Subgroups — C₂×#19136676 #25250983

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583339	C₂	1	2	yes	maximal
2	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136676	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582720	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582714	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₃×#443	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₂	—	2	1	no	—
8	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
9	C₂×#13063325	—	2	1	no	—
10	#14583339	—	2	4	no	—
11	#14583339	C₂²	1	8	yes	—
12	C₂×#13063271	—	2	2	no	—
13	C₂×#13063269	—	2	2	no	—
14	C₂×#13063265	C₂²	1	2	yes	—
15	#14583279	—	2	2	no	—
16	#14583279	C₂²	1	4	yes	complemented
17	#14582720	C₂²	1	4	yes	complemented
18	#14582714	C₂²	1	8	yes	complemented
19	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#131×C₄₃	—	2	1	no	—
21	C₄₃×#443	—	2	2	no	—
22	C₄₃×#443	C₂²	1	4	yes	—
23	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#104×C₄₃	—	2	1	no	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₆	—	2	2	no	—
27	C₂×Dic₁₇₂	—	2	2	no	—
28	C₂×Dic₁₇₂	—	2	2	no	—
29	C₂×Dic₁₇₂	—	2	2	no	—
30	C₂×Dic₁₇₂	—	2	2	no	—
31	C₂×Dic₁₇₂	—	2	4	no	—
32	C₂×Dic₁₇₂	—	2	2	no	—
33	C₂×Dic₁₇₂	—	4	1	no	—
34	C₂×Dic₁₇₂	—	4	1	no	—
35	C₂×Dic₁₇₂	—	4	2	no	—
36	C₂×Dic₁₇₂	C₂³	1	4	yes	—
37	#13063325	—	2	4	no	—
38	#13063325	—	4	2	no	—
39	C₂×C₄⋊Dic₄₃	—	4	2	no	—
40	#13063271	—	2	8	no	—
41	#13063269	—	2	8	no	—
42	#13063265	—	2	2	no	—
43	#13063265	C₂³	1	4	yes	—
44	C₂×C₄₃⋊C₁₆	—	2	2	no	—
45	C₂×#41×C₄₃	—	2	1	no	—
46	#135×C₄₃	—	2	1	no	—
47	#135×C₄₃	—	2	1	no	—
48	#135×C₄₃	D₄	1	2	yes	—
49	#135×C₄₃	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₄₃	D₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₄₃	—	2	1	no	—
52	C₂×D₈×C₄₃	D₄	1	1	yes	characteristic
53	#131×C₄₃	—	2	4	no	—
54	#131×C₄₃	—	4	2	no	—
55	C₂×M₁₆×C₄₃	—	2	1	no	—
56	C₂×M₁₆×C₄₃	—	4	1	no	—
57	C₂²×C₈×C₄₃	—	4	1	no	—
58	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
61	#108×C₄₃	—	2	1	no	—
62	#108×C₄₃	C₂³	1	2	yes	—
63	#104×C₄₃	—	2	4	no	—
64	C₄×C₈×C₄₃	D₄	1	2	yes	—
65	C₄×C₈×C₄₃	C₂³	1	2	yes	—
66	#41×C₄₃	—	2	2	no	—
67	#41×C₄₃	—	2	2	no	—
68	#41×C₄₃	—	4	1	no	—
69	#41×C₄₃	—	4	1	no	—
70	C₂×Q₈×C₄₃	—	2	1	no	—
71	C₂×D₄×C₄₃	—	2	1	no	—
72	C₂×D₄×C₄₃	—	4	1	no	—
73	C₂²×C₄×C₄₃	—	2	1	no	—
74	C₂²×C₄×C₄₃	—	4	1	no	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₃	—	4	2	no	—
77	C₂×Dic₈₆	—	2	4	no	—
78	C₂×Dic₈₆	—	4	2	no	—
79	C₂×Dic₈₆	—	4	2	no	—
80	C₂×Dic₈₆	—	4	2	no	—
81	C₂×Dic₈₆	—	4	2	no	—
82	C₂×Dic₈₆	—	4	2	no	—
83	Q₁₆×C₄₃	—	2	1	no	—
84	Q₁₆×C₄₃	C₂×D₄	1	2	yes	—
85	SD₁₆×C₄₃	—	2	2	no	—
86	SD₁₆×C₄₃	—	2	2	no	—
87	D₈×C₄₃	—	2	1	no	—
88	D₈×C₄₃	C₂×D₄	1	2	yes	—
89	M₁₆×C₄₃	—	2	2	no	—
90	M₁₆×C₄₃	—	4	1	no	—
91	M₁₆×C₄₃	—	4	4	no	—
92	C₂×C₈×C₄₃	—	4	1	no	—
93	C₂×C₈×C₄₃	—	4	4	no	—
94	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
95	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
96	C₂×C₈×C₄₃	D₈	1	2	yes	—
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₄²×C₄₃	—	2	1	no	—
105	C₄²×C₄₃	C₂×D₄	1	2	yes	—
106	C₄⋊Dic₄₃	—	4	8	no	—
107	Dic₁₇₂	—	2	4	no	—
108	Dic₁₇₂	—	2	4	no	—
109	Dic₁₇₂	—	4	2	no	—
110	Dic₁₇₂	—	4	2	no	—
111	Dic₁₇₂	—	4	4	no	—
112	Dic₁₇₂	—	4	4	no	—
113	C₄₃⋊C₁₆	—	2	4	no	—
114	C₂³×C₄₃	—	4	1	no	—
115	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	Q₈×C₄₃	—	2	2	no	—
117	Q₈×C₄₃	—	4	1	no	—
118	D₄×C₄₃	—	2	2	no	—
119	D₄×C₄₃	—	4	2	no	—
120	D₄×C₄₃	—	4	2	no	—
121	D₄×C₄₃	—	4	1	no	—
122	C₂×C₄×C₄₃	—	2	4	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	4	1	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₄₃	—	4	4	no	—
134	C₂×Dic₄₃	—	8	2	no	—
135	C₂×Dic₄₃	—	8	2	no	—
136	Dic₈₆	—	4	4	no	—
137	Dic₈₆	—	4	4	no	—
138	Dic₈₆	—	8	2	no	—
139	Dic₈₆	—	8	2	no	—
140	C₈×C₄₃	—	4	2	no	—
141	C₈×C₄₃	#121	1	1	yes	characteristic
142	C₈×C₄₃	#121	1	1	yes	characteristic
143	C₈×C₄₃	C₂×D₈	1	2	yes	—
144	C₈×C₄₃	C₂×D₈	1	2	yes	—
145	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₂×#2214	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	2	no	—
149	C₂²×C₄₃	—	4	2	no	—
150	C₂²×C₄₃	—	8	2	no	—
151	C₂²×C₄₃	#459	1	1	yes	characteristic
152	C₂²×C₄₃	#459	1	1	yes	characteristic
153	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₄×C₄₃	—	2	4	no	—
155	C₄×C₄₃	—	4	2	no	—
156	C₄×C₄₃	#459	1	1	yes	characteristic
157	C₄×C₄₃	#459	1	1	yes	characteristic
158	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
159	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
160	Dic₄₃	—	8	4	no	—
161	C₂×#510	—	43	2	no	—
162	C₂×#494	—	43	1	no	—
163	C₂×#443	D₄₃	1	1	yes	characteristic
164	#2214	—	43	8	no	—
165	C₂×#364	—	43	1	no	—
166	C₂×#362	—	43	2	no	—
167	C₂×C₄₃	—	2	1	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	—	8	2	no	—
170	C₂×C₄₃	#2214	1	2	yes	complemented
171	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
172	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
173	C₂²×Q₁₆	—	43	2	no	—
174	C₂²×Q₁₆	—	86	1	no	—
175	C₂×#131	—	2	1	no	—
176	C₂×C₄⋊Q₈	—	86	1	no	—
177	#510	—	43	8	no	—
178	#510	—	86	4	no	—
179	C₂×Q₃₂	—	86	2	no	—
180	C₂×SD₃₂	—	86	2	no	—
181	C₂×M₃₂	—	43	2	no	—
182	#494	—	43	4	no	—
183	#494	—	86	2	no	—
184	#443	—	2	2	no	—
185	#443	C₂×D₄₃	1	4	yes	—
186	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#104	—	2	1	no	—
188	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
189	#364	—	43	4	no	—
190	#362	—	43	8	no	—
191	C₄₃	C₂×#2214	1	1	yes	characteristic, complemented, Sylow
192	C₂×#41	—	2	1	no	—
193	C₂²×Q₈	—	86	2	no	—
194	#135	—	2	1	no	—
195	#135	—	2	1	no	—
196	#135	D₁₇₂	1	2	yes	—
197	#135	C₂²×D₄₃	1	2	yes	—
198	C₂×Q₁₆	—	43	4	no	—
199	C₂×Q₁₆	—	86	2	no	—
200	C₂×Q₁₆	—	86	2	no	—
201	C₂×Q₁₆	—	86	2	no	—
202	C₂×Q₁₆	—	86	2	no	—
203	C₂×Q₁₆	—	86	2	no	—
204	C₂×Q₁₆	—	86	4	no	—
205	C₂×Q₁₆	—	172	1	no	—
206	C₂×Q₁₆	—	172	1	no	—
207	C₂×Q₁₆	—	172	2	no	—
208	C₂×Q₁₆	D₁₇₂	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	1	no	—
210	C₂×D₈	D₁₇₂	1	1	yes	characteristic
211	#131	—	2	4	no	—
212	#131	—	4	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	4	1	no	—
215	C₂²×C₈	—	4	1	no	—
216	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	86	4	no	—
219	C₄⋊Q₈	—	172	2	no	—
220	C₂×C₄⋊C₄	—	172	2	no	—
221	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
222	Q₃₂	—	86	8	no	—
223	SD₃₂	—	86	8	no	—
224	M₃₂	—	43	4	no	—
225	M₃₂	—	86	2	no	—
226	C₂×C₁₆	—	86	2	no	—
227	#108	—	2	1	no	—
228	#108	C₂²×D₄₃	1	2	yes	—
229	#104	—	2	4	no	—
230	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
231	C₄×C₈	C₂²×D₄₃	1	2	yes	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	86	4	no	—
238	C₂×Q₈	—	172	2	no	—
239	C₂×Q₈	—	172	2	no	—
240	C₂×Q₈	—	172	2	no	—
241	C₂×Q₈	—	172	2	no	—
242	C₂×Q₈	—	172	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	172	2	no	—
248	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	86	4	no	—
251	Q₁₆	—	86	4	no	—
252	Q₁₆	—	172	2	no	—
253	Q₁₆	—	172	2	no	—
254	Q₁₆	—	172	4	no	—
255	Q₁₆	—	172	4	no	—
256	Q₁₆	C₂×D₁₇₂	1	2	yes	—
257	SD₁₆	—	2	2	no	—
258	SD₁₆	—	2	2	no	—
259	D₈	—	2	1	no	—
260	D₈	C₂×D₁₇₂	1	2	yes	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	#10621882	1	1	yes	characteristic
267	C₂×C₈	#10621882	1	1	yes	characteristic
268	C₂×C₈	#10621882	1	2	yes	—
269	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
270	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
271	C₂×C₈	#10621898	1	1	yes	characteristic
272	C₂×C₈	#10621898	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
275	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
276	C₄⋊C₄	—	172	8	no	—
277	C₄²	—	2	1	no	—
278	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
279	C₁₆	—	86	4	no	—
280	C₂³	—	4	1	no	—
281	C₂³	#13063350	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	344	2	no	—
285	Q₈	—	344	2	no	—
286	Q₈	—	172	4	no	—
287	Q₈	—	172	4	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	344	2	no	—
298	C₂×C₄	—	344	2	no	—
299	C₂×C₄	—	172	4	no	—
300	C₂×C₄	#13063299	1	1	yes	characteristic
301	C₂×C₄	#13063299	1	1	yes	characteristic
302	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
304	C₂×C₄	#13063350	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	#13063299	1	1	yes	characteristic
308	C₈	#13063299	1	1	yes	characteristic
309	C₈	C₂×#10621882	1	2	yes	—
310	C₈	C₂×#10621882	1	2	yes	—
311	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
312	C₈	C₂×#10621898	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	#14583094	1	1	yes	characteristic
317	C₂²	#14583094	1	1	yes	characteristic
318	C₂²	C₂×#13063350	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	344	4	no	—
322	C₄	#14583094	1	1	yes	characteristic
323	C₄	#14583094	1	1	yes	characteristic
324	C₄	C₂×#13063299	1	1	yes	characteristic
325	C₄	C₂²×#10621882	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	8	2	no	—
329	C₂	#19136676	1	2	yes	direct product factor
330	C₂	C₂×#14583094	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583094
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#443	C₂
socle	C₂²×C₄₃	#459
prime radical	C₂×#135×C₄₃	C₂²
hypercenter	C₂×#443	D₄₃
hypocenter	C₄₃	C₂×#2214
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. C₂×#14583279	C₂	semidirect
3. #19136676	C₂	direct (N × H)
4. C₂×#14582720	C₂	semidirect
5. C₂×#14582714	C₂	semidirect
16. #14583279	C₂²	semidirect
17. #14582720	C₂²	semidirect
18. #14582714	C₂²	semidirect
170. C₂×C₄₃	#2214	semidirect
191. C₄₃	C₂×#2214	semidirect
329. C₂	#19136676	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×#14583339	2
C₂×#14583279	2
#19136676	2
C₂×#14582720	2
C₂×#14582714	2
C₂×C₄₃×#443	2
C₂×#2214	43

Subgroups — C2×#19136676 #25250983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136676 #25250983