Subgroups — C₂×#19133955 #25078925

Order 11008.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583276	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583099	C₂	1	2	yes	maximal, complemented
3	#19133955	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582652	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582642	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10621884	—	2	2	no	—
8	C₂²×D₃₄₄	—	2	1	no	—
9	C₂²×D₃₄₄	—	2	1	no	—
10	C₂×#13063322	C₂²	1	2	yes	—
11	#14583276	—	2	2	no	—
12	#14583276	C₂²	1	4	yes	—
13	#14583099	—	2	4	no	—
14	#14583099	C₂²	1	8	yes	complemented
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13063245	—	2	2	no	—
17	#14582652	C₂²	1	4	yes	complemented
18	#14582642	C₂²	1	8	yes	complemented
19	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₄₃	—	2	2	no	—
21	C₈×Q₈×C₄₃	C₂²	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₃	—	2	1	no	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×D₁₇₂	—	2	2	no	—
27	C₂²×D₁₇₂	—	2	2	no	—
28	C₂²×D₁₇₂	—	2	2	no	—
29	C₂×#10621884	—	2	8	no	—
30	C₂×#10621884	—	4	8	no	—
31	C₂×D₃₄₄	—	2	4	no	—
32	C₂×D₃₄₄	—	2	4	no	—
33	C₂×D₃₄₄	—	4	4	no	—
34	C₂×D₃₄₄	—	4	4	no	—
35	C₂²×C₄₃⋊C₈	—	2	2	no	—
36	#13063322	—	2	2	no	—
37	#13063322	—	2	2	no	—
38	#13063322	—	2	2	no	—
39	#13063322	C₂³	1	4	yes	—
40	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
42	#13063245	—	2	8	no	—
43	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	—	2	1	no	—
45	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₄×Q₈×C₄₃	—	2	1	no	—
48	C₄×Q₈×C₄₃	—	2	1	no	—
49	C₄×Q₈×C₄₃	D₄	1	2	yes	—
50	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	—	2	1	no	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	—	2	4	no	—
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
58	C₄×C₈×C₄₃	—	2	4	no	—
59	C₄×C₈×C₄₃	D₄	1	2	yes	—
60	C₄×C₈×C₄₃	C₂³	1	2	yes	—
61	C₂³×D₄₃	—	4	2	no	—
62	C₂×Q₈×C₄₃	—	2	1	no	—
63	C₂×Q₈×C₄₃	—	2	2	no	—
64	C₂×Q₈×C₄₃	D₈	1	4	yes	complemented
65	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	—	2	1	no	—
67	C₂²×C₄×C₄₃	—	2	1	no	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
71	C₂×D₁₇₂	—	2	4	no	—
72	C₂×D₁₇₂	—	2	4	no	—
73	C₂×D₁₇₂	—	2	4	no	—
74	C₂×D₁₇₂	—	4	2	no	—
75	C₂×D₁₇₂	—	4	2	no	—
76	C₂×D₁₇₂	—	4	2	no	—
77	C₂×D₁₇₂	—	4	4	no	—
78	C₂×D₁₇₂	—	4	4	no	—
79	C₂×D₁₇₂	—	4	4	no	—
80	C₂×D₁₇₂	—	4	4	no	—
81	C₂×D₁₇₂	—	8	4	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	4	2	no	—
85	C₂×C₈×C₄₃	SD₁₆	1	4	yes	complemented
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₄⋊C₄×C₄₃	—	2	1	no	—
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	2	2	no	—
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₃	#41	1	1	yes	characteristic
97	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
98	#10621884	—	4	16	no	—
99	C₂×C₄₃⋊C₈	—	2	4	no	—
100	C₂×C₄₃⋊C₈	—	4	4	no	—
101	D₃₄₄	—	4	8	no	—
102	D₃₄₄	—	4	8	no	—
103	C₂³×C₄₃	#121	1	1	yes	characteristic
104	C₂²×D₄₃	—	4	4	no	—
105	C₂²×D₄₃	—	8	4	no	—
106	C₂²×D₄₃	—	8	4	no	—
107	C₂²×D₄₃	—	8	4	no	—
108	Q₈×C₄₃	—	2	2	no	—
109	Q₈×C₄₃	—	2	4	no	—
110	Q₈×C₄₃	C₂×D₈	1	4	yes	complemented
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₄₃	#135	1	2	yes	—
122	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
126	D₁₇₂	—	4	4	no	—
127	D₁₇₂	—	4	4	no	—
128	D₁₇₂	—	4	4	no	—
129	D₁₇₂	—	4	4	no	—
130	D₁₇₂	—	8	4	no	—
131	D₁₇₂	—	8	4	no	—
132	D₁₇₂	—	8	4	no	—
133	D₁₇₂	—	8	4	no	—
134	C₈×C₄₃	—	2	2	no	—
135	C₈×C₄₃	—	4	2	no	—
136	C₈×C₄₃	C₂×SD₁₆	1	4	yes	complemented
137	C₄₃⋊C₈	—	4	4	no	—
138	C₂×#1668	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	#460	1	2	yes	—
140	C₂²×C₄₃	#466	1	2	yes	—
141	C₂²×C₄₃	#472	1	2	yes	—
142	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
143	C₂×D₄₃	—	8	4	no	—
144	C₂×D₄₃	—	8	4	no	—
145	C₂×D₄₃	—	8	4	no	—
146	C₂×D₄₃	—	16	8	no	—
147	C₄×C₄₃	—	2	4	no	—
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	—	2	1	no	—
150	C₄×C₄₃	—	4	2	no	—
151	C₄×C₄₃	#460	1	2	yes	—
152	C₄×C₄₃	#466	1	2	yes	—
153	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
155	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
156	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×#493	—	43	1	no	—
158	C₂×#460	—	43	2	no	—
159	C₂×C₈×Q₈	D₄₃	1	1	yes	characteristic, complemented
160	#1668	—	43	8	no	—
161	C₂×#335	—	43	1	no	—
162	C₂×#331	—	43	2	no	—
163	C₂×C₄₃	#1668	1	4	yes	complemented
164	C₂×C₄₃	C₂×#460	1	1	yes	characteristic
165	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
166	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
167	D₄₃	—	16	4	no	—
168	C₂²×SD₁₆	—	86	2	no	—
169	C₂²×D₈	—	86	1	no	—
170	C₂²×D₈	—	86	1	no	—
171	C₂×#127	—	43	2	no	—
172	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
173	#493	—	43	4	no	—
174	#493	—	86	2	no	—
175	#460	—	43	8	no	—
176	#460	—	86	4	no	—
177	C₈×Q₈	—	2	2	no	—
178	C₈×Q₈	C₂×D₄₃	1	4	yes	complemented
179	C₂×C₄⋊C₈	—	2	1	no	—
180	C₂×C₄⋊C₈	—	43	2	no	—
181	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
182	C₂×#102	—	86	2	no	—
183	C₂×C₄×C₈	—	2	1	no	—
184	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
185	#335	—	43	4	no	—
186	#331	—	43	8	no	—
187	C₄₃	C₂×#1668	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
189	C₂²×D₄	—	86	2	no	—
190	C₂²×D₄	—	86	2	no	—
191	C₂²×D₄	—	86	2	no	—
192	C₂×SD₁₆	—	86	8	no	—
193	C₂×SD₁₆	—	172	8	no	—
194	C₂×D₈	—	86	4	no	—
195	C₂×D₈	—	86	4	no	—
196	C₂×D₈	—	172	4	no	—
197	C₂×D₈	—	172	4	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	86	2	no	—
200	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
202	#127	—	43	4	no	—
203	#127	—	86	2	no	—
204	#127	—	86	2	no	—
205	#127	—	86	2	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	D₁₇₂	1	2	yes	—
209	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	43	4	no	—
217	C₄⋊C₈	—	86	2	no	—
218	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
219	#102	—	86	8	no	—
220	C₄×C₈	—	2	4	no	—
221	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
222	C₄×C₈	C₂²×D₄₃	1	2	yes	—
223	C₂⁴	—	172	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	D₃₄₄	1	4	yes	complemented
227	C₂×Q₈	C₂×D₁₇₂	1	2	yes	—
228	C₂×D₄	—	86	4	no	—
229	C₂×D₄	—	86	4	no	—
230	C₂×D₄	—	86	4	no	—
231	C₂×D₄	—	344	4	no	—
232	C₂×D₄	—	172	2	no	—
233	C₂×D₄	—	172	2	no	—
234	C₂×D₄	—	172	2	no	—
235	C₂×D₄	—	172	4	no	—
236	C₂×D₄	—	172	4	no	—
237	C₂×D₄	—	172	4	no	—
238	C₂×D₄	—	172	4	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
242	C₂²×C₄	#10621898	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
244	SD₁₆	—	172	16	no	—
245	D₈	—	172	8	no	—
246	D₈	—	172	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	86	4	no	—
251	C₂×C₈	—	172	4	no	—
252	C₂×C₈	#10621884	1	4	yes	complemented
253	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	C₂×D₁₇₂	1	2	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
262	C₄²	#10621898	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₄²	C₂³×D₄₃	1	1	yes	characteristic
265	C₂³	—	344	4	no	—
266	C₂³	—	344	4	no	—
267	C₂³	—	344	4	no	—
268	C₂³	—	172	4	no	—
269	C₂³	#13063299	1	1	yes	characteristic
270	Q₈	—	2	4	no	—
271	Q₈	—	2	2	no	—
272	Q₈	C₂×D₃₄₄	1	4	yes	complemented
273	D₄	—	344	4	no	—
274	D₄	—	344	4	no	—
275	D₄	—	344	4	no	—
276	D₄	—	344	4	no	—
277	D₄	—	172	4	no	—
278	D₄	—	172	4	no	—
279	D₄	—	172	4	no	—
280	D₄	—	172	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	#13063299	1	1	yes	characteristic
287	C₂×C₄	#13063299	1	1	yes	characteristic
288	C₂×C₄	#13063299	1	1	yes	characteristic
289	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
290	C₂×C₄	C₂×#10621884	1	2	yes	—
291	C₂×C₄	#13063347	1	2	yes	—
292	C₂×C₄	#13063350	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
295	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₈	—	172	4	no	—
299	C₈	C₂×#10621884	1	4	yes	complemented
300	C₂²	—	344	4	no	—
301	C₂²	—	344	4	no	—
302	C₂²	—	344	4	no	—
303	C₂²	—	688	8	no	—
304	C₂²	#14583099	1	2	yes	—
305	C₂²	#14583135	1	2	yes	—
306	C₂²	#14583169	1	2	yes	—
307	C₂²	C₂×#13063299	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	#14583099	1	2	yes	—
313	C₄	#14583135	1	2	yes	—
314	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
315	C₄	C₂²×#10621884	1	1	yes	characteristic
316	C₄	C₂×#13063347	1	1	yes	characteristic
317	C₄	C₂×#13063350	1	1	yes	characteristic
318	C₂	—	688	4	no	—
319	C₂	#19133955	1	4	yes	direct product factor
320	C₂	C₂×#14583099	1	1	yes	characteristic
321	C₂	C₂×#14583135	1	1	yes	characteristic
322	C₂	C₂×#14583169	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063299
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂×#14583276	C₂
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	C₂×#1668
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14583099	C₂	semidirect
3. #19133955	C₂	direct (N × H)
4. C₂×#14582652	C₂	semidirect
5. C₂×#14582642	C₂	semidirect
6. C₂×C₈×Q₈×C₄₃	C₂	semidirect
14. #14583099	C₂²	semidirect
17. #14582652	C₂²	semidirect
18. #14582642	C₂²	semidirect
21. C₈×Q₈×C₄₃	C₂²	semidirect
64. C₂×Q₈×C₄₃	D₈	semidirect
85. C₂×C₈×C₄₃	SD₁₆	semidirect
110. Q₈×C₄₃	C₂×D₈	semidirect
136. C₈×C₄₃	C₂×SD₁₆	semidirect
159. C₂×C₈×Q₈	D₄₃	semidirect
163. C₂×C₄₃	#1668	semidirect
178. C₈×Q₈	C₂×D₄₃	semidirect
187. C₄₃	C₂×#1668	semidirect
226. C₂×Q₈	D₃₄₄	semidirect
252. C₂×C₈	#10621884	semidirect
272. Q₈	C₂×D₃₄₄	semidirect
299. C₈	C₂×#10621884	semidirect
319. C₂	#19133955	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×#14583276	2
C₂×#14583099	2
#19133955	2
C₂×#14582652	2
C₂×#14582642	2
C₂×C₈×Q₈×C₄₃	2
C₂×#1668	43

Subgroups — C2×#19133955 #25078925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19133955 #25078925