Subgroups — C₂×#19148380 #26005012

Order 11008.

11126 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063372	C₂	1	2	yes	maximal, complemented
2	#19148380	C₂	1	32	yes	maximal, direct product factor
3	C₂×Q₈×D₁₇₂	C₂	1	4	yes	maximal, complemented
4	C₂×#14583553	C₂	1	12	yes	maximal, complemented
5	C₂×#14583475	C₂	1	6	yes	maximal, complemented
6	C₂×#14583449	C₂	1	3	yes	maximal, complemented
7	C₂²×#13063325	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×#41×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063372	—	2	32	no	—
10	C₂×#13063372	C₂²	1	32	yes	complemented
11	C₂²×#10621898	C₂²	1	6	yes	—
12	C₂²×#10621900	C₂²	1	12	yes	—
13	C₂³×Dic₈₆	C₂²	1	6	yes	—
14	C₂²×Q₈×D₄₃	C₂²	1	4	yes	—
15	Q₈×D₁₇₂	C₂²	1	64	yes	complemented
16	#14583553	C₂²	1	192	yes	complemented
17	#14583475	C₂²	1	96	yes	complemented
18	#14583449	C₂²	1	48	yes	complemented
19	C₂×#13063325	C₂²	1	12	yes	—
20	C₂×#13063325	C₂²	1	3	yes	—
21	C₂×#13063325	C₂²	1	6	yes	—
22	C₂×#13063325	C₂²	1	48	yes	complemented
23	C₂×#13063322	C₂²	1	1	yes	characteristic
24	C₂×#13063313	C₂²	1	6	yes	—
25	C₂×#13063309	C₂²	1	24	yes	—
26	C₂×#13063305	C₂²	1	24	yes	—
27	C₂×#13063299	C₂²	1	12	yes	—
28	C₂×C₄×D₁₇₂	C₂²	1	12	yes	—
29	C₂²×C₄⋊Dic₄₃	C₂²	1	12	yes	—
30	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
31	C₄×#41×C₄₃	C₂²	1	16	yes	complemented
32	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₄₃	C₂²	1	6	yes	—
34	C₂×#117×C₄₃	C₂²	1	3	yes	—
35	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
36	#13063372	—	2	256	no	—
37	C₂×#10621898	—	2	48	no	—
38	C₂×#10621898	C₂³	1	48	yes	—
39	C₂×#10621900	—	2	96	no	—
40	C₂×#10621900	C₂³	1	96	yes	—
41	C₂²×Dic₈₆	—	2	48	no	—
42	C₂²×Dic₈₆	C₂³	1	12	yes	—
43	C₂²×Dic₈₆	C₂³	1	6	yes	—
44	C₂²×Dic₈₆	C₂³	1	48	yes	—
45	C₂×Q₈×D₄₃	—	2	32	no	—
46	C₂×Q₈×D₄₃	C₂³	1	32	yes	—
47	C₂²×D₁₇₂	C₂³	1	4	yes	—
48	C₂²×D₁₇₂	C₂³	1	2	yes	—
49	C₂²×C₂²⋊D₄₃	C₂³	1	12	yes	—
50	C₂³×Dic₄₃	C₂³	1	12	yes	—
51	C₂²×C₄×D₄₃	C₂³	1	12	yes	—
52	#13063325	C₂³	1	96	yes	—
53	#13063325	C₂³	1	24	yes	—
54	#13063325	C₂³	1	48	yes	—
55	#13063322	C₂³	1	8	yes	—
56	#13063313	C₂³	1	48	yes	—
57	#13063309	C₂³	1	192	yes	—
58	#13063305	C₂³	1	192	yes	—
59	#13063299	C₂³	1	96	yes	—
60	C₄×D₁₇₂	C₂³	1	96	yes	—
61	C₂×C₄⋊Dic₄₃	C₂³	1	96	yes	—
62	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
63	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
64	C₂×(C₂×D₄₃)⋊C₄	C₂³	1	24	yes	—
65	C₂×#41×C₄₃	D₄	1	16	yes	complemented
66	C₂×#41×C₄₃	C₂³	1	8	yes	—
67	C₂²×Q₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₄₃	C₂³	1	3	yes	—
69	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
70	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
71	C₄×Q₈×C₄₃	C₂³	1	16	yes	—
72	C₄×D₄×C₄₃	C₂³	1	48	yes	—
73	#117×C₄₃	C₂³	1	24	yes	—
74	C₂×C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
75	C₂×C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
76	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
78	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
79	C₂×C₄²×C₄₃	C₂³	1	24	yes	—
80	C₂⁴×C₄₃	C₂⁴	1	3	yes	—
81	C₂³×D₄₃	C₂⁴	1	4	yes	—
82	#41×C₄₃	C₂×D₄	1	64	yes	complemented
83	C₂×Q₈×C₄₃	C₂×D₄	1	8	yes	—
84	C₂×Q₈×C₄₃	C₂⁴	1	4	yes	—
85	C₂×D₄×C₄₃	C₂×D₄	1	24	yes	—
86	C₂×D₄×C₄₃	C₂⁴	1	12	yes	—
87	C₂²×C₄×C₄₃	—	2	24	no	—
88	C₂²×C₄×C₄₃	C₂×D₄	1	24	yes	—
89	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
90	C₂²×C₄×C₄₃	C₂⁴	1	24	yes	—
91	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
93	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
94	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
95	C₂×C₂²⋊D₄₃	—	2	48	no	—
96	C₂×C₂²⋊D₄₃	C₂⁴	1	48	yes	—
97	C₂²×Dic₄₃	—	2	48	no	—
98	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
99	C₂²×Dic₄₃	C₂⁴	1	48	yes	—
100	Q₈×D₄₃	—	2	128	no	—
101	#10621900	—	2	384	no	—
102	#10621898	—	2	192	no	—
103	C₂×D₁₇₂	—	2	8	no	—
104	C₂×D₁₇₂	—	2	16	no	—
105	C₂×D₁₇₂	C₂⁴	1	16	yes	—
106	C₂×D₁₇₂	C₂⁴	1	8	yes	—
107	C₂×C₄×D₄₃	—	2	48	no	—
108	C₂×C₄×D₄₃	C₂⁴	1	48	yes	—
109	C₂×Dic₈₆	—	2	48	no	—
110	C₂×Dic₈₆	—	2	24	no	—
111	C₂×Dic₈₆	—	2	192	no	—
112	C₂×Dic₈₆	C₂⁴	1	48	yes	—
113	C₂×Dic₈₆	C₂⁴	1	24	yes	—
114	C₄⋊C₄×C₄₃	C₂⁴	1	24	yes	—
115	C₂²⋊C₄×C₄₃	C₂⁴	1	24	yes	—
116	C₄²×C₄₃	C₂⁴	1	4	yes	—
117	C₄²×C₄₃	C₂⁴	1	12	yes	—
118	C₄²×C₄₃	C₂⁴	1	24	yes	—
119	(C₂×D₄₃)⋊C₄	C₂⁴	1	96	yes	—
120	C₄⋊Dic₄₃	C₂⁴	1	48	yes	—
121	C₄⋊Dic₄₃	C₂⁴	1	96	yes	—
122	C₂³×C₄₃	—	2	12	no	—
123	C₂³×C₄₃	C₂²×D₄	1	12	yes	—
124	C₂³×C₄₃	C₂⁵	1	6	yes	—
125	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
126	C₂²×D₄₃	—	2	8	no	—
127	C₂²×D₄₃	—	2	16	no	—
128	C₂²×D₄₃	C₂⁵	1	8	yes	—
129	Q₈×C₄₃	C₂²×D₄	1	16	yes	—
130	D₄×C₄₃	C₂²×D₄	1	48	yes	—
131	C₂×C₄×C₄₃	—	2	12	no	—
132	C₂×C₄×C₄₃	—	2	48	no	—
133	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
134	C₂×C₄×C₄₃	C₂²×D₄	1	12	yes	—
135	C₂×C₄×C₄₃	C₂²×D₄	1	48	yes	—
136	C₂×C₄×C₄₃	2₋¹⁺⁴	1	8	yes	complemented
137	C₂×C₄×C₄₃	C₂⁵	1	12	yes	—
138	C₂×C₄×C₄₃	C₂⁵	1	2	yes	—
139	C₂×C₄×C₄₃	C₂⁵	1	6	yes	—
140	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
141	C₂²⋊D₄₃	—	2	96	no	—
142	C₂×Dic₄₃	—	2	24	no	—
143	C₂×Dic₄₃	—	2	96	no	—
144	C₂×Dic₄₃	C₂⁵	1	24	yes	—
145	D₁₇₂	—	2	16	no	—
146	D₁₇₂	—	2	32	no	—
147	C₄×D₄₃	—	2	96	no	—
148	Dic₈₆	—	2	48	no	—
149	Dic₈₆	—	2	96	no	—
150	C₂×#3499	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	—	2	12	no	—
152	C₂²×C₄₃	—	2	24	no	—
153	C₂²×C₄₃	C₂³×D₄	1	12	yes	—
154	C₂²×C₄₃	C₂³×D₄	1	2	yes	—
155	C₂²×C₄₃	C₂×2₋¹⁺⁴	1	4	yes	—
156	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
157	C₂×D₄₃	—	2	8	no	—
158	C₂×D₄₃	—	2	16	no	—
159	C₂×D₄₃	—	4	16	no	—
160	C₄×C₄₃	—	2	12	no	—
161	C₄×C₄₃	C₂³×D₄	1	4	yes	—
162	C₄×C₄₃	C₂³×D₄	1	12	yes	—
163	C₄×C₄₃	C₂×2₋¹⁺⁴	1	8	yes	complemented
164	Dic₄₃	—	2	24	no	—
165	C₂²×2₋¹⁺⁴	—	43	2	no	—
166	#3499	—	43	32	no	—
167	C₂×D₄×Q₈	—	43	4	no	—
168	C₂×#547	—	43	12	no	—
169	C₂×#536	—	43	6	no	—
170	C₂×#532	—	43	3	no	—
171	C₂²×C₄⋊Q₈	—	43	3	no	—
172	C₂×C₄×#41	D₄₃	1	1	yes	characteristic, complemented
173	C₂×C₄₃	—	2	12	no	—
174	C₂×C₄₃	#3499	1	4	yes	complemented
175	C₂×C₄₃	C₂⁴×D₄	1	1	yes	characteristic
176	C₂×C₄₃	C₂²×2₋¹⁺⁴	1	2	yes	—
177	D₄₃	—	4	8	no	—
178	C₂×2₋¹⁺⁴	—	43	32	no	—
179	C₂×2₋¹⁺⁴	—	86	32	no	—
180	C₂²×#41	—	43	12	no	—
181	C₂²×#41	—	43	6	no	—
182	C₂²×#41	C₂×D₄₃	1	1	yes	characteristic
183	C₂³×Q₈	—	43	4	no	—
184	C₂³×Q₈	—	43	6	no	—
185	D₄×Q₈	—	43	64	no	—
186	#547	—	43	192	no	—
187	#536	—	43	96	no	—
188	#532	—	43	48	no	—
189	C₂×C₄⋊Q₈	—	43	3	no	—
190	C₂×C₄⋊Q₈	—	43	6	no	—
191	C₂×C₄⋊Q₈	—	43	12	no	—
192	C₂×C₄⋊Q₈	—	43	48	no	—
193	C₂×#127	—	43	1	no	—
194	C₂×#124	—	43	6	no	—
195	C₂×#123	—	43	24	no	—
196	C₂×C₂²⋊Q₈	—	43	24	no	—
197	C₂×#121	—	43	12	no	—
198	C₄×#41	C₂×D₄₃	1	16	yes	complemented
199	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
200	C₂×C₄×D₄	—	43	12	no	—
201	C₂×C₄×D₄	C₂×D₄₃	1	6	yes	—
202	C₂×#117	C₂×D₄₃	1	3	yes	—
203	C₂²×C₄⋊C₄	—	43	12	no	—
204	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
205	C₄₃	C₂×#3499	1	1	yes	characteristic, complemented, Sylow
206	2₋¹⁺⁴	—	86	256	no	—
207	C₂×#41	—	43	96	no	—
208	C₂×#41	—	43	48	no	—
209	C₂×#41	—	86	96	no	—
210	C₂×#41	—	86	48	no	—
211	C₂×#41	D₁₇₂	1	16	yes	complemented
212	C₂×#41	C₂²×D₄₃	1	8	yes	—
213	C₂²×Q₈	—	43	32	no	—
214	C₂²×Q₈	—	43	12	no	—
215	C₂²×Q₈	—	43	6	no	—
216	C₂²×Q₈	—	43	48	no	—
217	C₂²×Q₈	—	86	32	no	—
218	C₂²×Q₈	—	86	48	no	—
219	C₂²×Q₈	C₂²×D₄₃	1	1	yes	characteristic
220	C₂²×D₄	—	43	12	no	—
221	C₂²×D₄	—	43	4	no	—
222	C₂²×D₄	—	43	2	no	—
223	C₂²×D₄	C₂²×D₄₃	1	3	yes	—
224	C₂³×C₄	—	43	12	no	—
225	C₂³×C₄	—	43	12	no	—
226	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
227	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
228	C₄⋊Q₈	—	43	24	no	—
229	C₄⋊Q₈	—	43	48	no	—
230	C₄⋊Q₈	—	43	96	no	—
231	#127	—	43	8	no	—
232	#124	—	43	48	no	—
233	#123	—	43	192	no	—
234	C₂²⋊Q₈	—	43	192	no	—
235	#121	—	43	96	no	—
236	C₄×Q₈	C₂²×D₄₃	1	16	yes	—
237	C₄×D₄	—	43	96	no	—
238	C₄×D₄	C₂²×D₄₃	1	48	yes	—
239	#117	C₂²×D₄₃	1	24	yes	—
240	C₂×C₄⋊C₄	—	43	12	no	—
241	C₂×C₄⋊C₄	—	43	24	no	—
242	C₂×C₄⋊C₄	—	43	96	no	—
243	C₂×C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
244	C₂×C₂²⋊C₄	—	43	24	no	—
245	C₂×C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
246	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
247	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
248	C₂×C₄²	C₂²×D₄₃	1	24	yes	—
249	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
250	C₂⁴	—	43	4	no	—
251	C₂⁴	C₂³×D₄₃	1	3	yes	—
252	#41	—	86	384	no	—
253	#41	—	86	192	no	—
254	#41	C₂×D₁₇₂	1	64	yes	complemented
255	C₂×Q₈	—	43	48	no	—
256	C₂×Q₈	—	43	24	no	—
257	C₂×Q₈	—	86	128	no	—
258	C₂×Q₈	—	86	48	no	—
259	C₂×Q₈	—	86	24	no	—
260	C₂×Q₈	—	86	192	no	—
261	C₂×Q₈	C₂×D₁₇₂	1	8	yes	—
262	C₂×Q₈	C₂³×D₄₃	1	4	yes	—
263	C₂×D₄	—	43	48	no	—
264	C₂×D₄	—	43	16	no	—
265	C₂×D₄	—	43	8	no	—
266	C₂×D₄	—	86	48	no	—
267	C₂×D₄	—	86	8	no	—
268	C₂×D₄	—	86	16	no	—
269	C₂×D₄	C₂×D₁₇₂	1	24	yes	—
270	C₂×D₄	C₂³×D₄₃	1	12	yes	—
271	C₂²×C₄	—	2	24	no	—
272	C₂²×C₄	—	43	48	no	—
273	C₂²×C₄	—	43	12	no	—
274	C₂²×C₄	—	43	48	no	—
275	C₂²×C₄	—	86	48	no	—
276	C₂²×C₄	—	86	48	no	—
277	C₂²×C₄	C₂×D₁₇₂	1	24	yes	—
278	C₂²×C₄	C₂³×D₄₃	1	24	yes	—
279	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
280	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
281	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
282	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
283	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
284	C₄⋊C₄	—	43	48	no	—
285	C₄⋊C₄	—	43	96	no	—
286	C₄⋊C₄	C₂³×D₄₃	1	24	yes	—
287	C₂²⋊C₄	—	43	96	no	—
288	C₂²⋊C₄	C₂³×D₄₃	1	24	yes	—
289	C₄²	C₂³×D₄₃	1	4	yes	—
290	C₄²	C₂³×D₄₃	1	12	yes	—
291	C₄²	C₂³×D₄₃	1	24	yes	—
292	C₂³	—	2	12	no	—
293	C₂³	—	43	8	no	—
294	C₂³	—	86	8	no	—
295	C₂³	—	86	16	no	—
296	C₂³	C₂²×D₁₇₂	1	12	yes	—
297	C₂³	C₂⁴×D₄₃	1	6	yes	—
298	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
299	Q₈	—	86	96	no	—
300	Q₈	—	86	48	no	—
301	Q₈	C₂²×D₁₇₂	1	16	yes	—
302	D₄	—	86	96	no	—
303	D₄	—	86	16	no	—
304	D₄	—	86	32	no	—
305	D₄	C₂²×D₁₇₂	1	48	yes	—
306	C₂×C₄	—	2	12	no	—
307	C₂×C₄	—	2	48	no	—
308	C₂×C₄	—	43	24	no	—
309	C₂×C₄	—	86	24	no	—
310	C₂×C₄	—	86	96	no	—
311	C₂×C₄	—	86	96	no	—
312	C₂×C₄	C₂²×D₁₇₂	1	12	yes	—
313	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
314	C₂×C₄	C₂²×D₁₇₂	1	48	yes	—
315	C₂×C₄	#13063372	1	8	yes	complemented
316	C₂×C₄	C₂⁴×D₄₃	1	12	yes	—
317	C₂×C₄	C₂⁴×D₄₃	1	2	yes	—
318	C₂×C₄	C₂⁴×D₄₃	1	6	yes	—
319	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
320	C₂²	—	2	12	no	—
321	C₂²	—	2	24	no	—
322	C₂²	—	86	8	no	—
323	C₂²	—	86	16	no	—
324	C₂²	—	172	16	no	—
325	C₂²	C₂³×D₁₇₂	1	12	yes	—
326	C₂²	C₂³×D₁₇₂	1	2	yes	—
327	C₂²	C₂×#13063372	1	4	yes	—
328	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
329	C₄	—	2	12	no	—
330	C₄	—	86	24	no	—
331	C₄	C₂³×D₁₇₂	1	12	yes	—
332	C₄	C₂³×D₁₇₂	1	4	yes	—
333	C₄	C₂×#13063372	1	8	yes	complemented
334	C₂	—	2	12	no	—
335	C₂	—	172	8	no	—
336	C₂	#19148380	1	4	yes	direct product factor
337	C₂	C₂⁴×D₁₇₂	1	1	yes	characteristic
338	C₂	C₂²×#13063372	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₄₃
commutator, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂×C₄×#41×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁵
hypercenter	C₂×C₄×#41	D₄₃
hypocenter	C₄₃	C₂×#3499
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063372	C₂	semidirect
2. #19148380	C₂	direct (N × H)
3. C₂×Q₈×D₁₇₂	C₂	semidirect
4. C₂×#14583553	C₂	semidirect
5. C₂×#14583475	C₂	semidirect
6. C₂×#14583449	C₂	semidirect
7. C₂²×#13063325	C₂	semidirect
8. C₂×C₄×#41×C₄₃	C₂	semidirect
10. C₂×#13063372	C₂²	semidirect
15. Q₈×D₁₇₂	C₂²	semidirect
16. #14583553	C₂²	semidirect
17. #14583475	C₂²	semidirect
18. #14583449	C₂²	semidirect
22. C₂×#13063325	C₂²	semidirect
31. C₄×#41×C₄₃	C₂²	semidirect
65. C₂×#41×C₄₃	D₄	semidirect
82. #41×C₄₃	C₂×D₄	semidirect
136. C₂×C₄×C₄₃	2₋¹⁺⁴	semidirect
163. C₄×C₄₃	C₂×2₋¹⁺⁴	semidirect
172. C₂×C₄×#41	D₄₃	semidirect
174. C₂×C₄₃	#3499	semidirect
198. C₄×#41	C₂×D₄₃	semidirect
205. C₄₃	C₂×#3499	semidirect
211. C₂×#41	D₁₇₂	semidirect
254. #41	C₂×D₁₇₂	semidirect
315. C₂×C₄	#13063372	semidirect
333. C₄	C₂×#13063372	semidirect
336. C₂	#19148380	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×#13063372	2
#19148380	2
C₂×Q₈×D₁₇₂	2
C₂×#14583553	2
C₂×#14583475	2
C₂×#14583449	2
C₂²×#13063325	2
C₂×C₄×#41×C₄₃	2
C₂×#3499	43

Subgroups — C2×#19148380 #26005012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19148380 #26005012