Subgroups — C₂×#19134120 #25079228

Order 11008.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583191	C₂	1	2	yes	maximal
2	C₂×#14583029	C₂	1	1	yes	maximal, characteristic
3	#19134120	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582650	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582648	C₂	1	2	yes	maximal
6	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
8	#14583191	—	2	4	no	—
9	#14583191	C₂²	1	8	yes	—
10	#14583029	—	2	2	no	—
11	#14583029	C₂²	1	4	yes	—
12	C₂×#13063258	—	2	2	no	—
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₄₃⋊C₈	—	2	1	no	—
16	C₂×#13063248	—	2	2	no	—
17	C₂×#13063235	—	2	1	no	—
18	#14582650	C₂²	1	4	yes	—
19	#14582648	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
21	C₄₃×#500	—	2	2	no	—
22	C₄₃×#500	C₂²	1	4	yes	—
23	C₂×#107×C₄₃	—	2	1	no	—
24	C₂×#107×C₄₃	—	2	1	no	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
27	C₂²×C₄₃⋊C₈	—	2	1	no	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₄×Dic₈₆	—	2	1	no	—
30	C₄×Dic₈₆	—	2	1	no	—
31	C₄×Dic₈₆	D₄	1	2	yes	—
32	C₄×Dic₈₆	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
34	C₂×Dic₄₃⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₄₃	—	2	1	no	—
36	#13063258	—	2	8	no	—
37	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
38	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
40	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
41	Dic₄₃⋊C₈	—	2	4	no	—
42	#13063248	—	2	8	no	—
43	#13063235	—	2	4	no	—
44	C₂²×Q₈×C₄₃	—	2	2	no	—
45	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₃	—	2	2	no	—
48	C₄⋊Q₈×C₄₃	—	2	2	no	—
49	C₄⋊Q₈×C₄₃	—	2	2	no	—
50	C₄⋊Q₈×C₄₃	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	#107×C₄₃	—	2	4	no	—
55	#107×C₄₃	—	2	4	no	—
56	C₄×C₈×C₄₃	D₄	1	2	yes	—
57	C₄×C₈×C₄₃	C₂³	1	2	yes	—
58	C₂×Q₈×C₄₃	—	2	4	no	—
59	C₂×Q₈×C₄₃	—	4	2	no	—
60	C₂×Q₈×C₄₃	—	8	4	no	—
61	C₂²×C₄×C₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	C₂×Dic₈₆	—	2	4	no	—
68	C₂×Dic₈₆	—	2	1	no	—
69	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
72	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
74	C₄⋊C₄×C₄₃	—	2	4	no	—
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₄⋊C₄×C₄₃	—	4	2	no	—
77	C₄⋊C₄×C₄₃	—	4	2	no	—
78	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₃	#41	1	1	yes	characteristic
81	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₄⋊Dic₄₃	—	2	1	no	—
83	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
84	Dic₄₃⋊C₄	—	2	2	no	—
85	Dic₄₃⋊C₄	—	2	2	no	—
86	C₄×Dic₄₃	—	2	2	no	—
87	C₄×Dic₄₃	—	2	2	no	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂×C₄₃⋊C₈	—	4	4	no	—
92	C₂³×C₄₃	#121	1	1	yes	characteristic
93	Q₈×C₄₃	—	8	4	no	—
94	Q₈×C₄₃	—	8	4	no	—
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	—	4	4	no	—
97	C₂×C₄×C₄₃	—	8	4	no	—
98	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₄₃	#137	1	2	yes	—
103	C₂×C₄×C₄₃	#137	1	4	yes	—
104	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
107	C₂×Dic₄₃	—	2	2	no	—
108	C₂×Dic₄₃	—	2	2	no	—
109	C₂×Dic₄₃	—	4	2	no	—
110	C₂×Dic₄₃	—	4	2	no	—
111	Dic₈₆	—	2	4	no	—
112	Dic₈₆	—	4	2	no	—
113	C₈×C₄₃	—	2	2	no	—
114	C₈×C₄₃	C₂×Q₁₆	1	4	yes	—
115	C₄₃⋊C₈	—	4	2	no	—
116	C₄₃⋊C₈	—	4	4	no	—
117	C₂×#1696	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	#462	1	2	yes	—
119	C₂²×C₄₃	#470	1	2	yes	—
120	C₂²×C₄₃	#472	1	2	yes	—
121	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
122	C₄×C₄₃	—	2	1	no	—
123	C₄×C₄₃	—	2	1	no	—
124	C₄×C₄₃	—	8	4	no	—
125	C₄×C₄₃	#462	1	2	yes	—
126	C₄×C₄₃	#470	1	2	yes	—
127	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
128	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
129	C₄×C₄₃	C₂×#137	1	2	yes	—
130	Dic₄₃	—	4	2	no	—
131	Dic₄₃	—	4	2	no	—
132	C₂×#500	D₄₃	1	1	yes	characteristic
133	C₂×#477	—	43	2	no	—
134	C₂×#446	—	43	1	no	—
135	#1696	—	43	8	no	—
136	C₂×#334	—	43	1	no	—
137	C₂×#333	—	43	2	no	—
138	C₂×C₄₃	#1696	1	4	yes	complemented
139	C₂×C₄₃	C₂×#462	1	1	yes	characteristic
140	C₂×C₄₃	C₂×#470	1	1	yes	characteristic
141	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
142	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
143	C₂×C₄×Q₈	—	43	1	no	—
144	#500	—	2	2	no	—
145	#500	C₂×D₄₃	1	4	yes	—
146	#477	—	43	8	no	—
147	#477	—	86	4	no	—
148	#446	—	43	4	no	—
149	#446	—	86	2	no	—
150	C₂×#107	—	2	1	no	—
151	C₂×#107	—	2	1	no	—
152	C₂×#107	—	86	2	no	—
153	C₂×C₄⋊C₈	—	43	2	no	—
154	C₂×C₄⋊C₈	—	43	1	no	—
155	C₂×C₄⋊C₈	—	86	1	no	—
156	C₂×#103	—	86	2	no	—
157	C₂×#97	—	86	1	no	—
158	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
159	#334	—	43	4	no	—
160	#333	—	43	8	no	—
161	C₄₃	C₂×#1696	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	2	2	no	—
163	C₂²×Q₈	—	43	1	no	—
164	C₂²×C₈	—	86	2	no	—
165	C₂²×C₈	—	86	1	no	—
166	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	C₂²×D₄₃	1	4	yes	—
172	C₄×Q₈	—	43	2	no	—
173	C₄×Q₈	—	43	2	no	—
174	C₄×Q₈	—	86	1	no	—
175	C₄×Q₈	—	86	1	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	43	1	no	—
179	C₂×C₄⋊C₄	—	86	1	no	—
180	C₂×C₄²	—	86	1	no	—
181	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
182	#107	—	2	4	no	—
183	#107	—	2	4	no	—
184	#107	—	86	8	no	—
185	C₄⋊C₈	—	43	2	no	—
186	C₄⋊C₈	—	43	4	no	—
187	C₄⋊C₈	—	86	4	no	—
188	C₄⋊C₈	—	86	2	no	—
189	C₄⋊C₈	—	86	1	no	—
190	#103	—	86	8	no	—
191	#97	—	86	4	no	—
192	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
193	C₄×C₈	C₂²×D₄₃	1	2	yes	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	8	4	no	—
197	C₂×Q₈	—	43	2	no	—
198	C₂×Q₈	—	86	1	no	—
199	C₂×Q₈	—	86	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	86	1	no	—
202	C₂²×C₄	—	86	1	no	—
203	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
204	C₂²×C₄	#10621900	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	86	4	no	—
208	C₂×C₈	—	86	2	no	—
209	C₂×C₈	—	172	2	no	—
210	C₂×C₈	—	172	4	no	—
211	C₂×C₈	#10621885	1	4	yes	—
212	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
213	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	43	2	no	—
219	C₄⋊C₄	—	86	1	no	—
220	C₄⋊C₄	—	86	2	no	—
221	C₄⋊C₄	—	86	2	no	—
222	C₄²	—	86	2	no	—
223	C₄²	—	86	2	no	—
224	C₄²	D₄×D₄₃	1	1	yes	characteristic
225	C₄²	#10621900	1	1	yes	characteristic
226	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
227	C₄²	C₂³×D₄₃	1	1	yes	characteristic
228	C₂³	#13063296	1	1	yes	characteristic
229	Q₈	—	8	4	no	—
230	Q₈	—	8	4	no	—
231	Q₈	—	86	4	no	—
232	Q₈	—	172	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	86	2	no	—
237	C₂×C₄	—	86	2	no	—
238	C₂×C₄	—	172	2	no	—
239	C₂×C₄	—	172	2	no	—
240	C₂×C₄	#13063296	1	1	yes	characteristic
241	C₂×C₄	#13063296	1	1	yes	characteristic
242	C₂×C₄	#13063296	1	1	yes	characteristic
243	C₂×C₄	C₂×#10621885	1	2	yes	—
244	C₂×C₄	#13063356	1	2	yes	—
245	C₂×C₄	#13063357	1	4	yes	—
246	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	172	2	no	—
251	C₈	—	172	4	no	—
252	C₈	C₂×#10621885	1	4	yes	—
253	C₂²	#14583106	1	2	yes	—
254	C₂²	#14583154	1	2	yes	—
255	C₂²	#14583166	1	2	yes	—
256	C₂²	C₂×#13063296	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	8	4	no	—
260	C₄	—	172	2	no	—
261	C₄	—	172	2	no	—
262	C₄	#14583106	1	2	yes	—
263	C₄	#14583154	1	2	yes	—
264	C₄	C₂²×#10621885	1	1	yes	characteristic
265	C₄	C₂×#13063356	1	1	yes	characteristic
266	C₄	C₂×#13063357	1	2	yes	—
267	C₂	#19134120	1	4	yes	direct product factor
268	C₂	C₂×#14583106	1	1	yes	characteristic
269	C₂	C₂×#14583154	1	1	yes	characteristic
270	C₂	C₂×#14583166	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle, prime radical	C₂³×C₄₃	#121
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	C₂×#1696
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #19134120	C₂	direct (N × H)
138. C₂×C₄₃	#1696	semidirect
161. C₄₃	C₂×#1696	semidirect
267. C₂	#19134120	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×#14583191	2
C₂×#14583029	2
#19134120	2
C₂×#14582650	2
C₂×#14582648	2
C₂×C₄₃×#500	2
C₂×#1696	43

Subgroups — C2×#19134120 #25079228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134120 #25079228