503	1	2	yes	—
235	C₄	#15689507	1	2	yes	—
236	C₄	C₂²×#10625325	1	2	yes	—
237	C₄	C₂²×Dic₁₈₈	1	2	yes	—
238	C₄	C₂×#13094046	1	2	yes	—
239	C₂	#20242985	1	4	yes	direct product factor
240	C₂	C₂×#15689496	1	1	yes	characteristic
241	C₂	C₂×#15689503	1	1	yes	characteristic
242	C₂	C₂×#15689507	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#127
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	C₂×#1707
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
5. #20242985	C₂	direct (N × H)
67. C₂×C₈×C₄₇	Q₁₆	semidirect
104. C₈×C₄₇	C₂×Q₁₆	semidirect
124. C₂×C₄₇	#1707	semidirect
146. C₄₇	C₂×#1707	semidirect
190. C₂×C₈	Dic₁₈₈	semidirect
226. C₈	C₂×Dic₁₈₈	semidirect
239. C₂	#20242985	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×#15689528	2
C₂×#15689521	2
C₂×#15689503	2
C₂×#15689474	2
#20242985	2
C₂×C₈²×C₄₇	2
C₂×#1707	47

Subgroups — C2×#20242985 #26203927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242985 #26203927