Subgroups — C₂×#19136459 #25249886

Order 11008.

734 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063269	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136459	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582724	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582702	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582700	C₂	1	1	yes	maximal, characteristic
7	C₂×#14582675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	—	2	1	no	—
11	C₂×#13063269	C₄	1	8	yes	complemented
12	C₂×#13063269	C₂²	1	4	yes	—
13	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₇₂	—	2	2	no	—
16	C₄×Dic₁₇₂	C₂²	1	4	yes	complemented
17	C₂×#13063259	C₂²	1	1	yes	characteristic
18	C₂×#13063250	—	2	1	no	—
19	#14582724	C₂²	1	4	yes	complemented
20	#14582702	C₂²	1	4	yes	complemented
21	#14582700	C₂²	1	4	yes	—
22	#14582675	C₂²	1	4	yes	complemented
23	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₃	—	2	1	no	—
25	C₄×D₈×C₄₃	—	2	2	no	—
26	C₄×D₈×C₄₃	C₂²	1	4	yes	—
27	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₃	—	2	1	no	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂²×Dic₈₆	—	2	1	no	—
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	2	no	—
33	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	complemented
34	C₂×Dic₁₇₂	C₂³	1	2	yes	—
35	C₄×Dic₈₆	—	2	4	no	—
36	C₄×Dic₈₆	—	4	2	no	—
37	C₂×C₄⋊Dic₄₃	—	2	1	no	—
38	C₂×Dic₄₃⋊C₄	—	4	1	no	—
39	C₂×C₄×Dic₄₃	—	4	1	no	—
40	#13063269	C₂×C₄	1	16	yes	complemented
41	C₂×C₄₃⋊C₁₆	—	2	2	no	—
42	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
45	#13063259	—	2	1	no	—
46	#13063259	C₂³	1	2	yes	—
47	#13063250	—	2	4	no	—
48	C₂²×D₄×C₄₃	—	2	1	no	—
49	C₂³×C₄×C₄₃	—	4	1	no	—
50	C₂×D₈×C₄₃	—	2	1	no	—
51	C₂×D₈×C₄₃	—	2	2	no	—
52	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	—
53	C₂×D₈×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₄₃	—	2	4	no	—
57	C₄×D₄×C₄₃	—	4	2	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
60	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
61	#107×C₄₃	—	2	1	no	—
62	#107×C₄₃	C₂³	1	2	yes	—
63	#102×C₄₃	—	2	4	no	—
64	C₄×C₈×C₄₃	D₄	1	2	yes	—
65	C₄×C₈×C₄₃	C₂³	1	2	yes	—
66	C₂⁴×C₄₃	—	4	1	no	—
67	C₂×D₄×C₄₃	—	2	4	no	—
68	C₂×D₄×C₄₃	—	2	2	no	—
69	C₂×D₄×C₄₃	—	4	1	no	—
70	C₂×D₄×C₄₃	—	4	2	no	—
71	C₂²×C₄×C₄₃	—	4	4	no	—
72	C₂²×C₄×C₄₃	—	4	1	no	—
73	C₂²×C₄×C₄₃	—	8	4	no	—
74	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₃	—	4	1	no	—
78	C₂²×Dic₄₃	—	4	1	no	—
79	C₂×Dic₈₆	—	2	4	no	—
80	C₂×Dic₈₆	—	2	2	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	C₂×Dic₈₆	—	4	2	no	—
83	D₈×C₄₃	—	2	4	no	—
84	D₈×C₄₃	—	2	2	no	—
85	D₈×C₄₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	#41	1	2	yes	—
91	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	C₂²⋊C₄×C₄₃	—	4	4	no	—
95	C₄²×C₄₃	D₈	1	2	yes	—
96	C₄²×C₄₃	C₂×D₄	1	2	yes	—
97	C₄⋊Dic₄₃	—	2	2	no	—
98	C₄⋊Dic₄₃	—	4	1	no	—
99	Dic₄₃⋊C₄	—	4	4	no	—
100	C₄×Dic₄₃	—	4	4	no	—
101	Dic₁₇₂	—	2	4	no	—
102	Dic₁₇₂	—	2	2	no	—
103	Dic₁₇₂	C₂²×C₄	1	4	yes	complemented
104	C₄₃⋊C₁₆	—	2	2	no	—
105	C₄₃⋊C₁₆	C₂²×C₄	1	4	yes	complemented
106	C₂³×C₄₃	—	4	2	no	—
107	C₂³×C₄₃	—	4	4	no	—
108	C₂³×C₄₃	—	8	2	no	—
109	C₂³×C₄₃	—	8	2	no	—
110	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
111	D₄×C₄₃	—	2	4	no	—
112	D₄×C₄₃	—	4	2	no	—
113	D₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	8	8	no	—
118	C₂×C₄×C₄₃	—	8	2	no	—
119	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#135	1	2	yes	—
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	4	4	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	8	2	no	—
129	Dic₈₆	—	2	4	no	—
130	Dic₈₆	—	4	2	no	—
131	Dic₈₆	—	4	4	no	—
132	C₈×C₄₃	—	2	2	no	—
133	C₈×C₄₃	C₄×D₄	1	2	yes	—
134	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
135	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
136	C₂×#2175	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	—	4	4	no	—
138	C₂²×C₄₃	—	8	2	no	—
139	C₂²×C₄₃	—	8	2	no	—
140	C₂²×C₄₃	—	8	8	no	—
141	C₂²×C₄₃	C₄×D₈	1	2	yes	—
142	C₂²×C₄₃	#509	1	2	yes	—
143	C₂²×C₄₃	#510	1	2	yes	—
144	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₄₃	—	2	2	no	—
146	C₄×C₄₃	—	2	2	no	—
147	C₄×C₄₃	—	8	2	no	—
148	C₄×C₄₃	C₄×D₈	1	2	yes	—
149	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
151	Dic₄₃	—	4	4	no	—
152	Dic₄₃	—	8	2	no	—
153	C₂²×SD₃₂	—	43	1	no	—
154	C₂×C₄×Q₁₆	—	43	1	no	—
155	C₂×C₄×D₈	D₄₃	1	1	yes	characteristic
156	#2175	—	43	8	no	—
157	C₂×#366	—	43	1	no	—
158	C₂×#358	—	43	1	no	—
159	C₂×#357	—	43	1	no	—
160	C₂×#346	—	43	1	no	—
161	C₂×C₄₃	—	8	4	no	—
162	C₂×C₄₃	#2175	1	4	yes	complemented
163	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
164	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
165	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
166	C₂²×Q₁₆	—	43	1	no	—
167	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
168	C₂×C₄×Q₈	—	86	1	no	—
169	C₂×C₄×D₄	—	2	1	no	—
170	C₂×SD₃₂	—	43	4	no	—
171	C₂×SD₃₂	—	43	8	no	—
172	C₂²×C₁₆	—	43	1	no	—
173	C₂²×C₁₆	—	43	1	no	—
174	C₄×Q₁₆	—	43	4	no	—
175	C₄×Q₁₆	—	86	2	no	—
176	C₄×D₈	—	2	2	no	—
177	C₄×D₈	C₂×D₄₃	1	4	yes	—
178	C₂×#107	—	43	1	no	—
179	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
180	C₂×#103	—	86	1	no	—
181	C₂×#102	—	2	1	no	—
182	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
183	#366	—	43	4	no	—
184	#358	—	43	4	no	—
185	#357	—	43	4	no	—
186	#346	—	43	4	no	—
187	C₄₃	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	86	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂³×C₄	—	4	1	no	—
191	C₂×Q₁₆	—	43	4	no	—
192	C₂×Q₁₆	—	43	2	no	—
193	C₂×Q₁₆	—	86	2	no	—
194	C₂×Q₁₆	—	86	1	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	C₄×D₄₃	1	4	yes	—
198	C₂×D₈	C₂²×D₄₃	1	2	yes	—
199	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
201	C₄×Q₈	—	86	4	no	—
202	C₄×Q₈	—	172	2	no	—
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	4	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	86	1	no	—
207	C₂×C₄⋊C₄	—	172	1	no	—
208	C₂×C₂²⋊C₄	—	4	1	no	—
209	C₂×C₄²	—	172	1	no	—
210	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
211	SD₃₂	—	43	16	no	—
212	C₂×C₁₆	—	43	4	no	—
213	C₂×C₁₆	—	43	2	no	—
214	C₂×C₁₆	—	43	2	no	—
215	C₂×C₁₆	—	86	2	no	—
216	#107	—	2	1	no	—
217	#107	—	43	2	no	—
218	#107	—	86	1	no	—
219	#107	C₂²×D₄₃	1	2	yes	—
220	#103	—	86	4	no	—
221	#102	—	2	4	no	—
222	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
223	C₄×C₈	C₂²×D₄₃	1	2	yes	—
224	C₂⁴	—	4	1	no	—
225	C₂×Q₈	—	86	4	no	—
226	C₂×Q₈	—	86	2	no	—
227	C₂×Q₈	—	172	1	no	—
228	C₂×Q₈	—	172	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	8	4	no	—
236	C₂²×C₄	—	172	1	no	—
237	C₂²×C₄	—	172	1	no	—
238	C₂²×C₄	#10621882	1	1	yes	characteristic
239	C₂²×C₄	#10621882	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
241	Q₁₆	—	43	4	no	—
242	Q₁₆	—	86	4	no	—
243	Q₁₆	—	86	2	no	—
244	D₈	—	2	4	no	—
245	D₈	—	2	2	no	—
246	D₈	C₂×C₄×D₄₃	1	4	yes	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
249	C₂×C₈	#10621898	1	2	yes	—
250	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
251	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	86	2	no	—
256	C₄⋊C₄	—	172	1	no	—
257	C₄⋊C₄	—	172	4	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₄²	—	172	4	no	—
260	C₄²	#10621882	1	2	yes	—
261	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
262	C₁₆	—	43	4	no	—
263	C₁₆	—	86	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	8	2	no	—
267	C₂³	—	8	2	no	—
268	C₂³	C₂×#10621882	1	1	yes	characteristic
269	Q₈	—	86	4	no	—
270	Q₈	—	172	2	no	—
271	Q₈	—	172	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	8	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	344	2	no	—
281	C₂×C₄	—	172	2	no	—
282	C₂×C₄	—	172	2	no	—
283	C₂×C₄	—	172	4	no	—
284	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
285	C₂×C₄	C₂×#10621882	1	2	yes	—
286	C₂×C₄	C₂×#10621882	1	2	yes	—
287	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
288	C₂×C₄	#13063345	1	2	yes	—
289	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
292	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
293	C₈	C₂×#10621898	1	1	yes	characteristic
294	C₂²	—	4	4	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	8	no	—
298	C₂²	C₄×#10621882	1	2	yes	—
299	C₂²	#14583335	1	2	yes	—
300	C₂²	#14583339	1	2	yes	—
301	C₂²	C₂²×#10621882	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	344	2	no	—
306	C₄	—	172	4	no	—
307	C₄	C₄×#10621882	1	2	yes	—
308	C₄	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
309	C₄	C₂×#13063345	1	1	yes	characteristic
310	C₂	—	8	4	no	—
311	C₂	#19136459	1	4	yes	direct product factor
312	C₂	C₂×C₄×#10621882	1	1	yes	characteristic
313	C₂	C₂×#14583335	1	1	yes	characteristic
314	C₂	C₂×#14583339	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄×D₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₈×C₄₃	C₂²
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₃
hypocenter	C₄₃	C₂×#2175
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₁₇₂	C₂	semidirect
3. #19136459	C₂	direct (N × H)
4. C₂×#14582724	C₂	semidirect
5. C₂×#14582702	C₂	semidirect
7. C₂×#14582675	C₂	semidirect
11. C₂×#13063269	C₄	semidirect
16. C₄×Dic₁₇₂	C₂²	semidirect
19. #14582724	C₂²	semidirect
20. #14582702	C₂²	semidirect
22. #14582675	C₂²	semidirect
33. C₂×Dic₁₇₂	C₂×C₄	semidirect
40. #13063269	C₂×C₄	semidirect
42. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
103. Dic₁₇₂	C₂²×C₄	semidirect
105. C₄₃⋊C₁₆	C₂²×C₄	semidirect
162. C₂×C₄₃	#2175	semidirect
187. C₄₃	C₂×#2175	semidirect
311. C₂	#19136459	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×#13063269	2
C₂×C₄×Dic₁₇₂	2
#19136459	2
C₂×#14582724	2
C₂×#14582702	2
C₂×#14582700	2
C₂×#14582675	2
C₂×C₄×D₈×C₄₃	2
C₂×#2175	43

Subgroups — C2×#19136459 #25249886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136459 #25249886