Subgroups — C₂×#19136832 #25251904

Order 11008.

634 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	C₂×#14583304	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583283	C₂	1	1	yes	maximal, characteristic
3	#19136832	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582704	C₂	1	2	yes	maximal, complemented
5	C₂×#14582701	C₂	1	2	yes	maximal
6	C₂×C₄₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
8	C₂²×#10621873	—	2	1	no	—
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063325	—	2	1	no	—
11	C₂×#13063313	—	2	1	no	—
12	#14583304	—	2	2	no	—
13	#14583304	C₂²	1	4	yes	complemented
14	#14583283	—	2	2	no	—
15	#14583283	C₂²	1	4	yes	—
16	C₂×#13063259	—	2	1	no	—
17	C₂×#13063259	C₂²	1	2	yes	—
18	#14582704	C₂²	1	8	yes	complemented
19	#14582701	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#346	C₂²	1	4	yes	complemented
23	C₂²×Dic₈₆	—	2	1	no	—
24	C₂²×Dic₈₆	—	4	1	no	—
25	C₂²×D₁₇₂	—	2	1	no	—
26	C₂×Dic₁₇₂	—	2	4	no	—
27	C₂×Dic₁₇₂	—	2	1	no	—
28	C₂×Dic₁₇₂	C₂³	1	2	yes	—
29	C₂×#10621873	—	2	4	no	—
30	C₂×#10621873	—	4	4	no	—
31	C₂×D₃₄₄	—	2	4	no	—
32	C₂×D₃₄₄	—	2	1	no	—
33	C₂×D₃₄₄	C₂³	1	2	yes	—
34	#13063325	—	2	4	no	—
35	#13063325	—	4	2	no	—
36	#13063313	—	2	4	no	—
37	#13063313	—	4	2	no	—
38	C₂×C₄⋊Dic₄₃	—	2	2	no	—
39	C₂×C₄⋊Dic₄₃	—	4	1	no	—
40	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
41	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
42	#13063259	—	2	2	no	—
43	#13063259	—	2	4	no	—
44	#13063259	C₂³	1	4	yes	—
45	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₁₆×C₄₃	—	2	4	no	—
49	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
50	C₄×C₈×C₄₃	D₄	1	2	yes	—
51	C₄×C₈×C₄₃	C₂³	1	2	yes	—
52	C₂³×D₄₃	—	4	1	no	—
53	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₄₃	—	4	1	no	—
57	C₂²×Dic₄₃	—	4	2	no	—
58	C₂×D₁₇₂	—	2	2	no	—
59	C₂×D₁₇₂	—	4	1	no	—
60	C₂×D₁₇₂	—	4	2	no	—
61	C₂×D₁₇₂	—	4	2	no	—
62	C₂×Dic₈₆	—	2	2	no	—
63	C₂×Dic₈₆	—	4	2	no	—
64	C₂×Dic₈₆	—	4	2	no	—
65	C₂×Dic₈₆	—	4	1	no	—
66	C₂×Dic₈₆	—	4	4	no	—
67	C₂×Dic₈₆	—	8	4	no	—
68	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
69	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₃	#41	1	4	yes	—
72	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
73	C₄²×C₄₃	D₈	1	2	yes	—
74	C₄²×C₄₃	C₂×D₄	1	2	yes	—
75	(C₂×D₄₃)⋊C₄	—	4	4	no	—
76	(C₂×D₄₃)⋊C₄	—	4	4	no	—
77	C₄⋊Dic₄₃	—	2	4	no	—
78	C₄⋊Dic₄₃	—	4	2	no	—
79	C₄⋊Dic₄₃	—	4	4	no	—
80	Dic₁₇₂	—	2	4	no	—
81	Dic₁₇₂	—	4	2	no	—
82	D₃₄₄	—	2	4	no	—
83	D₃₄₄	—	4	2	no	—
84	#10621873	—	4	8	no	—
85	C₁₆×C₄₃	—	2	4	no	—
86	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
87	C₂²×D₄₃	—	4	2	no	—
88	C₂²×D₄₃	—	8	2	no	—
89	C₂²×D₄₃	—	8	2	no	—
90	C₂²×D₄₃	—	8	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	#124	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₄₃	—	4	4	no	—
100	C₂×Dic₄₃	—	4	2	no	—
101	C₂×Dic₄₃	—	8	2	no	—
102	C₂×Dic₄₃	—	8	4	no	—
103	D₁₇₂	—	4	2	no	—
104	D₁₇₂	—	4	2	no	—
105	D₁₇₂	—	8	2	no	—
106	Dic₈₆	—	4	2	no	—
107	Dic₈₆	—	4	2	no	—
108	Dic₈₆	—	8	8	no	—
109	Dic₈₆	—	8	2	no	—
110	C₈×C₄₃	#124	1	2	yes	—
111	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
112	C₈×C₄₃	C₂×#41	1	2	yes	—
113	C₂×#2243	—	43	1	no	maximal, Sylow
114	C₂²×C₄₃	#486	1	2	yes	—
115	C₂²×C₄₃	#509	1	2	yes	—
116	C₂²×C₄₃	#510	1	2	yes	—
117	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
118	C₂×D₄₃	—	8	2	no	—
119	C₂×D₄₃	—	8	2	no	—
120	C₂×D₄₃	—	8	2	no	—
121	C₂×D₄₃	—	16	4	no	—
122	C₄×C₄₃	—	2	2	no	—
123	C₄×C₄₃	—	2	2	no	—
124	C₄×C₄₃	#486	1	2	yes	—
125	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
126	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
127	Dic₄₃	—	8	2	no	—
128	Dic₄₃	—	8	4	no	—
129	C₂×#500	—	43	1	no	—
130	C₂×#495	—	43	1	no	—
131	#2243	—	43	8	no	—
132	C₂×#358	—	43	2	no	—
133	C₂×#357	—	43	2	no	—
134	C₂×#346	D₄₃	1	1	yes	characteristic, complemented
135	C₂×C₄₃	#2243	1	4	yes	complemented
136	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
137	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
138	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
139	D₄₃	—	16	2	no	—
140	C₂²×Q₁₆	—	43	1	no	—
141	C₂²×SD₁₆	—	86	1	no	—
142	C₂²×D₈	—	43	1	no	—
143	C₂×C₄⋊Q₈	—	86	1	no	—
144	C₂×#124	—	86	1	no	—
145	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
146	#500	—	43	4	no	—
147	#500	—	86	2	no	—
148	#495	—	43	4	no	—
149	#495	—	86	2	no	—
150	C₂×#107	—	43	2	no	—
151	C₂×#107	—	86	1	no	—
152	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
153	#358	—	43	8	no	—
154	#357	—	43	8	no	—
155	#346	C₂×D₄₃	1	4	yes	complemented
156	C₄₃	C₂×#2243	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	86	1	no	—
158	C₂²×Q₈	—	172	1	no	—
159	C₂²×D₄	—	86	1	no	—
160	C₂×Q₁₆	—	43	2	no	—
161	C₂×Q₁₆	—	86	1	no	—
162	C₂×Q₁₆	—	86	4	no	—
163	C₂×SD₁₆	—	86	4	no	—
164	C₂×SD₁₆	—	172	4	no	—
165	C₂×D₈	—	43	2	no	—
166	C₂×D₈	—	86	1	no	—
167	C₂×D₈	—	86	4	no	—
168	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	86	4	no	—
171	C₄⋊Q₈	—	172	2	no	—
172	#124	—	86	4	no	—
173	#124	—	172	2	no	—
174	C₂×C₄⋊C₄	—	86	2	no	—
175	C₂×C₄⋊C₄	—	172	1	no	—
176	C₂×C₂²⋊C₄	—	172	1	no	—
177	C₂×C₂²⋊C₄	—	172	1	no	—
178	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
181	#107	—	43	4	no	—
182	#107	—	86	2	no	—
183	#107	—	86	4	no	—
184	C₄×C₈	D₁₇₂	1	2	yes	—
185	C₄×C₈	C₂²×D₄₃	1	2	yes	—
186	C₂⁴	—	172	1	no	—
187	C₂×Q₈	—	86	2	no	—
188	C₂×Q₈	—	344	4	no	—
189	C₂×Q₈	—	172	2	no	—
190	C₂×Q₈	—	172	2	no	—
191	C₂×Q₈	—	172	1	no	—
192	C₂×Q₈	—	172	4	no	—
193	C₂×D₄	—	86	2	no	—
194	C₂×D₄	—	172	2	no	—
195	C₂×D₄	—	172	1	no	—
196	C₂×D₄	—	172	2	no	—
197	C₂²×C₄	—	172	1	no	—
198	C₂²×C₄	—	172	2	no	—
199	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
200	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
202	Q₁₆	—	86	4	no	—
203	Q₁₆	—	172	2	no	—
204	SD₁₆	—	172	8	no	—
205	D₈	—	86	4	no	—
206	D₈	—	172	2	no	—
207	C₂×C₈	#10621873	1	4	yes	—
208	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
209	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
210	C₂×C₈	#10621898	1	4	yes	—
211	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
212	C₄⋊C₄	—	86	4	no	—
213	C₄⋊C₄	—	172	2	no	—
214	C₄⋊C₄	—	172	4	no	—
215	C₂²⋊C₄	—	172	4	no	—
216	C₂²⋊C₄	—	172	4	no	—
217	C₄²	D₃₄₄	1	2	yes	—
218	C₄²	C₂×D₁₇₂	1	2	yes	—
219	C₁₆	—	2	4	no	—
220	C₂³	—	344	2	no	—
221	C₂³	—	344	2	no	—
222	C₂³	—	344	2	no	—
223	C₂³	—	172	2	no	—
224	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
225	Q₈	—	344	8	no	—
226	Q₈	—	344	2	no	—
227	Q₈	—	172	2	no	—
228	Q₈	—	172	2	no	—
229	D₄	—	344	2	no	—
230	D₄	—	172	2	no	—
231	D₄	—	172	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	344	4	no	—
235	C₂×C₄	—	344	2	no	—
236	C₂×C₄	—	172	2	no	—
237	C₂×C₄	—	172	4	no	—
238	C₂×C₄	#13063313	1	2	yes	—
239	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
240	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
241	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
242	C₂×C₄	C₂×#10621873	1	2	yes	—
243	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
244	C₈	#13063313	1	2	yes	—
245	C₈	C₂×#10621873	1	4	yes	—
246	C₈	C₂×#10621898	1	2	yes	—
247	C₂²	—	344	2	no	—
248	C₂²	—	344	2	no	—
249	C₂²	—	344	2	no	—
250	C₂²	—	688	4	no	—
251	C₂²	#14583246	1	2	yes	—
252	C₂²	#14583337	1	2	yes	—
253	C₂²	#14583342	1	2	yes	—
254	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	344	2	no	—
258	C₄	—	344	4	no	—
259	C₄	#14583246	1	2	yes	—
260	C₄	C₂×#13063313	1	1	yes	characteristic
261	C₄	C₂²×#10621873	1	1	yes	characteristic
262	C₂	—	688	2	no	—
263	C₂	#19136832	1	4	yes	direct product factor
264	C₂	C₂×#14583246	1	1	yes	characteristic
265	C₂	C₂×#14583337	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×#346	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
hypercenter	C₂×#346	D₄₃
hypocenter	C₄₃	C₂×#2243
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#14583304	C₂	semidirect
3. #19136832	C₂	direct (N × H)
4. C₂×#14582704	C₂	semidirect
6. C₂×C₄₃×#346	C₂	semidirect
13. #14583304	C₂²	semidirect
18. #14582704	C₂²	semidirect
22. C₄₃×#346	C₂²	semidirect
134. C₂×#346	D₄₃	semidirect
135. C₂×C₄₃	#2243	semidirect
155. #346	C₂×D₄₃	semidirect
156. C₄₃	C₂×#2243	semidirect
263. C₂	#19136832	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×#14583304	2
C₂×#14583283	2
#19136832	2
C₂×#14582704	2
C₂×#14582701	2
C₂×C₄₃×#346	2
C₂×#2243	43

Subgroups — C2×#19136832 #25251904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136832 #25251904