Subgroups — C₂×#19145023 #25966017

Order 11008.

3366 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063347	C₂	1	4	yes	maximal, complemented
2	C₂×#14583449	C₂	1	2	yes	maximal, complemented
3	#19145023	C₂	1	16	yes	maximal, direct product factor
4	C₂×#14583283	C₂	1	4	yes	maximal, complemented
5	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583276	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14583272	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621898	C₂²	1	4	yes	—
10	C₂×#13063347	—	2	32	no	—
11	C₂×#13063347	C₂²	1	32	yes	complemented
12	C₂²×Dic₁₇₂	C₂²	1	4	yes	—
13	C₂²×#10621873	C₂²	1	8	yes	—
14	C₂²×D₃₄₄	C₂²	1	4	yes	—
15	#14583449	—	2	8	no	—
16	#14583449	C₂²	1	16	yes	complemented
17	C₂×#13063325	C₂²	1	2	yes	—
18	C₂×#13063322	C₂²	1	2	yes	—
19	C₂×#13063313	C₂²	1	4	yes	—
20	C₂×#13063299	—	2	4	no	—
21	C₂×C₄×D₁₇₂	—	2	4	no	—
22	#14583283	C₂²	1	32	yes	complemented
23	#14583279	C₂²	1	8	yes	complemented
24	#14583276	C₂²	1	8	yes	complemented
25	#14583272	C₂²	1	16	yes	complemented
26	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	8	yes	complemented
32	C₂×#10621898	—	2	8	no	—
33	C₂×#10621898	—	2	32	no	—
34	C₂×#10621898	C₂³	1	16	yes	—
35	C₂²×Dic₈₆	C₂³	1	4	yes	—
36	C₂²×D₁₇₂	—	2	4	no	—
37	C₂²×D₁₇₂	C₂³	1	4	yes	—
38	C₂²×C₂²⋊D₄₃	—	2	4	no	—
39	C₂²×C₄×D₄₃	—	2	4	no	—
40	#13063347	—	2	128	no	—
41	C₂×Dic₁₇₂	—	2	16	no	—
42	C₂×Dic₁₇₂	C₂³	1	16	yes	—
43	C₂×#10621873	—	2	32	no	—
44	C₂×#10621873	C₂³	1	32	yes	—
45	C₂×D₃₄₄	—	2	16	no	—
46	C₂×D₃₄₄	C₂³	1	16	yes	—
47	#13063325	—	2	4	no	—
48	#13063325	C₂³	1	8	yes	—
49	#13063322	—	2	4	no	—
50	#13063322	C₂³	1	8	yes	—
51	#13063313	—	2	8	no	—
52	#13063313	C₂³	1	16	yes	—
53	#13063299	—	2	32	no	—
54	C₄×D₁₇₂	—	2	32	no	—
55	C₂×C₄⋊Dic₄₃	—	2	4	no	—
56	C₂×(C₂×D₄₃)⋊C₄	—	2	8	no	—
57	C₂³×C₄×C₄₃	D₄	1	2	yes	—
58	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	D₄	1	16	yes	complemented
60	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₃	C₂³	1	8	yes	—
63	C₂×C₄²×C₄₃	D₄	1	2	yes	—
64	C₂×C₄²×C₄₃	D₄	1	4	yes	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
68	C₄×C₈×C₄₃	C₂³	1	4	yes	—
69	C₄×C₈×C₄₃	C₂³	1	4	yes	—
70	C₄×C₈×C₄₃	C₂³	1	8	yes	—
71	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂³×D₄₃	—	2	4	no	—
73	C₂²×C₄×C₄₃	—	2	8	no	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
82	C₂×C₂²⋊D₄₃	—	2	16	no	—
83	C₂×C₂²⋊D₄₃	—	4	16	no	—
84	C₂²×Dic₄₃	—	2	4	no	—
85	#10621898	—	2	64	no	—
86	#10621898	—	4	32	no	—
87	C₂×D₁₇₂	—	2	16	no	—
88	C₂×D₁₇₂	—	2	4	no	—
89	C₂×D₁₇₂	—	2	16	no	—
90	C₂×D₁₇₂	—	4	16	no	—
91	C₂×D₁₇₂	C₂⁴	1	8	yes	—
92	C₂×C₄×D₄₃	—	2	16	no	—
93	C₂×C₄×D₄₃	—	4	16	no	—
94	C₂×Dic₈₆	—	2	16	no	—
95	C₂×Dic₈₆	—	2	4	no	—
96	C₂×Dic₈₆	C₂⁴	1	8	yes	—
97	C₂×C₈×C₄₃	C₂×D₄	1	8	yes	—
98	C₂×C₈×C₄₃	C₂×D₄	1	8	yes	—
99	C₂×C₈×C₄₃	C₂×D₄	1	32	yes	complemented
100	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
102	C₄²×C₄₃	C₂×D₄	1	8	yes	—
103	C₄²×C₄₃	C₂×D₄	1	2	yes	—
104	C₄²×C₄₃	C₂×D₄	1	2	yes	—
105	C₄²×C₄₃	C₂⁴	1	2	yes	—
106	C₄²×C₄₃	C₂⁴	1	2	yes	—
107	(C₂×D₄₃)⋊C₄	—	2	32	no	—
108	C₄⋊Dic₄₃	—	2	16	no	—
109	Dic₁₇₂	—	2	32	no	—
110	D₃₄₄	—	2	32	no	—
111	#10621873	—	2	64	no	—
112	C₂³×C₄₃	—	2	4	no	—
113	C₂³×C₄₃	#127	1	4	yes	—
114	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
115	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×D₄₃	—	2	8	no	—
117	C₂²×D₄₃	—	4	8	no	—
118	C₂²×D₄₃	—	4	16	no	—
119	C₂×C₄×C₄₃	—	2	4	no	—
120	C₂×C₄×C₄₃	—	2	16	no	—
121	C₂×C₄×C₄₃	#127	1	4	yes	—
122	C₂×C₄×C₄₃	#127	1	8	yes	—
123	C₂×C₄×C₄₃	#135	1	8	yes	complemented
124	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
128	C₂×C₄×C₄₃	C₂²×D₄	1	8	yes	—
129	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
131	C₂²⋊D₄₃	—	4	32	no	—
132	C₂×Dic₄₃	—	2	8	no	—
133	C₂×Dic₄₃	—	4	8	no	—
134	D₁₇₂	—	2	16	no	—
135	D₁₇₂	—	4	8	no	—
136	D₁₇₂	—	4	32	no	—
137	C₄×D₄₃	—	4	32	no	—
138	Dic₈₆	—	2	16	no	—
139	Dic₈₆	—	4	8	no	—
140	C₈×C₄₃	C₂²×D₄	1	8	yes	—
141	C₈×C₄₃	C₂²×D₄	1	8	yes	—
142	C₂×#3147	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	8	no	—
144	C₂²×C₄₃	—	2	4	no	—
145	C₂²×C₄₃	C₂×#127	1	4	yes	—
146	C₂²×C₄₃	C₂×#127	1	2	yes	—
147	C₂²×C₄₃	C₂×#135	1	4	yes	—
148	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
149	C₂×D₄₃	—	4	8	no	—
150	C₂×D₄₃	—	4	16	no	—
151	C₂×D₄₃	—	8	16	no	—
152	C₄×C₄₃	—	2	4	no	—
153	C₄×C₄₃	C₂×#127	1	2	yes	—
154	C₄×C₄₃	C₂×#127	1	4	yes	—
155	C₄×C₄₃	C₂×#135	1	8	yes	complemented
156	C₄×C₄₃	C₂³×D₄	1	2	yes	—
157	Dic₄₃	—	4	8	no	—
158	C₂²×#135	—	43	4	no	—
159	C₂×#532	—	43	2	no	—
160	#3147	—	43	16	no	—
161	C₂×#495	—	43	4	no	—
162	C₂×#494	—	43	1	no	—
163	C₂×#493	—	43	1	no	—
164	C₂×#492	—	43	2	no	—
165	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
166	C₂×C₄₃	—	2	4	no	—
167	C₂×C₄₃	#3147	1	4	yes	complemented
168	C₂×C₄₃	C₂²×#127	1	1	yes	characteristic
169	C₂×C₄₃	C₂²×#135	1	2	yes	—
170	D₄₃	—	8	8	no	—
171	C₂²×#41	—	43	4	no	—
172	C₂×#135	—	43	32	no	—
173	C₂×#135	—	86	32	no	—
174	C₂²×Q₁₆	—	43	4	no	—
175	C₂²×SD₁₆	—	43	8	no	—
176	C₂²×D₈	—	43	4	no	—
177	C₂³×C₈	C₂×D₄₃	1	2	yes	—
178	#532	—	43	16	no	—
179	#532	—	86	8	no	—
180	C₂×C₄⋊Q₈	—	43	2	no	—
181	C₂×#127	—	43	2	no	—
182	C₂×#124	—	43	4	no	—
183	C₂×#121	—	86	4	no	—
184	C₂×C₄×D₄	—	86	4	no	—
185	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
186	#495	—	43	32	no	—
187	#494	—	43	8	no	—
188	#493	—	43	8	no	—
189	#492	—	43	16	no	—
190	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
193	C₂×C₄×C₈	C₂×D₄₃	1	8	yes	complemented
194	C₄₃	C₂×#3147	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	—	43	16	no	—
196	C₂×#41	—	86	8	no	—
197	C₂×#41	—	86	32	no	—
198	C₂²×Q₈	—	43	4	no	—
199	C₂²×D₄	—	43	4	no	—
200	C₂²×D₄	—	86	4	no	—
201	C₂²×D₄	—	86	4	no	—
202	C₂³×C₄	—	86	4	no	—
203	C₂³×C₄	D₁₇₂	1	2	yes	—
204	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
205	#135	—	86	128	no	—
206	C₂×Q₁₆	—	43	16	no	—
207	C₂×Q₁₆	—	86	16	no	—
208	C₂×SD₁₆	—	43	32	no	—
209	C₂×SD₁₆	—	86	32	no	—
210	C₂×D₈	—	43	16	no	—
211	C₂×D₈	—	86	16	no	—
212	C₂²×C₈	D₁₇₂	1	16	yes	complemented
213	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
214	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
215	C₂²×C₈	C₂²×D₄₃	1	8	yes	—
216	C₄⋊Q₈	—	43	8	no	—
217	C₄⋊Q₈	—	86	4	no	—
218	#127	—	43	8	no	—
219	#127	—	86	4	no	—
220	#124	—	43	16	no	—
221	#124	—	86	8	no	—
222	#121	—	86	32	no	—
223	C₄×D₄	—	86	32	no	—
224	C₂×C₄⋊C₄	—	86	4	no	—
225	C₂×C₂²⋊C₄	—	86	8	no	—
226	C₂×C₄²	D₁₇₂	1	4	yes	—
227	C₂×C₄²	D₁₇₂	1	2	yes	—
228	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
231	C₄×C₈	C₂²×D₄₃	1	4	yes	—
232	C₄×C₈	C₂²×D₄₃	1	4	yes	—
233	C₄×C₈	C₂²×D₄₃	1	8	yes	—
234	C₂⁴	—	86	4	no	—
235	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
236	#41	—	86	64	no	—
237	#41	—	172	32	no	—
238	C₂×Q₈	—	43	8	no	—
239	C₂×Q₈	—	86	4	no	—
240	C₂×Q₈	—	86	16	no	—
241	C₂×D₄	—	43	8	no	—
242	C₂×D₄	—	86	16	no	—
243	C₂×D₄	—	86	16	no	—
244	C₂×D₄	—	86	16	no	—
245	C₂×D₄	—	86	4	no	—
246	C₂×D₄	—	172	16	no	—
247	C₂×D₄	—	172	16	no	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	—	86	4	no	—
250	C₂²×C₄	—	86	16	no	—
251	C₂²×C₄	—	172	16	no	—
252	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
253	C₂²×C₄	C₂×D₁₇₂	1	8	yes	—
254	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
255	C₂²×C₄	C₂×D₁₇₂	1	8	yes	—
256	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
257	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
258	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
260	Q₁₆	—	86	32	no	—
261	SD₁₆	—	86	64	no	—
262	D₈	—	86	32	no	—
263	C₂×C₈	C₂×D₁₇₂	1	8	yes	—
264	C₂×C₈	C₂×D₁₇₂	1	8	yes	—
265	C₂×C₈	C₂×D₁₇₂	1	32	yes	complemented
266	C₂×C₈	C₂³×D₄₃	1	4	yes	—
267	C₂×C₈	C₂³×D₄₃	1	4	yes	—
268	C₄⋊C₄	—	86	16	no	—
269	C₂²⋊C₄	—	86	32	no	—
270	C₄²	C₂×D₁₇₂	1	8	yes	—
271	C₄²	C₂×D₁₇₂	1	2	yes	—
272	C₄²	C₂×D₁₇₂	1	2	yes	—
273	C₄²	C₂³×D₄₃	1	2	yes	—
274	C₄²	C₂³×D₄₃	1	2	yes	—
275	C₂³	—	2	4	no	—
276	C₂³	—	86	8	no	—
277	C₂³	—	172	16	no	—
278	C₂³	—	172	8	no	—
279	C₂³	#13063322	1	4	yes	—
280	C₂³	C₂²×D₁₇₂	1	2	yes	—
281	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
282	Q₈	—	86	16	no	—
283	Q₈	—	172	8	no	—
284	D₄	—	86	16	no	—
285	D₄	—	172	32	no	—
286	D₄	—	172	32	no	—
287	D₄	—	172	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	16	no	—
290	C₂×C₄	—	86	8	no	—
291	C₂×C₄	—	172	8	no	—
292	C₂×C₄	—	172	32	no	—
293	C₂×C₄	#13063322	1	4	yes	—
294	C₂×C₄	#13063322	1	8	yes	—
295	C₂×C₄	#13063347	1	8	yes	complemented
296	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
297	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
298	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
299	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
301	C₂×C₄	C₂²×D₁₇₂	1	8	yes	—
302	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
303	C₈	C₂²×D₁₇₂	1	8	yes	—
304	C₈	C₂²×D₁₇₂	1	8	yes	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	344	16	no	—
308	C₂²	—	172	16	no	—
309	C₂²	—	172	8	no	—
310	C₂²	C₂×#13063322	1	4	yes	—
311	C₂²	C₂×#13063322	1	2	yes	—
312	C₂²	C₂×#13063347	1	4	yes	—
313	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	172	8	no	—
316	C₄	C₂×#13063322	1	4	yes	—
317	C₄	C₂×#13063322	1	2	yes	—
318	C₄	C₂×#13063347	1	8	yes	complemented
319	C₄	C₂³×D₁₇₂	1	2	yes	—
320	C₂	—	2	4	no	—
321	C₂	—	344	8	no	—
322	C₂	#19145023	1	4	yes	direct product factor
323	C₂	C₂²×#13063322	1	1	yes	characteristic
324	C₂	C₂²×#13063347	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂×#3147
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063347	C₂	semidirect
2. C₂×#14583449	C₂	semidirect
3. #19145023	C₂	direct (N × H)
4. C₂×#14583283	C₂	semidirect
5. C₂×#14583279	C₂	semidirect
6. C₂×#14583276	C₂	semidirect
7. C₂×#14583272	C₂	semidirect
8. C₂²×C₄×C₈×C₄₃	C₂	semidirect
11. C₂×#13063347	C₂²	semidirect
16. #14583449	C₂²	semidirect
22. #14583283	C₂²	semidirect
23. #14583279	C₂²	semidirect
24. #14583276	C₂²	semidirect
25. #14583272	C₂²	semidirect
31. C₂×C₄×C₈×C₄₃	C₂²	semidirect
59. C₂²×C₈×C₄₃	D₄	semidirect
99. C₂×C₈×C₄₃	C₂×D₄	semidirect
123. C₂×C₄×C₄₃	#135	semidirect
155. C₄×C₄₃	C₂×#135	semidirect
165. C₂²×C₄×C₈	D₄₃	semidirect
167. C₂×C₄₃	#3147	semidirect
193. C₂×C₄×C₈	C₂×D₄₃	semidirect
194. C₄₃	C₂×#3147	semidirect
212. C₂²×C₈	D₁₇₂	semidirect
265. C₂×C₈	C₂×D₁₇₂	semidirect
295. C₂×C₄	#13063347	semidirect
318. C₄	C₂×#13063347	semidirect
322. C₂	#19145023	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×#13063347	2
C₂×#14583449	2
#19145023	2
C₂×#14583283	2
C₂×#14583279	2
C₂×#14583276	2
C₂×#14583272	2
C₂²×C₄×C₈×C₄₃	2
C₂×#3147	43

Subgroups — C2×#19145023 #25966017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19145023 #25966017