Subgroups — C₂×#19145254 #25966441

Order 11008.

1842 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583511	C₂	1	2	yes	maximal
2	#19145254	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583301	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14583268	C₂	1	2	yes	maximal, complemented
6	C₂×#14583157	C₂	1	2	yes	maximal
7	C₂×#14583140	C₂	1	2	yes	maximal
8	C₂×#14583007	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Dic₁₇₂	C₂	1	2	yes	maximal, complemented
10	C₂×C₄₃×#405	C₂	1	1	yes	maximal, characteristic
11	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
12	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
13	#14583511	—	2	8	no	—
14	#14583511	C₂²	1	16	yes	—
15	C₂×#13063325	C₂²	1	2	yes	—
16	C₂×#13063318	—	2	4	no	—
17	C₂×#13063317	C₂²	1	2	yes	—
18	C₂×#13063315	—	2	4	no	—
19	C₂×#13063304	C₂²	1	4	yes	—
20	C₂×C₄×Dic₈₆	C₂²	1	4	yes	—
21	#14583301	C₂²	1	8	yes	complemented
22	#14583279	C₂²	1	8	yes	complemented
23	#14583268	C₂²	1	16	yes	complemented
24	#14583157	C₂²	1	16	yes	—
25	#14583140	C₂²	1	16	yes	—
26	#14583007	C₂²	1	16	yes	complemented
27	C₄×Dic₁₇₂	C₂²	1	16	yes	complemented
28	C₂×#13063259	C₂²	1	2	yes	—
29	C₂×#13063257	C₂²	1	2	yes	—
30	C₂×#13063250	C₂²	1	4	yes	—
31	C₂×#13063250	C₂²	1	4	yes	—
32	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
33	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#405	C₂²	1	8	yes	—
36	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₂²×Dic₈₆	C₂³	1	4	yes	—
41	C₂×Dic₁₇₂	—	2	8	no	—
42	C₂×Dic₁₇₂	—	2	8	no	—
43	C₂×Dic₁₇₂	C₂³	1	8	yes	—
44	C₂×Dic₁₇₂	C₂³	1	8	yes	—
45	#13063325	—	2	4	no	—
46	#13063325	C₂³	1	8	yes	—
47	#13063318	—	2	32	no	—
48	#13063317	—	2	4	no	—
49	#13063317	C₂³	1	8	yes	—
50	#13063315	—	2	32	no	—
51	#13063304	—	2	8	no	—
52	#13063304	C₂³	1	16	yes	—
53	C₄×Dic₈₆	—	2	8	no	—
54	C₄×Dic₈₆	C₂³	1	16	yes	—
55	C₂×C₄⋊Dic₄₃	—	2	4	no	—
56	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
57	C₂×Dic₄₃⋊C₄	—	2	4	no	—
58	C₂×Dic₄₃⋊C₄	—	2	4	no	—
59	C₂×Dic₄₃⋊C₄	—	2	4	no	—
60	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
61	C₂×C₄×Dic₄₃	—	2	4	no	—
62	#13063259	C₂³	1	8	yes	—
63	#13063257	C₂³	1	8	yes	—
64	#13063250	C₂³	1	16	yes	—
65	#13063250	C₂³	1	16	yes	—
66	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₄₃	—	2	4	no	—
68	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
69	C₂²×C₈×C₄₃	—	2	4	no	—
70	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	#117×C₄₃	D₄	1	4	yes	—
76	#117×C₄₃	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
79	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	#97×C₄₃	C₂³	1	4	yes	—
82	#97×C₄₃	C₂³	1	4	yes	—
83	C₄×C₈×C₄₃	C₂³	1	4	yes	—
84	C₄×C₈×C₄₃	C₂³	1	4	yes	—
85	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
93	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₃	—	2	4	no	—
95	C₂²×Dic₄₃	—	2	4	no	—
96	C₂×Dic₈₆	—	2	8	no	—
97	C₂×Dic₈₆	—	2	8	no	—
98	C₂×Dic₈₆	—	2	4	no	—
99	C₂×Dic₈₆	C₂⁴	1	8	yes	—
100	M₁₆×C₄₃	—	2	8	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	8	no	—
104	C₂×C₈×C₄₃	#41	1	4	yes	—
105	C₂×C₈×C₄₃	#41	1	4	yes	—
106	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
107	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
108	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
109	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
110	C₄⋊C₄×C₄₃	C₂×D₄	1	8	yes	—
111	C₂²⋊C₄×C₄₃	C₂×D₄	1	8	yes	—
112	C₄²×C₄₃	C₂×D₄	1	2	yes	—
113	C₄²×C₄₃	C₂×D₄	1	2	yes	—
114	C₄²×C₄₃	C₂⁴	1	2	yes	—
115	C₄²×C₄₃	C₂⁴	1	2	yes	—
116	C₂²⋊Dic₄₃	—	2	16	no	—
117	C₄⋊Dic₄₃	—	2	16	no	—
118	C₄⋊Dic₄₃	—	2	4	no	—
119	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
120	Dic₄₃⋊C₄	—	2	16	no	—
121	Dic₄₃⋊C₄	—	2	16	no	—
122	Dic₄₃⋊C₄	—	2	16	no	—
123	C₄×Dic₄₃	—	2	16	no	—
124	Dic₁₇₂	—	2	16	no	—
125	Dic₁₇₂	—	2	16	no	—
126	C₂³×C₄₃	—	2	2	no	—
127	C₂³×C₄₃	—	2	4	no	—
128	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
129	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	4	no	—
132	C₂×C₄×C₄₃	—	2	4	no	—
133	C₂×C₄×C₄₃	—	2	4	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
142	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
143	C₂×Dic₄₃	—	2	8	no	—
144	C₂×Dic₄₃	—	2	8	no	—
145	C₂×Dic₄₃	—	4	8	no	—
146	C₂×Dic₄₃	—	4	8	no	—
147	Dic₈₆	—	2	8	no	—
148	Dic₈₆	—	2	8	no	—
149	Dic₈₆	—	4	8	no	—
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	—	2	2	no	—
152	C₈×C₄₃	C₂×#41	1	4	yes	—
153	C₈×C₄₃	C₂×#41	1	4	yes	—
154	C₂×#3177	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	—	2	4	no	—
157	C₂²×C₄₃	—	4	4	no	—
158	C₂²×C₄₃	#532	1	2	yes	—
159	C₂²×C₄₃	#578	1	4	yes	—
160	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	4	no	—
163	C₄×C₄₃	—	2	4	no	—
164	C₄×C₄₃	#532	1	2	yes	—
165	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
166	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
167	Dic₄₃	—	4	8	no	—
168	Dic₄₃	—	4	8	no	—
169	C₂×#541	—	43	2	no	—
170	#3177	—	43	16	no	—
171	C₂×#499	—	43	1	no	—
172	C₂×#494	—	43	1	no	—
173	C₂×#491	—	43	2	no	—
174	C₂×#470	—	43	2	no	—
175	C₂×#467	—	43	2	no	—
176	C₂×#441	—	43	2	no	—
177	C₂×C₄×Q₁₆	—	43	2	no	—
178	C₂×#405	D₄₃	1	1	yes	characteristic
179	C₂×C₄₃	—	4	2	no	—
180	C₂×C₄₃	#3177	1	4	yes	complemented
181	C₂×C₄₃	C₂×#532	1	1	yes	characteristic
182	C₂×C₄₃	C₂×#578	1	2	yes	—
183	C₂²×Q₁₆	—	43	2	no	—
184	C₂²×Q₁₆	—	43	2	no	—
185	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
186	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
187	#541	—	43	16	no	—
188	#541	—	86	8	no	—
189	C₂×C₄⋊Q₈	—	43	2	no	—
190	C₂×#126	—	86	4	no	—
191	C₂×#125	—	43	2	no	—
192	C₂×#125	—	86	4	no	—
193	C₂×C₂²⋊Q₈	—	43	4	no	—
194	C₂×C₄×Q₈	—	43	4	no	—
195	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
196	#499	—	43	8	no	—
197	#494	—	43	8	no	—
198	#491	—	43	16	no	—
199	#470	—	43	16	no	—
200	#467	—	43	16	no	—
201	#441	—	43	16	no	—
202	C₄×Q₁₆	—	43	16	no	—
203	C₂×#107	—	43	2	no	—
204	C₂×#106	—	43	2	no	—
205	C₂×#103	—	43	4	no	—
206	C₂×#103	—	43	4	no	—
207	#405	C₂×D₄₃	1	8	yes	—
208	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
209	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
212	C₄₃	C₂×#3177	1	1	yes	characteristic, complemented, Sylow
213	C₂²×Q₈	—	43	4	no	—
214	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
215	C₂×Q₁₆	—	43	8	no	—
216	C₂×Q₁₆	—	43	8	no	—
217	C₂×Q₁₆	—	86	8	no	—
218	C₂×Q₁₆	—	86	8	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
221	C₂²×C₈	—	2	4	no	—
222	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
224	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
227	C₄⋊Q₈	—	43	8	no	—
228	C₄⋊Q₈	—	86	4	no	—
229	#126	—	86	32	no	—
230	#125	—	43	8	no	—
231	#125	—	86	32	no	—
232	#125	—	86	4	no	—
233	C₂²⋊Q₈	—	43	16	no	—
234	C₂²⋊Q₈	—	86	8	no	—
235	C₄×Q₈	—	43	16	no	—
236	C₄×Q₈	—	86	8	no	—
237	#117	D₁₇₂	1	4	yes	—
238	#117	C₂²×D₄₃	1	4	yes	—
239	C₂×C₄⋊C₄	—	43	4	no	—
240	C₂×C₄⋊C₄	—	86	4	no	—
241	C₂×C₄⋊C₄	—	86	4	no	—
242	C₂×C₄⋊C₄	—	86	4	no	—
243	C₂×C₄⋊C₄	—	86	4	no	—
244	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
245	C₂×C₂²⋊C₄	—	86	4	no	—
246	C₂×C₂²⋊C₄	D₁₇₂	1	2	yes	—
247	C₂×C₄²	—	86	4	no	—
248	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
250	#107	—	43	8	no	—
251	#106	—	43	8	no	—
252	#103	—	43	16	no	—
253	#103	—	43	16	no	—
254	#97	C₂²×D₄₃	1	4	yes	—
255	#97	C₂²×D₄₃	1	4	yes	—
256	C₄×C₈	C₂²×D₄₃	1	4	yes	—
257	C₄×C₈	C₂²×D₄₃	1	4	yes	—
258	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
259	C₂×Q₈	—	43	8	no	—
260	C₂×Q₈	—	86	4	no	—
261	C₂×Q₈	—	86	8	no	—
262	C₂×Q₈	—	86	8	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	86	4	no	—
266	C₂²×C₄	—	86	4	no	—
267	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
268	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
269	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
270	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
272	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
273	Q₁₆	—	86	16	no	—
274	Q₁₆	—	86	16	no	—
275	M₁₆	—	2	8	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	8	no	—
279	C₂×C₈	#10621898	1	4	yes	—
280	C₂×C₈	#10621898	1	4	yes	—
281	C₂×C₈	C₂³×D₄₃	1	2	yes	—
282	C₂×C₈	C₂³×D₄₃	1	2	yes	—
283	C₂×C₈	C₂³×D₄₃	1	2	yes	—
284	C₂×C₈	C₂³×D₄₃	1	2	yes	—
285	C₄⋊C₄	—	43	8	no	—
286	C₄⋊C₄	—	86	4	no	—
287	C₄⋊C₄	—	86	16	no	—
288	C₄⋊C₄	—	86	16	no	—
289	C₄⋊C₄	—	86	16	no	—
290	C₄⋊C₄	—	86	16	no	—
291	C₄⋊C₄	C₂×D₁₇₂	1	8	yes	—
292	C₂²⋊C₄	—	86	16	no	—
293	C₂²⋊C₄	C₂×D₁₇₂	1	8	yes	—
294	C₄²	—	86	16	no	—
295	C₄²	C₂×D₁₇₂	1	2	yes	—
296	C₄²	C₂×D₁₇₂	1	2	yes	—
297	C₄²	C₂³×D₄₃	1	2	yes	—
298	C₄²	C₂³×D₄₃	1	2	yes	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	4	no	—
301	C₂³	C₂²×D₁₇₂	1	2	yes	—
302	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
303	Q₈	—	86	8	no	—
304	Q₈	—	86	8	no	—
305	Q₈	—	172	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	86	8	no	—
313	C₂×C₄	—	86	8	no	—
314	C₂×C₄	—	172	8	no	—
315	C₂×C₄	—	172	8	no	—
316	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
317	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
318	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
319	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
320	C₂×C₄	C₂×#10621898	1	2	yes	—
321	C₂×C₄	C₂×#10621898	1	2	yes	—
322	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	C₂×#10621898	1	4	yes	—
326	C₈	C₂×#10621898	1	4	yes	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	4	no	—
330	C₂²	#14583449	1	2	yes	—
331	C₂²	#14583716	1	4	yes	—
332	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	2	2	no	—
336	C₄	—	172	8	no	—
337	C₄	—	172	8	no	—
338	C₄	#14583449	1	2	yes	—
339	C₄	C₂²×#10621898	1	1	yes	characteristic
340	C₄	C₂²×#10621898	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	#19145254	1	4	yes	direct product factor
343	C₂	C₂×#14583449	1	1	yes	characteristic
344	C₂	C₂×#14583716	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#405	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
hypercenter	C₂×#405	D₄₃
hypocenter	C₄₃	C₂×#3177
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #19145254	C₂	direct (N × H)
3. C₂×#14583301	C₂	semidirect
4. C₂×#14583279	C₂	semidirect
5. C₂×#14583268	C₂	semidirect
8. C₂×#14583007	C₂	semidirect
9. C₂×C₄×Dic₁₇₂	C₂	semidirect
21. #14583301	C₂²	semidirect
22. #14583279	C₂²	semidirect
23. #14583268	C₂²	semidirect
26. #14583007	C₂²	semidirect
27. C₄×Dic₁₇₂	C₂²	semidirect
180. C₂×C₄₃	#3177	semidirect
212. C₄₃	C₂×#3177	semidirect
342. C₂	#19145254	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×#14583511	2
#19145254	2
C₂×#14583301	2
C₂×#14583279	2
C₂×#14583268	2
C₂×#14583157	2
C₂×#14583140	2
C₂×#14583007	2
C₂×C₄×Dic₁₇₂	2
C₂×C₄₃×#405	2
C₂×#3177	43

Subgroups — C2×#19145254 #25966441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19145254 #25966441