Subgroups — C₂×#19137193 #25255179

Order 11008.

4142 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137193	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14582792	C₂	1	8	yes	maximal, complemented
3	C₂×#14582759	C₂	1	2	yes	maximal, complemented
4	C₂×#14582755	C₂	1	2	yes	maximal
5	C₂×C₄×C₂²⋊Dic₄₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₃	C₂²	1	4	yes	complemented
8	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
9	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
11	C₂²×C₄×Dic₄₃	C₂²	1	4	yes	complemented
12	#14582792	C₂²	1	64	yes	complemented
13	#14582759	C₂²	1	16	yes	complemented
14	#14582755	C₂²	1	16	yes	—
15	C₄×C₂²⋊Dic₄₃	C₂²	1	16	yes	complemented
16	C₂×C₄₃⋊#95	C₂²	1	4	yes	complemented
17	C₂×C₄₃⋊#95	C₂²	1	4	yes	complemented
18	C₂⁶×C₄₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₄₃	C₄	1	16	yes	complemented
22	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
23	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	8	yes	—
24	C₂³×Dic₄₃	C₂³	1	8	yes	—
25	C₂×C₄⋊Dic₄₃	—	4	8	no	—
26	C₂×C₄⋊Dic₄₃	C₂³	1	16	yes	complemented
27	C₂×C₂²⋊Dic₄₃	—	2	16	no	—
28	C₂×C₂²⋊Dic₄₃	—	4	8	no	—
29	C₂×C₂²⋊Dic₄₃	—	4	8	no	—
30	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
31	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
32	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
33	C₂×C₄×Dic₄₃	—	4	8	no	—
34	C₂×C₄×Dic₄₃	C₂³	1	16	yes	complemented
35	C₄₃⋊#95	C₂³	1	16	yes	complemented
36	C₄₃⋊#95	C₂³	1	16	yes	complemented
37	C₂⁵×C₄₃	—	2	8	no	—
38	C₂⁵×C₄₃	—	4	4	no	—
39	C₂⁵×C₄₃	—	4	4	no	—
40	C₂⁵×C₄₃	C₂×C₄	1	8	yes	—
41	C₂⁵×C₄₃	C₂³	1	2	yes	—
42	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
43	C₂⁵×C₄₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₄₃	—	4	4	no	—
45	C₂³×C₄×C₄₃	—	4	4	no	—
46	C₂³×C₄×C₄₃	C₂×C₄	1	8	yes	—
47	C₂³×C₄×C₄₃	C₂×C₄	1	8	yes	—
48	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
51	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	32	yes	—
53	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	64	yes	complemented
54	C₂×C₂²⋊C₄×C₄₃	C₂³	1	16	yes	—
55	C₂⁴×C₄₃	—	2	4	no	—
56	C₂⁴×C₄₃	—	2	4	no	—
57	C₂⁴×C₄₃	—	2	32	no	—
58	C₂⁴×C₄₃	—	2	16	no	—
59	C₂⁴×C₄₃	—	2	8	no	—
60	C₂⁴×C₄₃	—	2	8	no	—
61	C₂⁴×C₄₃	—	2	8	no	—
62	C₂⁴×C₄₃	—	4	16	no	—
63	C₂⁴×C₄₃	—	4	16	no	—
64	C₂⁴×C₄₃	—	4	8	no	—
65	C₂⁴×C₄₃	—	4	8	no	—
66	C₂⁴×C₄₃	—	4	64	no	—
67	C₂⁴×C₄₃	C₂²×C₄	1	8	yes	—
68	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
69	C₂⁴×C₄₃	C₂²×C₄	1	16	yes	—
70	C₂⁴×C₄₃	#41	1	8	yes	—
71	C₂⁴×C₄₃	C₂⁴	1	4	yes	—
72	C₂⁴×C₄₃	C₂⁴	1	2	yes	—
73	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	—	2	8	no	—
75	C₂²×C₄×C₄₃	—	2	8	no	—
76	C₂²×C₄×C₄₃	—	4	16	no	—
77	C₂²×C₄×C₄₃	—	4	16	no	—
78	C₂²×C₄×C₄₃	—	4	8	no	—
79	C₂²×C₄×C₄₃	—	4	8	no	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
81	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
82	C₂²×C₄×C₄₃	C₂²×C₄	1	16	yes	—
83	C₂²×C₄×C₄₃	C₂²×C₄	1	16	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
86	C₂²×Dic₄₃	—	2	16	no	—
87	C₂²×Dic₄₃	—	2	16	no	—
88	C₂²×Dic₄₃	—	2	16	no	—
89	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
90	C₂²⋊C₄×C₄₃	C₂²×C₄	1	64	yes	—
91	C₂²⋊Dic₄₃	—	2	32	no	—
92	C₂²⋊Dic₄₃	—	4	16	no	—
93	C₂²⋊Dic₄₃	—	4	16	no	—
94	C₄⋊Dic₄₃	—	4	16	no	—
95	C₄×Dic₄₃	—	4	16	no	—
96	C₂³×C₄₃	—	2	8	no	—
97	C₂³×C₄₃	—	2	8	no	—
98	C₂³×C₄₃	—	2	32	no	—
99	C₂³×C₄₃	—	2	4	no	—
100	C₂³×C₄₃	—	2	4	no	—
101	C₂³×C₄₃	—	2	32	no	—
102	C₂³×C₄₃	—	2	16	no	—
103	C₂³×C₄₃	—	2	16	no	—
104	C₂³×C₄₃	—	2	16	no	—
105	C₂³×C₄₃	—	2	8	no	—
106	C₂³×C₄₃	—	2	8	no	—
107	C₂³×C₄₃	—	2	8	no	—
108	C₂³×C₄₃	—	4	16	no	—
109	C₂³×C₄₃	—	4	16	no	—
110	C₂³×C₄₃	—	4	128	no	—
111	C₂³×C₄₃	—	4	64	no	—
112	C₂³×C₄₃	—	4	32	no	—
113	C₂³×C₄₃	#117	1	16	yes	—
114	C₂³×C₄₃	C₂³×C₄	1	8	yes	—
115	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
116	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
117	C₂³×C₄₃	C₂×#41	1	8	yes	—
118	C₂³×C₄₃	C₂×#41	1	4	yes	—
119	C₂³×C₄₃	H(5,𝔽₂)	1	4	yes	—
120	C₂³×C₄₃	H(5,𝔽₂)	1	4	yes	—
121	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
122	C₂×C₄×C₄₃	—	2	16	no	—
123	C₂×C₄×C₄₃	—	2	16	no	—
124	C₂×C₄×C₄₃	—	2	8	no	—
125	C₂×C₄×C₄₃	—	2	8	no	—
126	C₂×C₄×C₄₃	—	4	16	no	—
127	C₂×C₄×C₄₃	—	4	16	no	—
128	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
129	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
130	C₂×Dic₄₃	—	2	16	no	—
131	C₂×Dic₄₃	—	2	16	no	—
132	C₂×Dic₄₃	—	2	16	no	—
133	C₂×Dic₄₃	—	4	32	no	—
134	C₂×#2312	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	32	no	—
136	C₂²×C₄₃	—	2	4	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	2	16	no	—
139	C₂²×C₄₃	—	2	8	no	—
140	C₂²×C₄₃	—	2	8	no	—
141	C₂²×C₄₃	—	2	8	no	—
142	C₂²×C₄₃	—	4	64	no	—
143	C₂²×C₄₃	—	4	32	no	—
144	C₂²×C₄₃	—	4	16	no	—
145	C₂²×C₄₃	C₂×#117	1	8	yes	—
146	C₂²×C₄₃	C₂×#117	1	4	yes	—
147	C₂²×C₄₃	#518	1	4	yes	—
148	C₂²×C₄₃	#518	1	4	yes	—
149	C₂²×C₄₃	#538	1	16	yes	—
150	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
151	C₂²×C₄₃	C₂²×#41	1	2	yes	—
152	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
153	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
154	C₄×C₄₃	—	2	8	no	—
155	C₄×C₄₃	—	2	8	no	—
156	Dic₄₃	—	4	16	no	—
157	C₂³×C₂²⋊C₄	D₄₃	1	1	yes	characteristic
158	#2312	—	43	16	no	—
159	C₂×#388	—	43	8	no	—
160	C₂×#380	—	43	2	no	—
161	C₂×#379	—	43	2	no	—
162	C₂×C₄×C₂²⋊C₄	—	43	2	no	—
163	C₂×C₄₃	—	2	8	no	—
164	C₂×C₄₃	—	4	8	no	—
165	C₂×C₄₃	#2312	1	8	yes	complemented
166	C₂×C₄₃	C₂²×#117	1	1	yes	characteristic
167	C₂×C₄₃	C₂×#518	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#518	1	1	yes	characteristic
169	C₂×C₄₃	C₂×#538	1	4	yes	—
170	C₂⁶	C₂×D₄₃	1	1	yes	characteristic
171	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
172	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	—	43	4	no	—
174	C₂²×C₂²⋊C₄	—	43	4	no	—
175	C₂²×C₂²⋊C₄	—	43	4	no	—
176	C₂²×C₂²⋊C₄	—	43	4	no	—
177	C₂²×C₂²⋊C₄	Dic₄₃	1	16	yes	complemented
178	C₂²×C₂²⋊C₄	C₂×D₄₃	1	4	yes	—
179	C₂²×C₂²⋊C₄	C₂×D₄₃	1	8	yes	—
180	C₂²×C₄²	—	43	4	no	—
181	#388	—	43	64	no	—
182	#380	—	43	16	no	—
183	#379	—	43	16	no	—
184	C₄×C₂²⋊C₄	—	43	16	no	—
185	C₂×#95	—	43	4	no	—
186	C₂×#95	—	43	4	no	—
187	C₄₃	C₂×#2312	1	1	yes	characteristic, complemented, Sylow
188	C₂⁵	—	2	8	no	—
189	C₂⁵	—	4	4	no	—
190	C₂⁵	—	4	4	no	—
191	C₂⁵	C₂×Dic₄₃	1	8	yes	—
192	C₂⁵	C₂²×D₄₃	1	2	yes	—
193	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
194	C₂⁵	C₂²×D₄₃	1	4	yes	—
195	C₂³×C₄	—	4	4	no	—
196	C₂³×C₄	—	4	4	no	—
197	C₂³×C₄	—	43	8	no	—
198	C₂³×C₄	C₂×Dic₄₃	1	8	yes	—
199	C₂³×C₄	C₂×Dic₄₃	1	8	yes	—
200	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
201	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
202	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
203	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
204	C₂×C₄⋊C₄	—	43	16	no	—
205	C₂×C₄⋊C₄	—	172	8	no	—
206	C₂×C₂²⋊C₄	—	43	16	no	—
207	C₂×C₂²⋊C₄	—	43	16	no	—
208	C₂×C₂²⋊C₄	—	43	16	no	—
209	C₂×C₂²⋊C₄	—	86	16	no	—
210	C₂×C₂²⋊C₄	—	172	8	no	—
211	C₂×C₂²⋊C₄	—	172	8	no	—
212	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	32	yes	—
213	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	64	yes	complemented
214	C₂×C₂²⋊C₄	C₂²×D₄₃	1	16	yes	—
215	C₂×C₄²	—	43	16	no	—
216	C₂×C₄²	—	172	8	no	—
217	#95	—	43	16	no	—
218	#95	—	43	16	no	—
219	C₂⁴	—	2	4	no	—
220	C₂⁴	—	2	4	no	—
221	C₂⁴	—	2	32	no	—
222	C₂⁴	—	2	16	no	—
223	C₂⁴	—	2	8	no	—
224	C₂⁴	—	2	8	no	—
225	C₂⁴	—	2	8	no	—
226	C₂⁴	—	4	8	no	—
227	C₂⁴	—	4	8	no	—
228	C₂⁴	—	4	16	no	—
229	C₂⁴	—	4	16	no	—
230	C₂⁴	—	4	64	no	—
231	C₂⁴	#10621898	1	8	yes	—
232	C₂⁴	C₂²×Dic₄₃	1	8	yes	—
233	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
234	C₂⁴	C₂²×Dic₄₃	1	16	yes	—
235	C₂⁴	C₂³×D₄₃	1	4	yes	—
236	C₂⁴	C₂³×D₄₃	1	2	yes	—
237	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	—	2	8	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	4	8	no	—
241	C₂²×C₄	—	4	8	no	—
242	C₂²×C₄	—	4	16	no	—
243	C₂²×C₄	—	4	16	no	—
244	C₂²×C₄	—	43	16	no	—
245	C₂²×C₄	—	86	16	no	—
246	C₂²×C₄	—	86	16	no	—
247	C₂²×C₄	—	86	16	no	—
248	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
249	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
250	C₂²×C₄	C₂²×Dic₄₃	1	16	yes	—
251	C₂²×C₄	C₂²×Dic₄₃	1	16	yes	—
252	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
253	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
254	C₄⋊C₄	—	172	16	no	—
255	C₂²⋊C₄	—	86	32	no	—
256	C₂²⋊C₄	—	172	16	no	—
257	C₂²⋊C₄	—	172	16	no	—
258	C₂²⋊C₄	C₂²×Dic₄₃	1	64	yes	—
259	C₄²	—	172	16	no	—
260	C₂³	—	2	8	no	—
261	C₂³	—	2	8	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	32	no	—
265	C₂³	—	2	32	no	—
266	C₂³	—	2	16	no	—
267	C₂³	—	2	16	no	—
268	C₂³	—	2	16	no	—
269	C₂³	—	2	8	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	2	8	no	—
272	C₂³	—	4	16	no	—
273	C₂³	—	4	16	no	—
274	C₂³	—	4	64	no	—
275	C₂³	—	4	128	no	—
276	C₂³	—	4	32	no	—
277	C₂³	#13063284	1	16	yes	—
278	C₂³	C₂³×Dic₄₃	1	8	yes	—
279	C₂³	C₂³×Dic₄₃	1	4	yes	—
280	C₂³	C₂³×Dic₄₃	1	2	yes	—
281	C₂³	C₂×#10621898	1	8	yes	—
282	C₂³	C₂×#10621898	1	4	yes	—
283	C₂³	#13063370	1	4	yes	—
284	C₂³	#13063370	1	4	yes	—
285	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
286	C₂×C₄	—	2	16	no	—
287	C₂×C₄	—	2	16	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	86	16	no	—
293	C₂×C₄	—	86	16	no	—
294	C₂×C₄	—	86	16	no	—
295	C₂×C₄	—	172	32	no	—
296	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
297	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	16	no	—
301	C₂²	—	2	32	no	—
302	C₂²	—	2	8	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	8	no	—
305	C₂²	—	4	64	no	—
306	C₂²	—	4	16	no	—
307	C₂²	—	4	32	no	—
308	C₂²	C₂×#13063284	1	8	yes	—
309	C₂²	C₂×#13063284	1	4	yes	—
310	C₂²	#14583374	1	4	yes	—
311	C₂²	#14583374	1	4	yes	—
312	C₂²	#14583487	1	16	yes	—
313	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
314	C₂²	C₂²×#10621898	1	2	yes	—
315	C₂²	C₂×#13063370	1	2	yes	—
316	C₂²	C₂×#13063370	1	2	yes	—
317	C₄	—	2	8	no	—
318	C₄	—	2	8	no	—
319	C₄	—	172	16	no	—
320	C₂	—	2	8	no	—
321	C₂	—	4	8	no	—
322	C₂	#19137193	1	8	yes	direct product factor
323	C₂	C₂²×#13063284	1	1	yes	characteristic
324	C₂	C₂×#14583374	1	1	yes	characteristic
325	C₂	C₂×#14583374	1	1	yes	characteristic
326	C₂	C₂×#14583487	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₃
Fitting	C₂³×C₂²⋊C₄×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂⁴
prime radical	C₂⁶×C₄₃	C₂²
cosocle	C₂³×C₄₃	C₂⁵
hypercenter	C₂³×C₂²⋊C₄	D₄₃
hypocenter	C₄₃	C₂×#2312
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #19137193	C₂	direct (N × H)
2. C₂×#14582792	C₂	semidirect
3. C₂×#14582759	C₂	semidirect
5. C₂×C₄×C₂²⋊Dic₄₃	C₂	semidirect
7. C₂²×C₄⋊Dic₄₃	C₂²	semidirect
11. C₂²×C₄×Dic₄₃	C₂²	semidirect
12. #14582792	C₂²	semidirect
13. #14582759	C₂²	semidirect
15. C₄×C₂²⋊Dic₄₃	C₂²	semidirect
16. C₂×C₄₃⋊#95	C₂²	semidirect
17. C₂×C₄₃⋊#95	C₂²	semidirect
21. C₂²×C₂²⋊C₄×C₄₃	C₄	semidirect
26. C₂×C₄⋊Dic₄₃	C₂³	semidirect
34. C₂×C₄×Dic₄₃	C₂³	semidirect
35. C₄₃⋊#95	C₂³	semidirect
36. C₄₃⋊#95	C₂³	semidirect
53. C₂×C₂²⋊C₄×C₄₃	C₂×C₄	semidirect
165. C₂×C₄₃	#2312	semidirect
177. C₂²×C₂²⋊C₄	Dic₄₃	semidirect
187. C₄₃	C₂×#2312	semidirect
213. C₂×C₂²⋊C₄	C₂×Dic₄₃	semidirect
322. C₂	#19137193	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19137193	2
C₂×#14582792	2
C₂×#14582759	2
C₂×#14582755	2
C₂×C₄×C₂²⋊Dic₄₃	2
C₂³×C₂²⋊C₄×C₄₃	2
C₂×#2312	43

Subgroups — C2×#19137193 #25255179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19137193 #25255179