Subgroups — C₂×#19136646 #25250848

Order 11008.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₄₄	C₂	1	2	yes	maximal
2	C₂×#14583279	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
4	#19136646	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582704	C₂	1	2	yes	maximal
6	C₂×C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₂	—	2	1	no	—
8	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×#13063325	—	2	1	no	—
11	C₂×C₄×Dic₈₆	—	2	1	no	—
12	C₂×Dic₃₄₄	—	2	8	no	—
13	C₂×Dic₃₄₄	C₂²	1	8	yes	—
14	#14583279	—	2	2	no	—
15	#14583279	C₂²	1	4	yes	—
16	C₄×Dic₁₇₂	—	2	2	no	—
17	C₄×Dic₁₇₂	C₂²	1	4	yes	—
18	C₂×#13063259	C₂²	1	1	yes	characteristic
19	C₂×#13063250	—	2	1	no	—
20	#14582704	C₂²	1	8	yes	—
21	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×C₄⋊C₁₆	C₂²	1	4	yes	—
24	C₂²×Dic₈₆	—	2	1	no	—
25	C₂²×Dic₈₆	—	2	2	no	—
26	C₂×Dic₁₇₂	—	2	1	no	—
27	C₂×Dic₁₇₂	—	2	2	no	—
28	C₂×Dic₁₇₂	—	2	4	no	—
29	C₂×Dic₁₇₂	—	2	4	no	—
30	C₂×Dic₁₇₂	—	2	4	no	—
31	C₂×Dic₁₇₂	—	2	2	no	—
32	C₂×Dic₁₇₂	—	4	4	no	—
33	C₂×Dic₁₇₂	D₄	1	4	yes	—
34	C₂×Dic₁₇₂	C₂³	1	4	yes	—
35	C₂×Dic₁₇₂	C₂³	1	2	yes	—
36	#13063325	—	2	4	no	—
37	#13063325	—	4	2	no	—
38	C₄×Dic₈₆	—	2	4	no	—
39	C₄×Dic₈₆	—	4	2	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	4	1	no	—
42	C₂×C₄⋊Dic₄₃	—	4	1	no	—
43	C₂×Dic₄₃⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₄₃	—	4	1	no	—
45	Dic₃₄₄	—	2	16	no	—
46	#13063259	—	2	1	no	—
47	#13063259	C₂³	1	2	yes	—
48	#13063250	—	2	4	no	—
49	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₁₆×C₄₃	—	2	4	no	—
53	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
54	C₄×C₈×C₄₃	D₄	1	2	yes	—
55	C₄×C₈×C₄₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₄₃	—	4	1	no	—
60	C₂²×Dic₄₃	—	4	1	no	—
61	C₂²×Dic₄₃	—	4	2	no	—
62	C₂×Dic₈₆	—	2	4	no	—
63	C₂×Dic₈₆	—	2	2	no	—
64	C₂×Dic₈₆	—	2	4	no	—
65	C₂×Dic₈₆	—	4	1	no	—
66	C₂×Dic₈₆	—	4	2	no	—
67	C₂×Dic₈₆	—	4	4	no	—
68	C₂×Dic₈₆	—	4	4	no	—
69	C₂×Dic₈₆	—	4	2	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₄₃	#41	1	2	yes	—
75	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₄²×C₄₃	D₈	1	2	yes	—
77	C₄²×C₄₃	C₂×D₄	1	2	yes	—
78	C₄⋊Dic₄₃	—	2	2	no	—
79	C₄⋊Dic₄₃	—	4	1	no	—
80	C₄⋊Dic₄₃	—	4	4	no	—
81	C₄⋊Dic₄₃	—	4	4	no	—
82	Dic₄₃⋊C₄	—	4	4	no	—
83	C₄×Dic₄₃	—	4	4	no	—
84	Dic₁₇₂	—	2	4	no	—
85	Dic₁₇₂	—	2	4	no	—
86	Dic₁₇₂	—	2	2	no	—
87	Dic₁₇₂	—	2	4	no	—
88	Dic₁₇₂	—	4	8	no	—
89	Dic₁₇₂	—	4	4	no	—
90	Dic₁₇₂	C₂×D₄	1	4	yes	—
91	C₁₆×C₄₃	—	2	4	no	—
92	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	Q₃₂	1	4	yes	complemented
95	C₂×C₄×C₄₃	#121	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×Dic₄₃	—	4	2	no	—
102	C₂×Dic₄₃	—	4	4	no	—
103	C₂×Dic₄₃	—	4	2	no	—
104	C₂×Dic₄₃	—	4	4	no	—
105	C₂×Dic₄₃	—	8	4	no	—
106	C₂×Dic₄₃	—	8	2	no	—
107	Dic₈₆	—	2	4	no	—
108	Dic₈₆	—	4	2	no	—
109	Dic₈₆	—	4	4	no	—
110	Dic₈₆	—	4	4	no	—
111	Dic₈₆	—	4	4	no	—
112	Dic₈₆	—	8	4	no	—
113	C₈×C₄₃	—	2	2	no	—
114	C₈×C₄₃	#121	1	2	yes	—
115	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂×#2209	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	#459	1	2	yes	—
119	C₂²×C₄₃	C₂×Q₃₂	1	2	yes	—
120	C₂²×C₄₃	#510	1	2	yes	—
121	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₄₃	—	2	2	no	—
123	C₄×C₄₃	#459	1	2	yes	—
124	C₄×C₄₃	C₂×Q₃₂	1	4	yes	complemented
125	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
126	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
127	Dic₄₃	—	4	4	no	—
128	Dic₄₃	—	8	2	no	—
129	Dic₄₃	—	8	4	no	—
130	C₂²×Q₃₂	—	43	2	no	—
131	C₂×#494	—	43	1	no	—
132	C₂×C₄×Q₁₆	—	43	1	no	—
133	#2209	—	43	8	no	—
134	C₂×C₄⋊C₁₆	D₄₃	1	1	yes	characteristic
135	C₂×#358	—	43	2	no	—
136	C₂×C₄₃	#2209	1	4	yes	complemented
137	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
138	C₂×C₄₃	C₂²×Q₃₂	1	1	yes	characteristic
139	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
140	C₂²×Q₁₆	—	43	1	no	—
141	C₂²×Q₁₆	—	43	2	no	—
142	C₂²×Q₁₆	—	86	1	no	—
143	C₂×C₄⋊Q₈	—	86	1	no	—
144	C₂×C₄×Q₈	—	86	1	no	—
145	C₂×Q₃₂	—	43	8	no	—
146	C₂×Q₃₂	—	86	8	no	—
147	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
148	#494	—	43	4	no	—
149	#494	—	86	2	no	—
150	C₄×Q₁₆	—	43	4	no	—
151	C₄×Q₁₆	—	86	2	no	—
152	C₂×#107	—	43	1	no	—
153	C₂×#103	—	86	1	no	—
154	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
155	C₄⋊C₁₆	C₂×D₄₃	1	4	yes	—
156	#358	—	43	8	no	—
157	C₄₃	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	86	1	no	—
159	C₂²×Q₈	—	86	2	no	—
160	C₂×Q₁₆	—	43	4	no	—
161	C₂×Q₁₆	—	43	2	no	—
162	C₂×Q₁₆	—	43	4	no	—
163	C₂×Q₁₆	—	86	1	no	—
164	C₂×Q₁₆	—	86	2	no	—
165	C₂×Q₁₆	—	86	4	no	—
166	C₂×Q₁₆	—	86	4	no	—
167	C₂×Q₁₆	—	86	2	no	—
168	C₂×Q₁₆	—	86	4	no	—
169	C₂×Q₁₆	—	172	4	no	—
170	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
172	C₄⋊Q₈	—	86	4	no	—
173	C₄⋊Q₈	—	172	2	no	—
174	C₄×Q₈	—	86	4	no	—
175	C₄×Q₈	—	172	2	no	—
176	C₂×C₄⋊C₄	—	86	1	no	—
177	C₂×C₄⋊C₄	—	172	1	no	—
178	C₂×C₄⋊C₄	—	172	1	no	—
179	C₂×C₄⋊C₄	—	172	1	no	—
180	C₂×C₄²	—	172	1	no	—
181	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
182	Q₃₂	—	86	16	no	—
183	C₂×C₁₆	—	2	4	no	—
184	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
185	#107	—	43	2	no	—
186	#107	—	86	1	no	—
187	#103	—	86	4	no	—
188	C₄×C₈	D₁₇₂	1	2	yes	—
189	C₄×C₈	C₂²×D₄₃	1	2	yes	—
190	C₂×Q₈	—	86	4	no	—
191	C₂×Q₈	—	86	2	no	—
192	C₂×Q₈	—	86	4	no	—
193	C₂×Q₈	—	172	4	no	—
194	C₂×Q₈	—	172	4	no	—
195	C₂×Q₈	—	172	2	no	—
196	C₂×Q₈	—	172	1	no	—
197	C₂×Q₈	—	172	2	no	—
198	C₂²×C₄	—	172	1	no	—
199	C₂²×C₄	—	172	2	no	—
200	C₂²×C₄	—	172	1	no	—
201	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
202	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
204	Q₁₆	—	43	4	no	—
205	Q₁₆	—	86	4	no	—
206	Q₁₆	—	86	4	no	—
207	Q₁₆	—	86	2	no	—
208	Q₁₆	—	86	4	no	—
209	Q₁₆	—	172	8	no	—
210	Q₁₆	—	172	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
213	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
214	C₂×C₈	D₄×D₄₃	1	2	yes	—
215	C₂×C₈	#10621902	1	2	yes	—
216	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
217	C₄⋊C₄	—	86	2	no	—
218	C₄⋊C₄	—	172	1	no	—
219	C₄⋊C₄	—	172	4	no	—
220	C₄⋊C₄	—	172	4	no	—
221	C₄⋊C₄	—	172	4	no	—
222	C₄²	—	172	4	no	—
223	C₄²	D₃₄₄	1	2	yes	—
224	C₄²	C₂×D₁₇₂	1	2	yes	—
225	C₁₆	—	2	4	no	—
226	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
227	Q₈	—	86	4	no	—
228	Q₈	—	344	4	no	—
229	Q₈	—	172	4	no	—
230	Q₈	—	172	4	no	—
231	Q₈	—	172	4	no	—
232	Q₈	—	172	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	344	2	no	—
235	C₂×C₄	—	344	4	no	—
236	C₂×C₄	—	172	2	no	—
237	C₂×C₄	—	172	4	no	—
238	C₂×C₄	—	172	4	no	—
239	C₂×C₄	—	172	2	no	—
240	C₂×C₄	Dic₃₄₄	1	4	yes	complemented
241	C₂×C₄	#13063300	1	2	yes	—
242	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
243	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
244	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
245	C₂×C₄	#13063353	1	2	yes	—
246	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#13063300	1	2	yes	—
249	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
250	C₈	C₂×#10621902	1	1	yes	characteristic
251	C₂²	#14583095	1	2	yes	—
252	C₂²	C₂×Dic₃₄₄	1	2	yes	—
253	C₂²	#14583342	1	2	yes	—
254	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	344	4	no	—
257	C₄	—	344	2	no	—
258	C₄	—	172	4	no	—
259	C₄	#14583095	1	2	yes	—
260	C₄	C₂×Dic₃₄₄	1	4	yes	complemented
261	C₄	C₂×#13063300	1	1	yes	characteristic
262	C₄	C₂×#13063353	1	1	yes	characteristic
263	C₂	#19136646	1	4	yes	direct product factor
264	C₂	C₂×#14583095	1	1	yes	characteristic
265	C₂	C₂²×Dic₃₄₄	1	1	yes	characteristic
266	C₂	C₂×#14583342	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	C₂×C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	C₂×#2209
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #19136646	C₂	direct (N × H)
94. C₂×C₄×C₄₃	Q₃₂	semidirect
124. C₄×C₄₃	C₂×Q₃₂	semidirect
136. C₂×C₄₃	#2209	semidirect
157. C₄₃	C₂×#2209	semidirect
240. C₂×C₄	Dic₃₄₄	semidirect
260. C₄	C₂×Dic₃₄₄	semidirect
263. C₂	#19136646	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×Dic₃₄₄	2
C₂×#14583279	2
C₂×C₄×Dic₁₇₂	2
#19136646	2
C₂×#14582704	2
C₂×C₄₃×C₄⋊C₁₆	2
C₂×#2209	43

Subgroups — C2×#19136646 #25250848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136646 #25250848