Subgroups — C₂×#19148366 #26004773

Order 11008.

11210 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	C₂²×#13063373	C₂	1	2	yes	maximal, complemented
2	#19148366	C₂	1	32	yes	maximal, direct product factor
3	C₂×#14583571	C₂	1	4	yes	maximal, complemented
4	C₂×Q₈×D₁₇₂	C₂	1	4	yes	maximal, complemented
5	C₂×#14583561	C₂	1	8	yes	maximal, complemented
6	C₂×#14583480	C₂	1	6	yes	maximal, complemented
7	C₂×#14583449	C₂	1	6	yes	maximal, complemented
8	C₂²×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063373	—	2	32	no	—
10	C₂×#13063373	C₂²	1	32	yes	complemented
11	C₂²×#10621902	C₂²	1	8	yes	—
12	C₂²×#10621898	C₂²	1	12	yes	—
13	C₂²×Q₈×D₄₃	C₂²	1	8	yes	—
14	#14583571	C₂²	1	64	yes	complemented
15	Q₈×D₁₇₂	C₂²	1	64	yes	complemented
16	#14583561	C₂²	1	128	yes	complemented
17	#14583480	C₂²	1	96	yes	complemented
18	#14583449	C₂²	1	96	yes	complemented
19	C₂×#13063325	C₂²	1	6	yes	—
20	C₂×#13063322	C₂²	1	6	yes	—
21	C₂×#13063313	C₂²	1	12	yes	—
22	C₂×#13063305	C₂²	1	24	yes	—
23	C₂×#13063300	C₂²	1	24	yes	—
24	C₂×#13063299	C₂²	1	4	yes	—
25	C₂×#13063299	C₂²	1	12	yes	—
26	C₂×C₄×D₁₇₂	C₂²	1	12	yes	—
27	C₂×C₄×D₁₇₂	C₂²	1	12	yes	—
28	C₂³×Q₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₄₃	C₂²	1	4	yes	—
30	C₂×C₄×Q₈×C₄₃	C₂²	1	4	yes	—
31	C₂×C₄×Q₈×C₄₃	C₂²	1	16	yes	complemented
32	C₂²×C₄⋊C₄×C₄₃	C₂²	1	3	yes	—
33	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
34	#13063373	—	2	256	no	—
35	C₂×#10621902	—	2	64	no	—
36	C₂×#10621902	C₂³	1	64	yes	—
37	C₂×#10621898	—	2	96	no	—
38	C₂×#10621898	C₂³	1	96	yes	—
39	C₂²×Dic₈₆	C₂³	1	12	yes	—
40	C₂×Q₈×D₄₃	—	2	64	no	—
41	C₂×Q₈×D₄₃	C₂³	1	64	yes	—
42	C₂²×D₁₇₂	C₂³	1	4	yes	—
43	C₂²×D₁₇₂	C₂³	1	12	yes	—
44	C₂²×D₁₇₂	C₂³	1	12	yes	—
45	C₂²×C₂²⋊D₄₃	C₂³	1	8	yes	—
46	C₂²×C₄×D₄₃	C₂³	1	24	yes	—
47	#13063325	C₂³	1	48	yes	—
48	#13063322	C₂³	1	48	yes	—
49	#13063313	C₂³	1	96	yes	—
50	#13063305	C₂³	1	192	yes	—
51	#13063300	C₂³	1	192	yes	—
52	#13063299	C₂³	1	32	yes	—
53	#13063299	C₂³	1	96	yes	—
54	C₄×D₁₇₂	C₂³	1	96	yes	—
55	C₄×D₁₇₂	C₂³	1	96	yes	—
56	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
57	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
58	C₂×(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
59	C₂×(C₂×D₄₃)⋊C₄	C₂³	1	24	yes	—
60	C₂²×Q₈×C₄₃	D₄	1	16	yes	complemented
61	C₂²×Q₈×C₄₃	C₂³	1	4	yes	—
62	C₂²×Q₈×C₄₃	C₂³	1	8	yes	—
63	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
65	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
66	C₄×Q₈×C₄₃	C₂³	1	32	yes	—
67	C₄×Q₈×C₄₃	C₂³	1	32	yes	—
68	C₂×C₄⋊C₄×C₄₃	C₂³	1	12	yes	—
69	C₂×C₄⋊C₄×C₄₃	C₂³	1	24	yes	—
70	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
71	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
72	C₂×C₄²×C₄₃	C₂³	1	24	yes	—
73	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂³×D₄₃	C₂⁴	1	8	yes	—
75	C₂×Q₈×C₄₃	C₂×D₄	1	32	yes	—
76	C₂×Q₈×C₄₃	C₂×D₄	1	64	yes	complemented
77	C₂×Q₈×C₄₃	C₂⁴	1	16	yes	—
78	C₂²×C₄×C₄₃	—	2	4	no	—
79	C₂²×C₄×C₄₃	—	4	6	no	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	24	yes	—
81	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
86	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
87	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₂²⋊D₄₃	—	2	32	no	—
89	C₂×C₂²⋊D₄₃	C₂⁴	1	32	yes	—
90	C₂²×Dic₄₃	C₂⁴	1	8	yes	—
91	#10621902	—	2	256	no	—
92	Q₈×D₄₃	—	2	256	no	—
93	#10621898	—	2	384	no	—
94	C₂×D₁₇₂	—	2	48	no	—
95	C₂×D₁₇₂	—	2	16	no	—
96	C₂×D₁₇₂	—	4	24	no	—
97	C₂×D₁₇₂	C₂⁴	1	16	yes	—
98	C₂×D₁₇₂	C₂⁴	1	48	yes	—
99	C₂×D₁₇₂	C₂⁴	1	48	yes	—
100	C₂×C₄×D₄₃	—	2	96	no	—
101	C₂×C₄×D₄₃	C₂⁴	1	96	yes	—
102	C₂×Dic₈₆	—	2	48	no	—
103	C₂×Dic₈₆	C₂⁴	1	48	yes	—
104	C₄⋊C₄×C₄₃	C₂⁴	1	48	yes	—
105	C₄²×C₄₃	C₂⁴	1	24	yes	—
106	C₄²×C₄₃	C₂⁴	1	24	yes	—
107	(C₂×D₄₃)⋊C₄	C₂⁴	1	32	yes	—
108	(C₂×D₄₃)⋊C₄	C₂⁴	1	96	yes	—
109	C₄⋊Dic₄₃	C₂⁴	1	16	yes	—
110	C₄⋊Dic₄₃	C₂⁴	1	48	yes	—
111	C₂³×C₄₃	—	2	2	no	—
112	C₂³×C₄₃	—	2	2	no	—
113	C₂³×C₄₃	C₂²×D₄	1	4	yes	—
114	C₂³×C₄₃	C₂⁵	1	2	yes	—
115	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
116	C₂²×D₄₃	—	2	16	no	—
117	C₂²×D₄₃	—	2	16	no	—
118	C₂²×D₄₃	—	2	16	no	—
119	C₂²×D₄₃	C₂⁵	1	16	yes	—
120	Q₈×C₄₃	C₂²×D₄	1	64	yes	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	12	no	—
123	C₂×C₄×C₄₃	—	2	8	no	—
124	C₂×C₄×C₄₃	—	4	12	no	—
125	C₂×C₄×C₄₃	C₂²×D₄	1	24	yes	—
126	C₂×C₄×C₄₃	C₂²×D₄	1	48	yes	—
127	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	complemented
128	C₂×C₄×C₄₃	C₂⁵	1	12	yes	—
129	C₂×C₄×C₄₃	C₂⁵	1	2	yes	—
130	C₂×C₄×C₄₃	C₂⁵	1	12	yes	—
131	C₂×C₄×C₄₃	C₂⁵	1	2	yes	—
132	C₂²⋊D₄₃	—	2	64	no	—
133	C₂×Dic₄₃	—	2	16	no	—
134	C₂×Dic₄₃	C₂⁵	1	16	yes	—
135	D₁₇₂	—	2	96	no	—
136	D₁₇₂	—	2	32	no	—
137	D₁₇₂	—	4	48	no	—
138	C₄×D₄₃	—	2	192	no	—
139	Dic₈₆	—	2	96	no	—
140	C₂×#3497	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	2	8	no	—
144	C₂²×C₄₃	C₂³×D₄	1	4	yes	—
145	C₂²×C₄₃	C₂³×D₄	1	2	yes	—
146	C₂²×C₄₃	C₂×2₋¹⁺⁴	1	2	yes	—
147	C₂²×C₄₃	#585	1	2	yes	—
148	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
149	C₂×D₄₃	—	2	16	no	—
150	C₂×D₄₃	—	2	16	no	—
151	C₂×D₄₃	—	2	16	no	—
152	C₂×D₄₃	—	4	32	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	2	12	no	—
155	C₄×C₄₃	C₂³×D₄	1	24	yes	—
156	C₄×C₄₃	C₂×2₋¹⁺⁴	1	4	yes	complemented
157	Dic₄₃	—	2	16	no	—
158	C₂²×2₋¹⁺⁴	—	43	2	no	—
159	#3497	—	43	32	no	—
160	C₂×#550	—	43	4	no	—
161	C₂×D₄×Q₈	—	43	4	no	—
162	C₂×#548	—	43	8	no	—
163	C₂×#537	—	43	6	no	—
164	C₂×#532	—	43	6	no	—
165	C₂²×C₄×Q₈	D₄₃	1	1	yes	characteristic, complemented
166	C₂×C₄₃	—	2	4	no	—
167	C₂×C₄₃	#3497	1	4	yes	complemented
168	C₂×C₄₃	C₂⁴×D₄	1	1	yes	characteristic
169	C₂×C₄₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
170	C₂×C₄₃	C₂×#585	1	1	yes	characteristic
171	D₄₃	—	4	16	no	—
172	C₂×2₋¹⁺⁴	—	43	32	no	—
173	C₂×2₋¹⁺⁴	—	86	32	no	—
174	C₂²×#41	—	43	8	no	—
175	C₂²×#41	—	43	12	no	—
176	C₂³×Q₈	—	43	8	no	—
177	C₂³×Q₈	C₂×D₄₃	1	1	yes	characteristic
178	#550	—	43	64	no	—
179	D₄×Q₈	—	43	64	no	—
180	#548	—	43	128	no	—
181	#537	—	43	96	no	—
182	#532	—	43	96	no	—
183	C₂×C₄⋊Q₈	—	43	6	no	—
184	C₂×#127	—	43	6	no	—
185	C₂×#124	—	43	12	no	—
186	C₂×C₂²⋊Q₈	—	43	24	no	—
187	C₂×#121	—	43	4	no	—
188	C₂×#121	—	43	12	no	—
189	C₂×#121	—	43	24	no	—
190	C₂×C₄×Q₈	C₂×D₄₃	1	4	yes	—
191	C₂×C₄×Q₈	C₂×D₄₃	1	4	yes	—
192	C₂×C₄×Q₈	C₂×D₄₃	1	16	yes	complemented
193	C₂×C₄×D₄	—	43	12	no	—
194	C₂×C₄×D₄	—	43	12	no	—
195	C₂²×C₄⋊C₄	C₂×D₄₃	1	3	yes	—
196	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
197	C₄₃	C₂×#3497	1	1	yes	characteristic, complemented, Sylow
198	2₋¹⁺⁴	—	86	256	no	—
199	C₂×#41	—	43	64	no	—
200	C₂×#41	—	43	96	no	—
201	C₂×#41	—	86	64	no	—
202	C₂×#41	—	86	96	no	—
203	C₂²×Q₈	—	43	64	no	—
204	C₂²×Q₈	—	43	12	no	—
205	C₂²×Q₈	—	86	64	no	—
206	C₂²×Q₈	D₁₇₂	1	16	yes	complemented
207	C₂²×Q₈	C₂²×D₄₃	1	4	yes	—
208	C₂²×Q₈	C₂²×D₄₃	1	8	yes	—
209	C₂²×D₄	—	43	8	no	—
210	C₂²×D₄	—	43	4	no	—
211	C₂²×D₄	—	43	12	no	—
212	C₂²×D₄	—	43	12	no	—
213	C₂³×C₄	—	43	24	no	—
214	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
215	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
216	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
217	C₄⋊Q₈	—	43	48	no	—
218	#127	—	43	48	no	—
219	#124	—	43	96	no	—
220	C₂²⋊Q₈	—	43	192	no	—
221	#121	—	43	32	no	—
222	#121	—	43	96	no	—
223	#121	—	43	192	no	—
224	C₄×Q₈	C₂²×D₄₃	1	32	yes	—
225	C₄×Q₈	C₂²×D₄₃	1	32	yes	—
226	C₄×D₄	—	43	96	no	—
227	C₄×D₄	—	43	96	no	—
228	C₂×C₄⋊C₄	—	43	4	no	—
229	C₂×C₄⋊C₄	—	43	12	no	—
230	C₂×C₄⋊C₄	C₂²×D₄₃	1	12	yes	—
231	C₂×C₄⋊C₄	C₂²×D₄₃	1	24	yes	—
232	C₂×C₂²⋊C₄	—	43	8	no	—
233	C₂×C₂²⋊C₄	—	43	24	no	—
234	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
235	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
236	C₂×C₄²	C₂²×D₄₃	1	24	yes	—
237	C₂⁴	—	43	8	no	—
238	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
239	#41	—	86	256	no	—
240	#41	—	86	384	no	—
241	C₂×Q₈	—	43	48	no	—
242	C₂×Q₈	—	86	256	no	—
243	C₂×Q₈	—	86	48	no	—
244	C₂×Q₈	C₂×D₁₇₂	1	32	yes	—
245	C₂×Q₈	C₂×D₁₇₂	1	64	yes	complemented
246	C₂×Q₈	C₂³×D₄₃	1	16	yes	—
247	C₂×D₄	—	43	32	no	—
248	C₂×D₄	—	43	16	no	—
249	C₂×D₄	—	43	48	no	—
250	C₂×D₄	—	43	48	no	—
251	C₂×D₄	—	86	32	no	—
252	C₂×D₄	—	86	48	no	—
253	C₂×D₄	—	86	16	no	—
254	C₂×D₄	—	172	24	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	4	6	no	—
257	C₂²×C₄	—	43	8	no	—
258	C₂²×C₄	—	43	96	no	—
259	C₂²×C₄	—	86	96	no	—
260	C₂²×C₄	C₂×D₁₇₂	1	24	yes	—
261	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
262	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
264	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
265	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
266	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
267	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
268	C₄⋊C₄	—	43	16	no	—
269	C₄⋊C₄	—	43	48	no	—
270	C₄⋊C₄	C₂³×D₄₃	1	48	yes	—
271	C₂²⋊C₄	—	43	32	no	—
272	C₂²⋊C₄	—	43	96	no	—
273	C₄²	C₂³×D₄₃	1	24	yes	—
274	C₄²	C₂³×D₄₃	1	24	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	43	16	no	—
278	C₂³	—	86	16	no	—
279	C₂³	—	86	16	no	—
280	C₂³	—	86	16	no	—
281	C₂³	C₂²×D₁₇₂	1	4	yes	—
282	C₂³	C₂⁴×D₄₃	1	2	yes	—
283	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
284	Q₈	—	86	96	no	—
285	Q₈	C₂²×D₁₇₂	1	64	yes	—
286	D₄	—	86	64	no	—
287	D₄	—	86	96	no	—
288	D₄	—	86	32	no	—
289	D₄	—	172	48	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	12	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	4	12	no	—
294	C₂×C₄	—	43	16	no	—
295	C₂×C₄	—	86	16	no	—
296	C₂×C₄	—	86	192	no	—
297	C₂×C₄	C₂²×D₁₇₂	1	24	yes	—
298	C₂×C₄	C₂²×D₁₇₂	1	48	yes	—
299	C₂×C₄	#13063373	1	4	yes	complemented
300	C₂×C₄	C₂⁴×D₄₃	1	2	yes	—
301	C₂×C₄	C₂⁴×D₄₃	1	12	yes	—
302	C₂×C₄	C₂⁴×D₄₃	1	12	yes	—
303	C₂×C₄	C₂⁴×D₄₃	1	2	yes	—
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	86	16	no	—
308	C₂²	—	86	16	no	—
309	C₂²	—	86	16	no	—
310	C₂²	—	172	32	no	—
311	C₂²	C₂³×D₁₇₂	1	4	yes	—
312	C₂²	C₂³×D₁₇₂	1	2	yes	—
313	C₂²	C₂×#13063373	1	2	yes	—
314	C₂²	#14583735	1	2	yes	—
315	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	12	no	—
318	C₄	—	86	16	no	—
319	C₄	C₂³×D₁₇₂	1	24	yes	—
320	C₄	C₂×#13063373	1	4	yes	complemented
321	C₂	—	2	4	no	—
322	C₂	—	172	16	no	—
323	C₂	#19148366	1	4	yes	direct product factor
324	C₂	C₂⁴×D₁₇₂	1	1	yes	characteristic
325	C₂	C₂²×#13063373	1	1	yes	characteristic
326	C₂	C₂×#14583735	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₄₃
commutator, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂²×C₄×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₄₃
hypocenter	C₄₃	C₂×#3497
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063373	C₂	semidirect
2. #19148366	C₂	direct (N × H)
3. C₂×#14583571	C₂	semidirect
4. C₂×Q₈×D₁₇₂	C₂	semidirect
5. C₂×#14583561	C₂	semidirect
6. C₂×#14583480	C₂	semidirect
7. C₂×#14583449	C₂	semidirect
8. C₂²×C₄×Q₈×C₄₃	C₂	semidirect
10. C₂×#13063373	C₂²	semidirect
14. #14583571	C₂²	semidirect
15. Q₈×D₁₇₂	C₂²	semidirect
16. #14583561	C₂²	semidirect
17. #14583480	C₂²	semidirect
18. #14583449	C₂²	semidirect
31. C₂×C₄×Q₈×C₄₃	C₂²	semidirect
60. C₂²×Q₈×C₄₃	D₄	semidirect
76. C₂×Q₈×C₄₃	C₂×D₄	semidirect
127. C₂×C₄×C₄₃	2₋¹⁺⁴	semidirect
156. C₄×C₄₃	C₂×2₋¹⁺⁴	semidirect
165. C₂²×C₄×Q₈	D₄₃	semidirect
167. C₂×C₄₃	#3497	semidirect
192. C₂×C₄×Q₈	C₂×D₄₃	semidirect
197. C₄₃	C₂×#3497	semidirect
206. C₂²×Q₈	D₁₇₂	semidirect
245. C₂×Q₈	C₂×D₁₇₂	semidirect
299. C₂×C₄	#13063373	semidirect
320. C₄	C₂×#13063373	semidirect
323. C₂	#19148366	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×#13063373	2
#19148366	2
C₂×#14583571	2
C₂×Q₈×D₁₇₂	2
C₂×#14583561	2
C₂×#14583480	2
C₂×#14583449	2
C₂²×C₄×Q₈×C₄₃	2
C₂×#3497	43

Subgroups — C2×#19148366 #26004773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19148366 #26004773