Subgroups — C₂×#19136461 #25249888

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×#13063270	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
4	#19136461	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582725	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582704	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582701	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₆	—	2	1	no	—
12	C₂×C₄×D₁₇₂	—	2	1	no	—
13	C₂×#13063270	C₄	1	8	yes	complemented
14	C₂×#13063270	C₂²	1	4	yes	—
15	C₄×Dic₁₇₂	—	2	2	no	—
16	C₄×Dic₁₇₂	C₂²	1	4	yes	complemented
17	C₄×D₃₄₄	—	2	2	no	—
18	C₄×D₃₄₄	C₂²	1	4	yes	—
19	C₂×#13063259	C₂²	1	1	yes	characteristic
20	C₂×#13063259	C₂²	1	1	yes	characteristic
21	C₂×#13063250	—	2	1	no	—
22	C₂×#13063247	—	2	1	no	—
23	#14582725	C₂²	1	4	yes	complemented
24	#14582704	C₂²	1	4	yes	complemented
25	#14582701	C₂²	1	4	yes	—
26	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#346	C₂²	1	4	yes	complemented
30	C₂²×Dic₈₆	—	2	1	no	—
31	C₂²×D₁₇₂	—	2	1	no	—
32	C₂²×C₄×D₄₃	—	4	1	no	—
33	C₂×Dic₁₇₂	—	2	1	no	—
34	C₂×Dic₁₇₂	—	2	2	no	—
35	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	complemented
36	C₂×Dic₁₇₂	C₂³	1	2	yes	—
37	C₂×D₃₄₄	—	2	1	no	—
38	C₂×D₃₄₄	—	2	2	no	—
39	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
40	C₂×D₃₄₄	C₂³	1	2	yes	—
41	C₄×Dic₈₆	—	2	4	no	—
42	C₄×Dic₈₆	—	4	2	no	—
43	C₄×D₁₇₂	—	2	4	no	—
44	C₄×D₁₇₂	—	4	2	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	2	1	no	—
47	C₂×Dic₄₃⋊C₄	—	4	1	no	—
48	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
49	C₂×C₄×Dic₄₃	—	4	1	no	—
50	#13063270	C₂×C₄	1	16	yes	complemented
51	#13063259	—	2	1	no	—
52	#13063259	—	2	1	no	—
53	#13063259	C₂³	1	2	yes	—
54	#13063259	C₂³	1	2	yes	—
55	#13063250	—	2	4	no	—
56	#13063247	—	2	4	no	—
57	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
60	C₂×C₁₆×C₄₃	—	2	2	no	—
61	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
62	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
64	C₄×C₈×C₄₃	D₄	1	2	yes	—
65	C₄×C₈×C₄₃	C₂³	1	2	yes	—
66	C₂³×D₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	4	1	no	—
71	C₂²×Dic₄₃	—	4	1	no	—
72	C₂²×Dic₄₃	—	4	1	no	—
73	C₂×D₁₇₂	—	2	4	no	—
74	C₂×D₁₇₂	—	2	2	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	—	4	2	no	—
77	C₂×C₄×D₄₃	—	4	4	no	—
78	C₂×C₄×D₄₃	—	8	4	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	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	#41	1	2	yes	—
88	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
89	C₄²×C₄₃	D₈	1	2	yes	—
90	C₄²×C₄₃	C₂×D₄	1	2	yes	—
91	(C₂×D₄₃)⋊C₄	—	4	4	no	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	4	1	no	—
95	C₄⋊Dic₄₃	—	4	1	no	—
96	Dic₄₃⋊C₄	—	4	4	no	—
97	C₄×Dic₄₃	—	4	4	no	—
98	Dic₁₇₂	—	2	4	no	—
99	Dic₁₇₂	—	2	2	no	—
100	Dic₁₇₂	C₂²×C₄	1	4	yes	complemented
101	D₃₄₄	—	2	4	no	—
102	D₃₄₄	—	2	2	no	—
103	D₃₄₄	C₂²×C₄	1	4	yes	—
104	C₁₆×C₄₃	—	2	2	no	—
105	C₁₆×C₄₃	C₂²×C₄	1	4	yes	complemented
106	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₄₃	—	4	2	no	—
108	C₂²×D₄₃	—	4	4	no	—
109	C₂²×D₄₃	—	8	2	no	—
110	C₂²×D₄₃	—	8	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#135	1	2	yes	—
118	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₄₃	—	4	2	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	8	2	no	—
124	C₂×Dic₄₃	—	8	2	no	—
125	D₁₇₂	—	2	4	no	—
126	D₁₇₂	—	4	2	no	—
127	D₁₇₂	—	4	4	no	—
128	C₄×D₄₃	—	8	8	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	C₄×D₈	1	2	yes	—
138	C₂²×C₄₃	#509	1	2	yes	—
139	C₂²×C₄₃	#510	1	2	yes	—
140	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₄₃	—	4	4	no	—
142	C₂×D₄₃	—	8	2	no	—
143	C₂×D₄₃	—	8	2	no	—
144	C₂×D₄₃	—	8	8	no	—
145	C₄×C₄₃	—	2	2	no	—
146	C₄×C₄₃	—	2	2	no	—
147	C₄×C₄₃	C₄×D₈	1	2	yes	—
148	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
150	Dic₄₃	—	4	4	no	—
151	Dic₄₃	—	8	2	no	—
152	Dic₄₃	—	8	2	no	—
153	C₂²×SD₃₂	—	43	1	no	—
154	C₂×C₄×Q₁₆	—	43	1	no	—
155	C₂×C₄×D₈	—	43	1	no	—
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	D₄₃	1	1	yes	characteristic, complemented
161	C₂×C₄₃	#2175	1	4	yes	complemented
162	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
164	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
165	D₄₃	—	8	4	no	—
166	C₂²×Q₁₆	—	43	1	no	—
167	C₂²×D₈	—	43	1	no	—
168	C₂×C₄×Q₈	—	86	1	no	—
169	C₂×C₄×D₄	—	86	1	no	—
170	C₂×SD₃₂	—	43	4	no	—
171	C₂×SD₃₂	—	43	8	no	—
172	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
173	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
174	C₄×Q₁₆	—	43	4	no	—
175	C₄×Q₁₆	—	86	2	no	—
176	C₄×D₈	—	43	4	no	—
177	C₄×D₈	—	86	2	no	—
178	C₂×#107	—	43	1	no	—
179	C₂×#107	—	43	1	no	—
180	C₂×#103	—	86	1	no	—
181	C₂×#102	—	86	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	C₂×D₄₃	1	4	yes	complemented
187	C₄₃	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	86	1	no	—
189	C₂²×D₄	—	86	1	no	—
190	C₂³×C₄	—	172	1	no	—
191	C₂×Q₁₆	—	43	4	no	—
192	C₂×Q₁₆	—	43	2	no	—
193	C₂×Q₁₆	—	86	1	no	—
194	C₂×Q₁₆	—	86	2	no	—
195	C₂×D₈	—	43	4	no	—
196	C₂×D₈	—	43	2	no	—
197	C₂×D₈	—	86	1	no	—
198	C₂×D₈	—	86	2	no	—
199	C₂²×C₈	D₁₇₂	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	—	86	4	no	—
204	C₄×D₄	—	172	2	no	—
205	C₂×C₄⋊C₄	—	86	1	no	—
206	C₂×C₄⋊C₄	—	86	1	no	—
207	C₂×C₄⋊C₄	—	172	1	no	—
208	C₂×C₂²⋊C₄	—	172	1	no	—
209	C₂×C₄²	—	172	1	no	—
210	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
211	SD₃₂	—	43	16	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
214	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
215	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
216	#107	—	43	2	no	—
217	#107	—	43	2	no	—
218	#107	—	86	1	no	—
219	#107	—	86	1	no	—
220	#103	—	86	4	no	—
221	#102	—	86	4	no	—
222	C₄×C₈	D₁₇₂	1	2	yes	—
223	C₄×C₈	C₂²×D₄₃	1	2	yes	—
224	C₂⁴	—	172	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	—	86	4	no	—
230	C₂×D₄	—	86	2	no	—
231	C₂×D₄	—	172	1	no	—
232	C₂×D₄	—	172	2	no	—
233	C₂²×C₄	—	344	4	no	—
234	C₂²×C₄	—	172	1	no	—
235	C₂²×C₄	—	172	1	no	—
236	C₂²×C₄	—	172	4	no	—
237	C₂²×C₄	—	172	1	no	—
238	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
239	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
241	Q₁₆	—	43	4	no	—
242	Q₁₆	—	86	4	no	—
243	Q₁₆	—	86	2	no	—
244	D₈	—	43	4	no	—
245	D₈	—	86	4	no	—
246	D₈	—	86	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
249	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
250	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
251	C₂×C₈	#10621898	1	2	yes	—
252	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
253	C₄⋊C₄	—	86	2	no	—
254	C₄⋊C₄	—	86	2	no	—
255	C₄⋊C₄	—	172	1	no	—
256	C₄⋊C₄	—	172	1	no	—
257	C₄⋊C₄	—	172	4	no	—
258	C₂²⋊C₄	—	172	4	no	—
259	C₄²	—	172	4	no	—
260	C₄²	D₃₄₄	1	2	yes	—
261	C₄²	C₂×D₁₇₂	1	2	yes	—
262	C₁₆	—	2	2	no	—
263	C₁₆	C₂×C₄×D₄₃	1	4	yes	complemented
264	C₂³	—	344	2	no	—
265	C₂³	—	344	2	no	—
266	C₂³	—	172	2	no	—
267	C₂³	—	172	4	no	—
268	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
269	Q₈	—	86	4	no	—
270	Q₈	—	172	2	no	—
271	Q₈	—	172	4	no	—
272	D₄	—	86	4	no	—
273	D₄	—	172	2	no	—
274	D₄	—	172	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	344	8	no	—
278	C₂×C₄	—	344	2	no	—
279	C₂×C₄	—	344	2	no	—
280	C₂×C₄	—	172	2	no	—
281	C₂×C₄	—	172	2	no	—
282	C₂×C₄	—	172	2	no	—
283	C₂×C₄	—	172	4	no	—
284	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
285	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
286	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
287	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
288	C₂×C₄	#13063347	1	2	yes	—
289	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×D₁₇₂	1	2	yes	—
292	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
293	C₈	C₂×#10621898	1	1	yes	characteristic
294	C₂²	—	344	2	no	—
295	C₂²	—	344	8	no	—
296	C₂²	—	344	2	no	—
297	C₂²	—	172	4	no	—
298	C₂²	C₄×D₃₄₄	1	2	yes	—
299	C₂²	#14583337	1	2	yes	—
300	C₂²	#14583342	1	2	yes	—
301	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	344	2	no	—
305	C₄	—	344	2	no	—
306	C₄	—	172	4	no	—
307	C₄	C₄×D₃₄₄	1	2	yes	—
308	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
309	C₄	C₂×#13063347	1	1	yes	characteristic
310	C₂	—	344	4	no	—
311	C₂	#19136461	1	4	yes	direct product factor
312	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic
313	C₂	C₂×#14583337	1	1	yes	characteristic
314	C₂	C₂×#14583342	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#346	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#346	D₄₃
hypocenter	C₄₃	C₂×#2175
LCS	G ▷ 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
2. C₂×C₄×Dic₁₇₂	C₂	semidirect
4. #19136461	C₂	direct (N × H)
5. C₂×#14582725	C₂	semidirect
6. C₂×#14582704	C₂	semidirect
8. C₂×C₄₃×#346	C₂	semidirect
13. C₂×#13063270	C₄	semidirect
16. C₄×Dic₁₇₂	C₂²	semidirect
23. #14582725	C₂²	semidirect
24. #14582704	C₂²	semidirect
29. C₄₃×#346	C₂²	semidirect
35. C₂×Dic₁₇₂	C₂×C₄	semidirect
50. #13063270	C₂×C₄	semidirect
61. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
100. Dic₁₇₂	C₂²×C₄	semidirect
105. C₁₆×C₄₃	C₂²×C₄	semidirect
160. C₂×#346	D₄₃	semidirect
161. C₂×C₄₃	#2175	semidirect
186. #346	C₂×D₄₃	semidirect
187. C₄₃	C₂×#2175	semidirect
213. C₂×C₁₆	C₄×D₄₃	semidirect
263. C₁₆	C₂×C₄×D₄₃	semidirect
311. C₂	#19136461	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×#13063270	2
C₂×C₄×Dic₁₇₂	2
C₂×C₄×D₃₄₄	2
#19136461	2
C₂×#14582725	2
C₂×#14582704	2
C₂×#14582701	2
C₂×C₄₃×#346	2
C₂×#2175	43

Subgroups — C2×#19136461 #25249888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136461 #25249888