Subgroups — C₂×#19147742 #26003116

Order 11008.

2074 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583716	C₂	1	1	yes	maximal, characteristic, complemented
2	#19147742	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583339	C₂	1	3	yes	maximal, complemented
4	C₂×#14583335	C₂	1	3	yes	maximal, complemented
5	C₂×#14583329	C₂	1	3	yes	maximal, complemented
6	C₂²×#13063269	C₂	1	3	yes	maximal, complemented
7	C₂×#14583316	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063372	—	2	1	no	—
10	#14583716	—	2	4	no	—
11	#14583716	C₂²	1	8	yes	complemented
12	C₂×#13063358	—	2	3	no	—
13	C₂×#13063347	C₂²	1	3	yes	complemented
14	C₂²×Dic₁₇₂	C₂²	1	3	yes	complemented
15	#14583339	C₂²	1	24	yes	complemented
16	#14583335	C₂²	1	24	yes	complemented
17	#14583329	C₂²	1	24	yes	complemented
18	C₂×#13063271	C₂²	1	3	yes	complemented
19	C₂×#13063269	C₂²	1	3	yes	complemented
20	C₂×#13063269	C₂²	1	6	yes	complemented
21	C₂×#13063269	C₂²	1	24	yes	complemented
22	C₂×#13063268	C₂²	1	1	yes	characteristic, complemented
23	C₂×#13063266	C₂²	1	3	yes	complemented
24	#14583316	C₂²	1	8	yes	complemented
25	C₂×#13063265	C₂²	1	3	yes	complemented
26	C₂²×C₄₃⋊C₁₆	C₂²	1	3	yes	complemented
27	C₂×H(5,𝔽₂)×C₄₃	—	2	1	no	—
28	C₄₃×#576	—	2	4	no	—
29	C₄₃×#576	C₂²	1	8	yes	complemented
30	C₂×AGL(1,ℤ/8)×C₄₃	—	2	3	no	—
31	C₂×#135×C₄₃	C₂²	1	3	yes	—
32	C₂²×D₈×C₄₃	C₂²	1	3	yes	—
33	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
34	#13063372	—	2	8	no	—
35	#13063372	—	4	4	no	—
36	C₂×#10621898	—	2	3	no	—
37	C₂×#10621900	—	4	3	no	—
38	C₂²×Dic₈₆	—	2	3	no	—
39	C₂×Q₈×D₄₃	—	4	1	no	—
40	#13063358	—	2	24	no	—
41	#13063347	—	2	6	no	—
42	#13063347	C₂³	1	12	yes	complemented
43	C₂×Dic₁₇₂	—	2	3	no	—
44	C₂×Dic₁₇₂	—	2	6	no	—
45	C₂×Dic₁₇₂	C₂³	1	3	yes	—
46	C₂×Dic₁₇₂	C₂³	1	12	yes	complemented
47	C₂×#10621873	—	2	3	no	—
48	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
49	#13063271	C₂³	1	12	yes	complemented
50	#13063269	C₂³	1	12	yes	complemented
51	#13063269	C₂³	1	24	yes	complemented
52	#13063268	C₂³	1	4	yes	complemented
53	#13063266	C₂³	1	12	yes	complemented
54	#13063265	C₂³	1	12	yes	complemented
55	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₄₃⋊C₁₆	C₂³	1	3	yes	—
57	C₂×C₄₃⋊C₁₆	C₂³	1	12	yes	complemented
58	H(5,𝔽₂)×C₄₃	—	2	8	no	—
59	H(5,𝔽₂)×C₄₃	—	4	4	no	—
60	C₂×#41×C₄₃	—	2	3	no	—
61	C₂×#41×C₄₃	—	4	1	no	—
62	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
63	C₂²×D₄×C₄₃	—	2	3	no	—
64	C₂²×D₄×C₄₃	—	4	3	no	—
65	AGL(1,ℤ/8)×C₄₃	—	2	24	no	—
66	#135×C₄₃	—	2	6	no	—
67	#135×C₄₃	C₂³	1	12	yes	—
68	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
69	C₂×SD₁₆×C₄₃	—	2	3	no	—
70	C₂×D₈×C₄₃	—	2	3	no	—
71	C₂×D₈×C₄₃	—	2	6	no	—
72	C₂×D₈×C₄₃	C₂³	1	3	yes	—
73	C₂×D₈×C₄₃	C₂³	1	12	yes	—
74	#131×C₄₃	D₄	1	4	yes	—
75	#131×C₄₃	C₂³	1	4	yes	—
76	C₂×M₁₆×C₄₃	D₄	1	3	yes	—
77	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
78	C₂⁴×C₄₃	—	4	3	no	—
79	#41×C₄₃	—	2	12	no	—
80	#41×C₄₃	—	4	8	no	—
81	#41×C₄₃	—	4	6	no	—
82	#41×C₄₃	D₈	1	4	yes	—
83	#41×C₄₃	C₂×D₄	1	4	yes	—
84	C₂×Q₈×C₄₃	—	2	1	no	—
85	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
86	C₂×D₄×C₄₃	—	2	3	no	—
87	C₂×D₄×C₄₃	—	2	12	no	—
88	C₂×D₄×C₄₃	—	4	24	no	—
89	C₂×D₄×C₄₃	—	4	3	no	—
90	C₂×D₄×C₄₃	—	4	6	no	—
91	C₂×D₄×C₄₃	—	4	3	no	—
92	C₂×D₄×C₄₃	D₈	1	3	yes	—
93	C₂²×C₄×C₄₃	—	4	3	no	—
94	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
95	C₂×C₂²⋊D₄₃	—	4	3	no	—
96	C₂²×Dic₄₃	—	4	3	no	—
97	Q₈×D₄₃	—	4	8	no	—
98	#10621900	—	4	24	no	—
99	#10621898	—	2	12	no	—
100	#10621898	—	4	6	no	—
101	C₂×D₁₇₂	—	2	1	no	—
102	C₂×C₄×D₄₃	—	4	3	no	—
103	C₂×Dic₈₆	—	2	3	no	—
104	C₂×Dic₈₆	—	2	12	no	—
105	C₂×Dic₈₆	—	4	3	no	—
106	C₂×Dic₈₆	—	4	6	no	—
107	Q₁₆×C₄₃	—	2	1	no	—
108	Q₁₆×C₄₃	C₂⁴	1	2	yes	—
109	SD₁₆×C₄₃	—	2	12	no	—
110	D₈×C₄₃	—	2	3	no	—
111	D₈×C₄₃	—	2	12	no	—
112	D₈×C₄₃	C₂⁴	1	6	yes	—
113	M₁₆×C₄₃	C₂×D₄	1	12	yes	—
114	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
115	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
116	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
118	Dic₁₇₂	—	2	3	no	—
119	Dic₁₇₂	—	2	12	no	—
120	Dic₁₇₂	C₂⁴	1	6	yes	—
121	D₃₄₄	—	2	1	no	—
122	D₃₄₄	C₂⁴	1	2	yes	—
123	#10621873	—	2	12	no	—
124	C₄₃⋊C₁₆	C₂⁴	1	2	yes	—
125	C₄₃⋊C₁₆	C₂⁴	1	6	yes	—
126	C₂³×C₄₃	—	4	3	no	—
127	C₂³×C₄₃	—	4	12	no	—
128	C₂³×C₄₃	—	16	6	no	—
129	C₂³×C₄₃	C₂×D₈	1	3	yes	—
130	C₂²×D₄₃	—	4	1	no	—
131	Q₈×C₄₃	—	2	2	no	—
132	Q₈×C₄₃	—	4	1	no	—
133	Q₈×C₄₃	C₂×D₈	1	4	yes	—
134	D₄×C₄₃	—	2	6	no	—
135	D₄×C₄₃	—	4	12	no	—
136	D₄×C₄₃	—	4	3	no	—
137	D₄×C₄₃	—	4	12	no	—
138	D₄×C₄₃	C₂×D₈	1	12	yes	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	12	no	—
141	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
142	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
145	C₂²⋊D₄₃	—	4	12	no	—
146	C₂×Dic₄₃	—	4	3	no	—
147	C₂×Dic₄₃	—	4	12	no	—
148	D₁₇₂	—	2	2	no	—
149	D₁₇₂	—	4	1	no	—
150	C₄×D₄₃	—	4	12	no	—
151	Dic₈₆	—	2	6	no	—
152	Dic₈₆	—	4	3	no	—
153	Dic₈₆	—	4	12	no	—
154	C₈×C₄₃	C₂²×D₄	1	6	yes	—
155	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
156	C₈×C₄₃	C₂⁵	1	1	yes	characteristic
157	C₂×#3416	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	6	no	—
159	C₂²×C₄₃	—	4	6	no	—
160	C₂²×C₄₃	—	8	6	no	—
161	C₂²×C₄₃	—	16	12	no	—
162	C₂²×C₄₃	C₂²×D₈	1	6	yes	—
163	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
164	C₂×D₄₃	—	4	2	no	—
165	C₂×D₄₃	—	8	2	no	—
166	C₄×C₄₃	—	4	2	no	—
167	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₄₃	C₂²×D₈	1	6	yes	—
169	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
170	Dic₄₃	—	4	6	no	—
171	C₂×#578	—	43	1	no	—
172	C₂×#576	D₄₃	1	1	yes	characteristic, complemented
173	#3416	—	43	16	no	—
174	C₂×#510	—	43	3	no	—
175	C₂×#509	—	43	3	no	—
176	C₂×#508	—	43	3	no	—
177	C₂²×SD₃₂	—	43	3	no	—
178	C₂×#504	—	43	1	no	—
179	C₂×C₄₃	—	2	6	no	—
180	C₂×C₄₃	—	8	6	no	—
181	C₂×C₄₃	#3416	1	2	yes	complemented
182	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
183	D₄₃	—	8	2	no	—
184	C₂×2₋¹⁺⁴	—	86	1	no	—
185	C₂×H(5,𝔽₂)	—	2	1	no	—
186	#578	—	43	8	no	—
187	#578	—	86	4	no	—
188	#576	—	2	4	no	—
189	#576	C₂×D₄₃	1	8	yes	complemented
190	C₂×#137	—	86	3	no	—
191	C₂×AGL(1,ℤ/8)	—	2	3	no	—
192	C₂×#135	—	43	3	no	—
193	C₂×#135	C₂×D₄₃	1	3	yes	—
194	C₂²×Q₁₆	—	43	3	no	—
195	C₂²×D₈	C₂×D₄₃	1	3	yes	—
196	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
197	#510	—	43	24	no	—
198	#509	—	43	24	no	—
199	#508	—	43	24	no	—
200	C₂×Q₃₂	—	43	3	no	—
201	C₂×SD₃₂	—	43	1	no	—
202	C₂×SD₃₂	—	43	3	no	—
203	C₂×SD₃₂	—	43	6	no	—
204	C₂×SD₃₂	—	43	24	no	—
205	C₂×D₁₆	—	43	3	no	—
206	#504	—	43	8	no	—
207	C₂×M₃₂	—	43	3	no	—
208	C₂²×C₁₆	—	43	3	no	—
209	C₄₃	C₂×#3416	1	1	yes	characteristic, complemented, Sylow
210	2₋¹⁺⁴	—	86	8	no	—
211	2₋¹⁺⁴	—	172	4	no	—
212	H(5,𝔽₂)	—	2	8	no	—
213	H(5,𝔽₂)	—	4	4	no	—
214	C₂×#41	—	2	3	no	—
215	C₂×#41	—	4	1	no	—
216	C₂×#41	—	86	3	no	—
217	C₂×#41	—	172	3	no	—
218	C₂×#41	C₂²⋊D₄₃	1	1	yes	characteristic
219	C₂²×Q₈	—	86	3	no	—
220	C₂²×Q₈	—	172	1	no	—
221	C₂²×D₄	—	2	3	no	—
222	C₂²×D₄	—	4	3	no	—
223	#137	—	86	24	no	—
224	AGL(1,ℤ/8)	—	2	24	no	—
225	#135	—	2	6	no	—
226	#135	—	43	12	no	—
227	#135	—	86	6	no	—
228	#135	C₂²×D₄₃	1	12	yes	—
229	C₂×Q₁₆	—	43	3	no	—
230	C₂×Q₁₆	—	43	12	no	—
231	C₂×Q₁₆	—	86	3	no	—
232	C₂×Q₁₆	—	86	6	no	—
233	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
234	C₂×SD₁₆	—	2	3	no	—
235	C₂×SD₁₆	—	86	3	no	—
236	C₂×D₈	—	2	3	no	—
237	C₂×D₈	—	2	6	no	—
238	C₂×D₈	—	43	1	no	—
239	C₂×D₈	C₂²×D₄₃	1	3	yes	—
240	C₂×D₈	C₂²×D₄₃	1	12	yes	—
241	#131	C₂²⋊D₄₃	1	4	yes	—
242	#131	C₂²×D₄₃	1	4	yes	—
243	C₂×M₁₆	C₂²⋊D₄₃	1	3	yes	—
244	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
245	Q₃₂	—	43	12	no	—
246	SD₃₂	—	43	4	no	—
247	SD₃₂	—	43	12	no	—
248	SD₃₂	—	43	24	no	—
249	D₁₆	—	43	12	no	—
250	M₃₂	—	43	12	no	—
251	C₂×C₁₆	—	43	1	no	—
252	C₂×C₁₆	—	43	3	no	—
253	C₂×C₁₆	—	43	12	no	—
254	C₂⁴	—	4	3	no	—
255	#41	—	2	12	no	—
256	#41	—	4	8	no	—
257	#41	—	4	6	no	—
258	#41	—	86	12	no	—
259	#41	—	172	24	no	—
260	#41	—	172	6	no	—
261	#41	#10621882	1	4	yes	—
262	#41	C₂×C₂²⋊D₄₃	1	4	yes	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	86	3	no	—
265	C₂×Q₈	—	86	12	no	—
266	C₂×Q₈	—	172	8	no	—
267	C₂×Q₈	—	172	3	no	—
268	C₂×Q₈	—	172	6	no	—
269	C₂×Q₈	#10621882	1	1	yes	characteristic
270	C₂×D₄	—	2	3	no	—
271	C₂×D₄	—	2	12	no	—
272	C₂×D₄	—	4	3	no	—
273	C₂×D₄	—	4	24	no	—
274	C₂×D₄	—	4	3	no	—
275	C₂×D₄	—	4	6	no	—
276	C₂×D₄	—	86	1	no	—
277	C₂×D₄	—	172	3	no	—
278	C₂×D₄	#10621882	1	3	yes	—
279	C₂²×C₄	—	4	3	no	—
280	C₂²×C₄	—	172	3	no	—
281	C₂²×C₄	—	172	3	no	—
282	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
283	Q₁₆	—	2	1	no	—
284	Q₁₆	—	43	6	no	—
285	Q₁₆	—	86	3	no	—
286	Q₁₆	—	86	12	no	—
287	Q₁₆	C₂³×D₄₃	1	2	yes	—
288	SD₁₆	—	2	12	no	—
289	SD₁₆	—	86	12	no	—
290	D₈	—	2	3	no	—
291	D₈	—	2	12	no	—
292	D₈	—	43	2	no	—
293	D₈	—	86	1	no	—
294	D₈	C₂³×D₄₃	1	6	yes	—
295	M₁₆	C₂×C₂²⋊D₄₃	1	12	yes	—
296	C₂×C₈	C₂×C₂²⋊D₄₃	1	3	yes	—
297	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
298	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₄₃	1	6	yes	—
300	C₁₆	—	43	2	no	—
301	C₁₆	—	43	6	no	—
302	C₂³	—	4	3	no	—
303	C₂³	—	4	12	no	—
304	C₂³	—	16	6	no	—
305	C₂³	—	172	1	no	—
306	C₂³	C₂×#10621882	1	3	yes	—
307	Q₈	—	2	2	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	86	6	no	—
310	Q₈	—	172	12	no	—
311	Q₈	—	172	3	no	—
312	Q₈	C₂×#10621882	1	4	yes	—
313	D₄	—	2	6	no	—
314	D₄	—	4	12	no	—
315	D₄	—	4	12	no	—
316	D₄	—	4	3	no	—
317	D₄	—	86	2	no	—
318	D₄	—	172	12	no	—
319	D₄	—	172	1	no	—
320	D₄	C₂×#10621882	1	12	yes	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	12	no	—
323	C₂×C₄	—	172	12	no	—
324	C₂×C₄	—	172	3	no	—
325	C₂×C₄	—	172	12	no	—
326	C₂×C₄	C₂×#10621882	1	6	yes	—
327	C₂×C₄	C₂×#10621882	1	3	yes	—
328	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
329	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
330	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
331	C₈	C₂²×C₂²⋊D₄₃	1	6	yes	—
332	C₈	C₂⁴×D₄₃	1	1	yes	characteristic
333	C₂²	—	2	6	no	—
334	C₂²	—	4	6	no	—
335	C₂²	—	8	6	no	—
336	C₂²	—	16	12	no	—
337	C₂²	—	344	2	no	—
338	C₂²	—	172	2	no	—
339	C₂²	C₂²×#10621882	1	6	yes	—
340	C₂²	C₂²×#10621882	1	1	yes	characteristic
341	C₄	—	4	2	no	—
342	C₄	—	172	6	no	—
343	C₄	C₂²×#10621882	1	6	yes	—
344	C₄	C₂²×#10621882	1	1	yes	characteristic
345	C₄	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
346	C₂	—	2	6	no	—
347	C₂	—	8	6	no	—
348	C₂	—	344	2	no	—
349	C₂	#19147742	1	2	yes	direct product factor
350	C₂	C₂³×#10621882	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₈×C₄₃	C₂⁵
Frattini	C₈	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#576	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₂×#576	D₄₃
hypocenter	C₄₃	C₂×#3416
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#576

Complements

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

N	H ≅ G/N	type
1. C₂×#14583716	C₂	semidirect
2. #19147742	C₂	direct (N × H)
3. C₂×#14583339	C₂	semidirect
4. C₂×#14583335	C₂	semidirect
5. C₂×#14583329	C₂	semidirect
6. C₂²×#13063269	C₂	semidirect
7. C₂×#14583316	C₂	semidirect
8. C₂×C₄₃×#576	C₂	semidirect
11. #14583716	C₂²	semidirect
13. C₂×#13063347	C₂²	semidirect
14. C₂²×Dic₁₇₂	C₂²	semidirect
15. #14583339	C₂²	semidirect
16. #14583335	C₂²	semidirect
17. #14583329	C₂²	semidirect
18. C₂×#13063271	C₂²	semidirect
19. C₂×#13063269	C₂²	semidirect
20. C₂×#13063269	C₂²	semidirect
21. C₂×#13063269	C₂²	semidirect
22. C₂×#13063268	C₂²	semidirect
23. C₂×#13063266	C₂²	semidirect
24. #14583316	C₂²	semidirect
25. C₂×#13063265	C₂²	semidirect
26. C₂²×C₄₃⋊C₁₆	C₂²	semidirect
29. C₄₃×#576	C₂²	semidirect
42. #13063347	C₂³	semidirect
46. C₂×Dic₁₇₂	C₂³	semidirect
49. #13063271	C₂³	semidirect
50. #13063269	C₂³	semidirect
51. #13063269	C₂³	semidirect
52. #13063268	C₂³	semidirect
53. #13063266	C₂³	semidirect
54. #13063265	C₂³	semidirect
57. C₂×C₄₃⋊C₁₆	C₂³	semidirect
172. C₂×#576	D₄₃	semidirect
181. C₂×C₄₃	#3416	semidirect
189. #576	C₂×D₄₃	semidirect
209. C₄₃	C₂×#3416	semidirect
349. C₂	#19147742	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×#14583716	2
#19147742	2
C₂×#14583339	2
C₂×#14583335	2
C₂×#14583329	2
C₂²×#13063269	2
C₂×#14583316	2
C₂×C₄₃×#576	2
C₂×#3416	43

Subgroups — C2×#19147742 #26003116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147742 #26003116