Subgroups — C₂×#19136469 #25249898

Order 11008.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 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
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136469	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582726	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	—	2	1	no	—
11	C₂×#13063266	C₄	1	8	yes	complemented
12	C₂×#13063266	C₂²	1	4	yes	—
13	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×D₃₄₄	—	2	2	no	—
16	C₄×D₃₄₄	C₂²	1	4	yes	complemented
17	C₂×#13063259	C₂²	1	1	yes	characteristic
18	C₂×#13063247	—	2	1	no	—
19	#14582726	C₂²	1	4	yes	complemented
20	#14582700	C₂²	1	4	yes	complemented
21	#14582699	C₂²	1	4	yes	complemented
22	#14582675	C₂²	1	4	yes	complemented
23	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₃	—	2	1	no	—
25	C₄×D₈×C₄₃	—	2	2	no	—
26	C₄×D₈×C₄₃	C₂²	1	4	yes	complemented
27	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₃	—	2	1	no	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂²×D₁₇₂	—	2	1	no	—
31	C₂²×C₄×D₄₃	—	4	1	no	—
32	C₂×D₃₄₄	—	2	1	no	—
33	C₂×D₃₄₄	—	2	2	no	—
34	C₂×D₃₄₄	C₂×C₄	1	4	yes	complemented
35	C₂×D₃₄₄	C₂³	1	2	yes	—
36	C₄×D₁₇₂	—	2	4	no	—
37	C₄×D₁₇₂	—	4	2	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
40	#13063266	C₂×C₄	1	16	yes	complemented
41	C₂×C₄₃⋊C₁₆	—	2	2	no	—
42	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
45	#13063259	—	2	1	no	—
46	#13063259	C₂³	1	2	yes	—
47	#13063247	—	2	4	no	—
48	C₂²×D₄×C₄₃	—	2	1	no	—
49	C₂³×C₄×C₄₃	—	4	1	no	—
50	C₂×D₈×C₄₃	—	2	1	no	—
51	C₂×D₈×C₄₃	—	2	2	no	—
52	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	complemented
53	C₂×D₈×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₄₃	—	2	4	no	—
57	C₄×D₄×C₄₃	—	4	2	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
60	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
61	#107×C₄₃	—	2	1	no	—
62	#107×C₄₃	C₂³	1	2	yes	—
63	#102×C₄₃	—	2	4	no	—
64	C₄×C₈×C₄₃	D₄	1	2	yes	—
65	C₄×C₈×C₄₃	C₂³	1	2	yes	—
66	C₂⁴×C₄₃	—	4	1	no	—
67	C₂³×D₄₃	—	4	1	no	—
68	C₂×D₄×C₄₃	—	2	4	no	—
69	C₂×D₄×C₄₃	—	2	2	no	—
70	C₂×D₄×C₄₃	—	4	1	no	—
71	C₂×D₄×C₄₃	—	4	2	no	—
72	C₂²×C₄×C₄₃	—	4	4	no	—
73	C₂²×C₄×C₄₃	—	4	1	no	—
74	C₂²×C₄×C₄₃	—	8	4	no	—
75	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₄₃	—	4	1	no	—
79	C₂×D₁₇₂	—	2	4	no	—
80	C₂×D₁₇₂	—	2	2	no	—
81	C₂×D₁₇₂	—	4	1	no	—
82	C₂×D₁₇₂	—	4	2	no	—
83	C₂×C₄×D₄₃	—	4	4	no	—
84	C₂×C₄×D₄₃	—	8	4	no	—
85	D₈×C₄₃	—	2	4	no	—
86	D₈×C₄₃	—	2	2	no	—
87	D₈×C₄₃	C₂²×C₄	1	4	yes	complemented
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	#41	1	2	yes	—
93	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	4	1	no	—
96	C₂²⋊C₄×C₄₃	—	4	4	no	—
97	C₄²×C₄₃	D₈	1	2	yes	—
98	C₄²×C₄₃	C₂×D₄	1	2	yes	—
99	(C₂×D₄₃)⋊C₄	—	4	4	no	—
100	C₄⋊Dic₄₃	—	2	2	no	—
101	C₄⋊Dic₄₃	—	4	1	no	—
102	D₃₄₄	—	2	4	no	—
103	D₃₄₄	—	2	2	no	—
104	D₃₄₄	C₂²×C₄	1	4	yes	complemented
105	C₄₃⋊C₁₆	—	2	2	no	—
106	C₄₃⋊C₁₆	C₂²×C₄	1	4	yes	complemented
107	C₂³×C₄₃	—	4	2	no	—
108	C₂³×C₄₃	—	4	4	no	—
109	C₂³×C₄₃	—	8	2	no	—
110	C₂³×C₄₃	—	8	2	no	—
111	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	4	4	no	—
114	C₂²×D₄₃	—	8	2	no	—
115	C₂²×D₄₃	—	8	2	no	—
116	D₄×C₄₃	—	2	4	no	—
117	D₄×C₄₃	—	4	2	no	—
118	D₄×C₄₃	—	4	4	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	8	8	no	—
123	C₂×C₄×C₄₃	—	8	2	no	—
124	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#135	1	2	yes	—
129	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	8	2	no	—
132	D₁₇₂	—	2	4	no	—
133	D₁₇₂	—	4	2	no	—
134	D₁₇₂	—	4	4	no	—
135	C₄×D₄₃	—	8	8	no	—
136	C₈×C₄₃	—	2	2	no	—
137	C₈×C₄₃	C₄×D₄	1	2	yes	—
138	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₂×#2177	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	—	4	4	no	—
142	C₂²×C₄₃	—	8	2	no	—
143	C₂²×C₄₃	—	8	2	no	—
144	C₂²×C₄₃	—	8	8	no	—
145	C₂²×C₄₃	C₄×D₈	1	2	yes	—
146	C₂²×C₄₃	#509	1	2	yes	—
147	C₂²×C₄₃	#509	1	2	yes	—
148	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₄₃	—	4	4	no	—
150	C₂×D₄₃	—	8	2	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₂×D₄₃	—	8	8	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	—	8	2	no	—
156	C₄×C₄₃	C₄×D₈	1	2	yes	—
157	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
159	Dic₄₃	—	8	2	no	—
160	C₂²×D₁₆	—	43	1	no	—
161	C₂×C₄×D₈	—	43	1	no	—
162	C₂×C₄×D₈	D₄₃	1	1	yes	characteristic, complemented
163	#2177	—	43	8	no	—
164	C₂×#367	—	43	1	no	—
165	C₂×#357	—	43	1	no	—
166	C₂×#357	—	43	1	no	—
167	C₂×#346	—	43	1	no	—
168	C₂×C₄₃	—	8	4	no	—
169	C₂×C₄₃	#2177	1	4	yes	complemented
170	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
171	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
173	D₄₃	—	8	4	no	—
174	C₂²×D₈	—	43	1	no	—
175	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂×C₄×D₄	—	2	1	no	—
177	C₂×C₄×D₄	—	86	1	no	—
178	C₂×D₁₆	—	43	4	no	—
179	C₂×D₁₆	—	43	8	no	—
180	C₂²×C₁₆	—	43	1	no	—
181	C₂²×C₁₆	—	43	1	no	—
182	C₄×D₈	—	2	2	no	—
183	C₄×D₈	—	43	4	no	—
184	C₄×D₈	—	86	2	no	—
185	C₄×D₈	C₂×D₄₃	1	4	yes	complemented
186	C₂×#107	—	43	1	no	—
187	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
188	C₂×#102	—	2	1	no	—
189	C₂×#102	—	86	1	no	—
190	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
191	#367	—	43	4	no	—
192	#357	—	43	4	no	—
193	#357	—	43	4	no	—
194	#346	—	43	4	no	—
195	C₄₃	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	86	1	no	—
198	C₂³×C₄	—	4	1	no	—
199	C₂³×C₄	—	172	1	no	—
200	C₂×D₈	—	2	2	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	43	4	no	—
203	C₂×D₈	—	43	2	no	—
204	C₂×D₈	—	86	2	no	—
205	C₂×D₈	—	86	1	no	—
206	C₂×D₈	C₄×D₄₃	1	4	yes	complemented
207	C₂×D₈	C₂²×D₄₃	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
210	C₄×D₄	—	2	4	no	—
211	C₄×D₄	—	4	2	no	—
212	C₄×D₄	—	86	4	no	—
213	C₄×D₄	—	172	2	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	86	1	no	—
216	C₂×C₂²⋊C₄	—	4	1	no	—
217	C₂×C₂²⋊C₄	—	172	1	no	—
218	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
219	D₁₆	—	43	16	no	—
220	C₂×C₁₆	—	43	4	no	—
221	C₂×C₁₆	—	43	2	no	—
222	C₂×C₁₆	—	43	2	no	—
223	C₂×C₁₆	—	86	2	no	—
224	#107	—	2	1	no	—
225	#107	—	43	2	no	—
226	#107	—	86	1	no	—
227	#107	C₂²×D₄₃	1	2	yes	—
228	#102	—	2	4	no	—
229	#102	—	86	4	no	—
230	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
231	C₄×C₈	C₂²×D₄₃	1	2	yes	—
232	C₂⁴	—	4	1	no	—
233	C₂⁴	—	172	1	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	86	4	no	—
239	C₂×D₄	—	86	2	no	—
240	C₂×D₄	—	172	1	no	—
241	C₂×D₄	—	172	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	8	4	no	—
245	C₂²×C₄	—	344	4	no	—
246	C₂²×C₄	—	172	1	no	—
247	C₂²×C₄	—	172	4	no	—
248	C₂²×C₄	#10621882	1	1	yes	characteristic
249	C₂²×C₄	#10621882	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	D₈	—	2	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	43	4	no	—
254	D₈	—	86	4	no	—
255	D₈	—	86	2	no	—
256	D₈	C₂×C₄×D₄₃	1	4	yes	complemented
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
259	C₂×C₈	#10621898	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
261	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	86	2	no	—
266	C₄⋊C₄	—	172	1	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₂²⋊C₄	—	172	4	no	—
269	C₄²	#10621882	1	2	yes	—
270	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
271	C₁₆	—	43	4	no	—
272	C₁₆	—	86	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	4	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	—	344	2	no	—
278	C₂³	—	344	2	no	—
279	C₂³	—	172	2	no	—
280	C₂³	—	172	4	no	—
281	C₂³	C₂×#10621882	1	1	yes	characteristic
282	D₄	—	2	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	4	no	—
285	D₄	—	86	4	no	—
286	D₄	—	172	2	no	—
287	D₄	—	172	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	8	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	344	8	no	—
294	C₂×C₄	—	344	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
297	C₂×C₄	C₂×#10621882	1	2	yes	—
298	C₂×C₄	C₂×#10621882	1	2	yes	—
299	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
300	C₂×C₄	#13063345	1	2	yes	—
301	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
304	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
305	C₈	C₂×#10621898	1	1	yes	characteristic
306	C₂²	—	4	4	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	8	no	—
310	C₂²	—	344	2	no	—
311	C₂²	—	344	2	no	—
312	C₂²	—	344	8	no	—
313	C₂²	—	172	4	no	—
314	C₂²	C₄×#10621882	1	2	yes	—
315	C₂²	#14583334	1	2	yes	—
316	C₂²	#14583335	1	2	yes	—
317	C₂²	C₂²×#10621882	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	344	2	no	—
322	C₄	C₄×#10621882	1	2	yes	—
323	C₄	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
324	C₄	C₂×#13063345	1	1	yes	characteristic
325	C₂	—	8	4	no	—
326	C₂	—	344	4	no	—
327	C₂	#19136469	1	4	yes	direct product factor
328	C₂	C₂×C₄×#10621882	1	1	yes	characteristic
329	C₂	C₂×#14583334	1	1	yes	characteristic
330	C₂	C₂×#14583335	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄×D₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×#13063266	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₃
hypocenter	C₄₃	C₂×#2177
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₄₄	C₂	semidirect
3. #19136469	C₂	direct (N × H)
4. C₂×#14582726	C₂	semidirect
5. C₂×#14582700	C₂	semidirect
6. C₂×#14582699	C₂	semidirect
7. C₂×#14582675	C₂	semidirect
8. C₂×C₄×D₈×C₄₃	C₂	semidirect
11. C₂×#13063266	C₄	semidirect
16. C₄×D₃₄₄	C₂²	semidirect
19. #14582726	C₂²	semidirect
20. #14582700	C₂²	semidirect
21. #14582699	C₂²	semidirect
22. #14582675	C₂²	semidirect
26. C₄×D₈×C₄₃	C₂²	semidirect
34. C₂×D₃₄₄	C₂×C₄	semidirect
40. #13063266	C₂×C₄	semidirect
42. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
52. C₂×D₈×C₄₃	C₂×C₄	semidirect
87. D₈×C₄₃	C₂²×C₄	semidirect
104. D₃₄₄	C₂²×C₄	semidirect
106. C₄₃⋊C₁₆	C₂²×C₄	semidirect
162. C₂×C₄×D₈	D₄₃	semidirect
169. C₂×C₄₃	#2177	semidirect
185. C₄×D₈	C₂×D₄₃	semidirect
195. C₄₃	C₂×#2177	semidirect
206. C₂×D₈	C₄×D₄₃	semidirect
256. D₈	C₂×C₄×D₄₃	semidirect
327. C₂	#19136469	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
C₂×C₄×D₃₄₄	2
#19136469	2
C₂×#14582726	2
C₂×#14582700	2
C₂×#14582699	2
C₂×#14582675	2
C₂×C₄×D₈×C₄₃	2
C₂×#2177	43

Subgroups — C2×#19136469 #25249898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136469 #25249898