Subgroups — C₂×#19136679 #25251013

Order 11008.

806 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063266	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14583135	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136679	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582724	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582700	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582699	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063299	—	2	1	no	—
11	C₂×#13063266	—	2	4	no	—
12	C₂×#13063266	C₂²	1	4	yes	complemented
13	C₂²×C₄₃⋊C₁₆	—	2	4	no	—
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₁₆	C₂²	1	4	yes	complemented
17	#14583135	—	2	2	no	—
18	#14583135	C₂²	1	4	yes	complemented
19	C₂×#13063259	C₂²	1	1	yes	characteristic
20	C₂×#13063247	—	2	1	no	—
21	#14582724	C₂²	1	4	yes	complemented
22	#14582700	C₂²	1	4	yes	complemented
23	#14582699	C₂²	1	4	yes	complemented
24	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#121×C₄₃	—	2	1	no	—
27	C₄₃×#466	—	2	2	no	—
28	C₄₃×#466	C₂²	1	4	yes	complemented
29	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#102×C₄₃	—	2	1	no	—
31	C₂²×D₁₇₂	—	2	1	no	—
32	C₂²×C₂²⋊D₄₃	—	4	1	no	—
33	C₂×D₃₄₄	—	2	2	no	—
34	C₂×D₃₄₄	—	2	2	no	—
35	C₂×D₃₄₄	—	2	1	no	—
36	C₂×D₃₄₄	C₂³	1	2	yes	—
37	#13063299	—	2	4	no	—
38	#13063299	—	4	2	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
41	#13063266	—	2	8	no	—
42	C₂×C₄₃⋊C₁₆	—	2	2	no	—
43	C₂×C₄₃⋊C₁₆	—	2	8	no	—
44	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	complemented
45	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
46	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
47	#13063259	—	2	1	no	—
48	#13063259	C₂³	1	2	yes	—
49	#13063247	—	2	4	no	—
50	C₂²×D₄×C₄₃	—	2	1	no	—
51	C₂²×D₄×C₄₃	—	4	1	no	—
52	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
53	C₂×D₈×C₄₃	—	2	2	no	—
54	C₂×D₈×C₄₃	—	2	1	no	—
55	C₂×D₈×C₄₃	—	2	2	no	—
56	C₂×D₈×C₄₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₄₃	—	2	2	no	—
58	C₂²×C₈×C₄₃	—	2	2	no	—
59	C₂²×C₈×C₄₃	D₄	1	2	yes	—
60	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	#121×C₄₃	—	2	4	no	—
64	#121×C₄₃	—	4	2	no	—
65	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
67	#107×C₄₃	—	2	1	no	—
68	#107×C₄₃	C₂³	1	2	yes	—
69	#102×C₄₃	—	2	4	no	—
70	C₂⁴×C₄₃	—	4	1	no	—
71	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
72	C₂³×D₄₃	—	4	1	no	—
73	C₂×D₄×C₄₃	—	2	2	no	—
74	C₂×D₄×C₄₃	—	4	1	no	—
75	C₂×D₄×C₄₃	—	4	4	no	—
76	C₂×D₄×C₄₃	—	4	2	no	—
77	C₂×D₄×C₄₃	—	4	2	no	—
78	C₂×D₄×C₄₃	—	8	4	no	—
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	4	1	no	—
82	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	D₈	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₂²⋊D₄₃	—	4	4	no	—
87	C₂×C₂²⋊D₄₃	—	8	4	no	—
88	C₂²×Dic₄₃	—	4	1	no	—
89	C₂×D₁₇₂	—	2	2	no	—
90	C₂×D₁₇₂	—	4	1	no	—
91	C₂×D₁₇₂	—	4	2	no	—
92	C₂×D₁₇₂	—	4	2	no	—
93	D₈×C₄₃	—	2	2	no	—
94	D₈×C₄₃	—	2	2	no	—
95	D₈×C₄₃	—	4	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	4	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	#41	1	2	yes	—
104	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	4	1	no	—
107	C₂²⋊C₄×C₄₃	—	4	4	no	—
108	(C₂×D₄₃)⋊C₄	—	4	4	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	4	1	no	—
111	D₃₄₄	—	2	2	no	—
112	D₃₄₄	—	2	2	no	—
113	D₃₄₄	—	4	2	no	—
114	C₄₃⋊C₁₆	—	2	2	no	—
115	C₄₃⋊C₁₆	C₂×D₄	1	4	yes	complemented
116	C₂³×C₄₃	—	2	2	no	—
117	C₂³×C₄₃	—	2	2	no	—
118	C₂³×C₄₃	—	4	2	no	—
119	C₂³×C₄₃	—	8	2	no	—
120	C₂³×C₄₃	—	8	2	no	—
121	C₂³×C₄₃	—	8	2	no	—
122	C₂³×C₄₃	D₁₆	1	4	yes	complemented
123	C₂³×C₄₃	C₂×D₈	1	2	yes	—
124	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₄₃	—	4	2	no	—
126	C₂²×D₄₃	—	8	2	no	—
127	C₂²×D₄₃	—	8	2	no	—
128	C₂²×D₄₃	—	8	2	no	—
129	D₄×C₄₃	—	4	2	no	—
130	D₄×C₄₃	—	4	2	no	—
131	D₄×C₄₃	—	8	8	no	—
132	D₄×C₄₃	—	8	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	4	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	8	2	no	—
139	C₂×C₄×C₄₃	#121	1	2	yes	—
140	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#135	1	2	yes	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₄₃	—	8	8	no	—
145	C₂×Dic₄₃	—	4	2	no	—
146	C₂×Dic₄₃	—	8	2	no	—
147	D₁₇₂	—	4	2	no	—
148	D₁₇₂	—	4	2	no	—
149	D₁₇₂	—	8	2	no	—
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	#121	1	2	yes	—
152	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
154	C₂×#2215	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	—	2	8	no	—
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	8	2	no	—
159	C₂²×C₄₃	—	8	2	no	—
160	C₂²×C₄₃	—	8	2	no	—
161	C₂²×C₄₃	—	16	4	no	—
162	C₂²×C₄₃	#466	1	2	yes	—
163	C₂²×C₄₃	C₂×D₁₆	1	2	yes	—
164	C₂²×C₄₃	C₂×D₁₆	1	4	yes	complemented
165	C₂²×C₄₃	#508	1	2	yes	—
166	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₂×D₄₃	—	8	2	no	—
170	C₂×D₄₃	—	16	4	no	—
171	C₄×C₄₃	—	2	2	no	—
172	C₄×C₄₃	—	8	2	no	—
173	C₄×C₄₃	#466	1	2	yes	—
174	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
175	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
176	Dic₄₃	—	8	2	no	—
177	C₂²×D₁₆	—	43	1	no	—
178	C₂³×C₁₆	—	43	1	no	—
179	C₂×#466	—	43	1	no	—
180	C₂×#466	D₄₃	1	1	yes	characteristic, complemented
181	#2215	—	43	8	no	—
182	C₂×#366	—	43	1	no	—
183	C₂×#357	—	43	1	no	—
184	C₂×#357	—	43	1	no	—
185	C₂×C₄₃	—	2	4	no	—
186	C₂×C₄₃	—	16	2	no	—
187	C₂×C₄₃	#2215	1	4	yes	complemented
188	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
189	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
190	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
191	D₄₃	—	16	2	no	—
192	C₂²×D₈	—	43	1	no	—
193	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
194	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
195	C₂×#121	—	2	1	no	—
196	C₂×#121	—	86	1	no	—
197	C₂×D₁₆	—	43	4	no	—
198	C₂×D₁₆	—	86	4	no	—
199	C₂²×C₁₆	—	43	1	no	—
200	C₂²×C₁₆	—	43	1	no	—
201	C₂²×C₁₆	—	43	4	no	—
202	C₂²×C₁₆	—	86	4	no	—
203	#466	—	2	2	no	—
204	#466	—	43	4	no	—
205	#466	—	86	2	no	—
206	#466	C₂×D₄₃	1	4	yes	complemented
207	C₂×#107	—	43	1	no	—
208	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
209	C₂×#102	—	2	1	no	—
210	C₂×#102	—	86	1	no	—
211	#366	—	43	4	no	—
212	#357	—	43	4	no	—
213	#357	—	43	4	no	—
214	C₄₃	C₂×#2215	1	1	yes	characteristic, complemented, Sylow
215	C₂²×D₄	—	2	1	no	—
216	C₂²×D₄	—	4	1	no	—
217	C₂²×D₄	—	86	1	no	—
218	C₂²×D₄	—	172	1	no	—
219	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	43	2	no	—
224	C₂×D₈	—	86	2	no	—
225	C₂×D₈	—	86	2	no	—
226	C₂×D₈	—	86	1	no	—
227	C₂×D₈	C₂²×D₄₃	1	2	yes	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
232	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
233	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	#121	—	2	4	no	—
235	#121	—	4	2	no	—
236	#121	—	86	4	no	—
237	#121	—	172	2	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	86	1	no	—
240	C₂×C₂²⋊C₄	—	4	1	no	—
241	C₂×C₂²⋊C₄	—	172	1	no	—
242	D₁₆	—	86	8	no	—
243	C₂×C₁₆	—	43	4	no	—
244	C₂×C₁₆	—	43	2	no	—
245	C₂×C₁₆	—	43	2	no	—
246	C₂×C₁₆	—	86	8	no	—
247	C₂×C₁₆	—	86	2	no	—
248	#107	—	2	1	no	—
249	#107	—	43	2	no	—
250	#107	—	86	1	no	—
251	#107	C₂²×D₄₃	1	2	yes	—
252	#102	—	2	4	no	—
253	#102	—	86	4	no	—
254	C₂⁴	—	4	1	no	—
255	C₂⁴	—	172	1	no	—
256	C₂⁴	#10621882	1	1	yes	characteristic
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	8	4	no	—
263	C₂×D₄	—	86	2	no	—
264	C₂×D₄	—	344	4	no	—
265	C₂×D₄	—	172	4	no	—
266	C₂×D₄	—	172	2	no	—
267	C₂×D₄	—	172	1	no	—
268	C₂×D₄	—	172	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	172	1	no	—
273	C₂²×C₄	#10621882	1	2	yes	—
274	C₂²×C₄	#10621882	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
276	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
277	D₈	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	4	2	no	—
280	D₈	—	86	2	no	—
281	D₈	—	86	2	no	—
282	D₈	—	172	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	D₄×D₄₃	1	2	yes	—
288	C₂×C₈	#10621900	1	2	yes	—
289	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
290	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
291	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	86	2	no	—
295	C₄⋊C₄	—	172	1	no	—
296	C₂²⋊C₄	—	4	4	no	—
297	C₂²⋊C₄	—	172	4	no	—
298	C₁₆	—	43	4	no	—
299	C₁₆	—	86	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	8	2	no	—
304	C₂³	—	8	2	no	—
305	C₂³	—	8	2	no	—
306	C₂³	—	344	2	no	—
307	C₂³	—	344	2	no	—
308	C₂³	—	344	2	no	—
309	C₂³	—	172	2	no	—
310	C₂³	#13063266	1	4	yes	complemented
311	C₂³	C₂×#10621882	1	2	yes	—
312	C₂³	C₂×#10621882	1	1	yes	characteristic
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	8	no	—
316	D₄	—	8	2	no	—
317	D₄	—	344	8	no	—
318	D₄	—	344	2	no	—
319	D₄	—	172	2	no	—
320	D₄	—	172	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	344	2	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	#13063297	1	2	yes	—
330	C₂×C₄	C₂×#10621882	1	2	yes	—
331	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
332	C₂×C₄	#13063345	1	2	yes	—
333	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	#13063297	1	2	yes	—
336	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
337	C₈	C₂×#10621900	1	1	yes	characteristic
338	C₂²	—	2	8	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	16	4	no	—
345	C₂²	—	344	2	no	—
346	C₂²	—	344	2	no	—
347	C₂²	—	344	2	no	—
348	C₂²	—	688	4	no	—
349	C₂²	#14583133	1	2	yes	—
350	C₂²	C₂×#13063266	1	2	yes	—
351	C₂²	C₂×#13063266	1	4	yes	complemented
352	C₂²	#14583329	1	2	yes	—
353	C₂²	C₂²×#10621882	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	8	2	no	—
356	C₄	—	344	2	no	—
357	C₄	#14583133	1	2	yes	—
358	C₄	C₂×#13063297	1	1	yes	characteristic
359	C₄	C₂×#13063345	1	1	yes	characteristic
360	C₂	—	2	4	no	—
361	C₂	—	16	2	no	—
362	C₂	—	688	2	no	—
363	C₂	#19136679	1	4	yes	direct product factor
364	C₂	C₂×#14583133	1	1	yes	characteristic
365	C₂	C₂²×#13063266	1	1	yes	characteristic
366	C₂	C₂×#14583329	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#466	C₂
socle	C₂³×C₄₃	C₂×D₈
hypercenter	C₂×#466	D₄₃
hypocenter	C₄₃	C₂×#2215
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063266	C₂	semidirect
2. C₂³×C₄₃⋊C₁₆	C₂	semidirect
3. C₂×#14583135	C₂	semidirect
4. #19136679	C₂	direct (N × H)
5. C₂×#14582724	C₂	semidirect
6. C₂×#14582700	C₂	semidirect
7. C₂×#14582699	C₂	semidirect
8. C₂×C₄₃×#466	C₂	semidirect
12. C₂×#13063266	C₂²	semidirect
16. C₂²×C₄₃⋊C₁₆	C₂²	semidirect
18. #14583135	C₂²	semidirect
21. #14582724	C₂²	semidirect
22. #14582700	C₂²	semidirect
23. #14582699	C₂²	semidirect
28. C₄₃×#466	C₂²	semidirect
44. C₂×C₄₃⋊C₁₆	D₄	semidirect
115. C₄₃⋊C₁₆	C₂×D₄	semidirect
122. C₂³×C₄₃	D₁₆	semidirect
164. C₂²×C₄₃	C₂×D₁₆	semidirect
180. C₂×#466	D₄₃	semidirect
187. C₂×C₄₃	#2215	semidirect
206. #466	C₂×D₄₃	semidirect
214. C₄₃	C₂×#2215	semidirect
310. C₂³	#13063266	semidirect
351. C₂²	C₂×#13063266	semidirect
363. C₂	#19136679	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×#13063266	2
C₂³×C₄₃⋊C₁₆	2
C₂×#14583135	2
#19136679	2
C₂×#14582724	2
C₂×#14582700	2
C₂×#14582699	2
C₂×C₄₃×#466	2
C₂×#2215	43

Subgroups — C2×#19136679 #25251013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136679 #25251013