Subgroups — C₂×#20243026 #26204017

Order 12032.

966 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689474	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689469	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15689469	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15689469	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243026	C₂	1	8	yes	maximal, direct product factor
7	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625335	—	2	2	no	—
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13094024	C₂²	1	1	yes	characteristic
12	#15689495	—	2	2	no	—
13	#15689495	C₂²	1	4	yes	complemented
14	#15689474	—	2	2	no	—
15	#15689474	C₂²	1	4	yes	complemented
16	#15689469	—	2	2	no	—
17	#15689469	—	2	2	no	—
18	#15689469	—	2	2	no	—
19	#15689469	C₂²	1	4	yes	complemented
20	#15689469	C₂²	1	4	yes	complemented
21	#15689469	C₂²	1	4	yes	complemented
22	C₂×#13093949	—	2	2	no	—
23	C₂×#13093945	—	2	2	no	—
24	C₂×#13093945	—	2	2	no	—
25	C₂×#13093945	—	2	2	no	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	#15688845	C₂²	1	4	yes	complemented
29	C₂²×D₈×C₄₇	—	2	2	no	—
30	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#493	—	2	2	no	—
33	C₄₇×#493	C₂²	1	4	yes	—
34	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂²×Dic₉₄	—	2	1	no	—
37	C₂×#10625335	—	2	8	no	—
38	C₂×#10625335	—	4	8	no	—
39	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
40	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
41	#13094024	—	2	1	no	—
42	#13094024	—	2	1	no	—
43	#13094024	—	2	1	no	—
44	#13094024	—	2	1	no	—
45	#13094024	—	2	1	no	—
46	#13094024	—	2	1	no	—
47	#13094024	C₂³	1	2	yes	—
48	#13094024	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	1	no	—
53	#13093949	—	2	8	no	—
54	#13093945	—	2	8	no	—
55	#13093945	—	2	8	no	—
56	#13093945	—	2	8	no	—
57	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
58	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
59	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
60	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
61	C₂²×D₄×C₄₇	—	2	1	no	—
62	C₂²×D₄×C₄₇	—	2	1	no	—
63	C₂²×D₄×C₄₇	—	2	1	no	—
64	C₂²×D₄×C₄₇	—	2	1	no	—
65	C₂²×D₄×C₄₇	—	2	1	no	—
66	C₂²×D₄×C₄₇	—	2	1	no	—
67	C₂×D₈×C₄₇	—	2	8	no	—
68	C₂×D₈×C₄₇	—	4	8	no	—
69	C₂²×C₈×C₄₇	D₄	1	2	yes	—
70	#127×C₄₇	—	2	1	no	—
71	#127×C₄₇	—	2	1	no	—
72	#127×C₄₇	—	2	1	no	—
73	#127×C₄₇	—	2	1	no	—
74	#127×C₄₇	—	2	1	no	—
75	#127×C₄₇	—	2	1	no	—
76	#127×C₄₇	C₂³	1	2	yes	—
77	#127×C₄₇	C₂³	1	2	yes	—
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	D₄	1	2	yes	—
80	C₄×C₈×C₄₇	C₂³	1	2	yes	—
81	C₂⁴×C₄₇	—	4	1	no	—
82	C₂⁴×C₄₇	—	4	1	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	2	2	no	—
86	C₂×D₄×C₄₇	—	2	2	no	—
87	C₂×D₄×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	—	4	1	no	—
90	C₂×D₄×C₄₇	—	4	1	no	—
91	C₂×D₄×C₄₇	—	4	1	no	—
92	C₂×D₄×C₄₇	—	4	1	no	—
93	C₂×D₄×C₄₇	—	4	1	no	—
94	C₂×D₄×C₄₇	—	4	1	no	—
95	C₂×D₄×C₄₇	—	4	4	no	—
96	C₂×D₄×C₄₇	—	4	4	no	—
97	C₂×D₄×C₄₇	—	4	4	no	—
98	C₂×D₄×C₄₇	—	4	4	no	—
99	C₂×D₄×C₄₇	—	4	4	no	—
100	C₂×D₄×C₄₇	—	4	4	no	—
101	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×Dic₄₇	—	4	1	no	—
105	C₂²×Dic₄₇	—	4	1	no	—
106	C₂×Dic₉₄	—	2	2	no	—
107	C₂×Dic₉₄	—	2	2	no	—
108	C₂×Dic₉₄	—	4	4	no	—
109	C₂×Dic₉₄	—	4	4	no	—
110	C₂×Dic₉₄	—	4	1	no	—
111	C₂×Dic₉₄	—	4	1	no	—
112	D₈×C₄₇	—	4	16	no	—
113	C₂×C₈×C₄₇	—	2	4	no	—
114	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
119	#10625335	—	4	16	no	—
120	C₄⋊Dic₄₇	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	2	no	—
122	C₄⋊Dic₄₇	—	2	2	no	—
123	C₄⋊Dic₄₇	—	2	2	no	—
124	C₄⋊Dic₄₇	—	4	1	no	—
125	C₄⋊Dic₄₇	—	4	1	no	—
126	C₄⋊Dic₄₇	—	4	1	no	—
127	C₄⋊Dic₄₇	—	4	1	no	—
128	C₂×C₄₇⋊C₈	—	2	4	no	—
129	C₂×C₄₇⋊C₈	—	2	4	no	—
130	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
131	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
132	C₂³×C₄₇	—	4	2	no	—
133	C₂³×C₄₇	—	4	2	no	—
134	C₂³×C₄₇	—	8	2	no	—
135	C₂³×C₄₇	—	8	2	no	—
136	C₂³×C₄₇	—	8	2	no	—
137	C₂³×C₄₇	—	8	2	no	—
138	C₂³×C₄₇	—	8	2	no	—
139	C₂³×C₄₇	—	8	2	no	—
140	C₂³×C₄₇	#127	1	1	yes	characteristic
141	D₄×C₄₇	—	4	4	no	—
142	D₄×C₄₇	—	4	4	no	—
143	D₄×C₄₇	—	4	4	no	—
144	D₄×C₄₇	—	4	4	no	—
145	D₄×C₄₇	—	4	4	no	—
146	D₄×C₄₇	—	4	4	no	—
147	D₄×C₄₇	—	8	2	no	—
148	D₄×C₄₇	—	8	2	no	—
149	D₄×C₄₇	—	8	2	no	—
150	D₄×C₄₇	—	8	2	no	—
151	D₄×C₄₇	—	8	2	no	—
152	D₄×C₄₇	—	8	2	no	—
153	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
154	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
155	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
156	C₂×C₄×C₄₇	#135	1	4	yes	—
157	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
158	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
159	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×Dic₄₇	—	4	2	no	—
163	C₂×Dic₄₇	—	4	2	no	—
164	C₂×Dic₄₇	—	8	2	no	—
165	C₂×Dic₄₇	—	8	2	no	—
166	Dic₉₄	—	4	4	no	—
167	Dic₉₄	—	4	4	no	—
168	Dic₉₄	—	8	2	no	—
169	Dic₉₄	—	8	2	no	—
170	C₈×C₄₇	—	2	4	no	—
171	C₄₇⋊C₈	—	2	4	no	—
172	C₄₇⋊C₈	—	2	4	no	—
173	C₂×#1717	—	47	1	no	maximal, Sylow
174	C₂²×C₄₇	—	8	2	no	—
175	C₂²×C₄₇	—	8	2	no	—
176	C₂²×C₄₇	—	8	2	no	—
177	C₂²×C₄₇	—	8	2	no	—
178	C₂²×C₄₇	—	8	2	no	—
179	C₂²×C₄₇	—	8	2	no	—
180	C₂²×C₄₇	—	16	4	no	—
181	C₂²×C₄₇	—	16	4	no	—
182	C₂²×C₄₇	#495	1	2	yes	—
183	C₂²×C₄₇	#496	1	2	yes	—
184	C₂²×C₄₇	#496	1	2	yes	—
185	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
186	C₄×C₄₇	#495	1	2	yes	—
187	C₄×C₄₇	#496	1	2	yes	—
188	C₄×C₄₇	#496	1	2	yes	—
189	C₄×C₄₇	C₂×#135	1	2	yes	—
190	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
191	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
192	Dic₄₇	—	8	2	no	—
193	Dic₄₇	—	8	2	no	—
194	C₂×#493	D₄₇	1	1	yes	characteristic
195	C₂×#492	—	47	1	no	—
196	C₂×#487	—	47	1	no	—
197	C₂×#486	—	47	1	no	—
198	C₂×#486	—	47	1	no	—
199	C₂×#486	—	47	1	no	—
200	#1717	—	47	8	no	—
201	C₂×#322	—	47	1	no	—
202	C₂×C₄₇	—	16	2	no	—
203	C₂×C₄₇	—	16	2	no	—
204	C₂×C₄₇	#1717	1	4	yes	complemented
205	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
206	C₂×C₄₇	C₂×#496	1	1	yes	characteristic
207	C₂×C₄₇	C₂×#496	1	1	yes	characteristic
208	C₂²×SD₁₆	—	94	2	no	—
209	C₂²×D₈	—	2	2	no	—
210	C₂×C₄⋊Q₈	—	47	1	no	—
211	C₂×C₄⋊Q₈	—	47	1	no	—
212	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
213	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
214	#493	—	2	2	no	—
215	#493	C₂×D₄₇	1	4	yes	—
216	#492	—	47	4	no	—
217	#492	—	94	2	no	—
218	#487	—	47	4	no	—
219	#487	—	94	2	no	—
220	#486	—	47	4	no	—
221	#486	—	47	4	no	—
222	#486	—	47	4	no	—
223	#486	—	94	2	no	—
224	#486	—	94	2	no	—
225	#486	—	94	2	no	—
226	C₂×#103	—	94	2	no	—
227	C₂×#102	—	94	2	no	—
228	C₂×#102	—	94	2	no	—
229	C₂×#102	—	94	2	no	—
230	C₂×C₄×C₈	—	47	1	no	—
231	C₂×C₄×C₈	—	47	1	no	—
232	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
233	#322	—	47	4	no	—
234	C₄₇	C₂×#1717	1	1	yes	characteristic, complemented, Sylow
235	C₂²×Q₈	—	94	1	no	—
236	C₂²×Q₈	—	94	1	no	—
237	C₂²×D₄	—	2	1	no	—
238	C₂²×D₄	—	2	1	no	—
239	C₂²×D₄	—	2	1	no	—
240	C₂²×D₄	—	2	1	no	—
241	C₂²×D₄	—	2	1	no	—
242	C₂²×D₄	—	2	1	no	—
243	C₂×SD₁₆	—	94	8	no	—
244	C₂×SD₁₆	—	188	8	no	—
245	C₂×D₈	—	2	8	no	—
246	C₂×D₈	—	4	8	no	—
247	C₂²×C₈	—	47	2	no	—
248	C₂²×C₈	—	47	2	no	—
249	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
250	C₄⋊Q₈	—	47	2	no	—
251	C₄⋊Q₈	—	47	2	no	—
252	C₄⋊Q₈	—	94	1	no	—
253	C₄⋊Q₈	—	94	1	no	—
254	C₄⋊Q₈	—	94	1	no	—
255	C₄⋊Q₈	—	94	1	no	—
256	C₄⋊Q₈	—	94	1	no	—
257	C₄⋊Q₈	—	94	1	no	—
258	#127	—	2	1	no	—
259	#127	—	2	1	no	—
260	#127	—	2	1	no	—
261	#127	—	2	1	no	—
262	#127	—	2	1	no	—
263	#127	—	2	1	no	—
264	#127	C₂²×D₄₇	1	2	yes	—
265	#127	C₂²×D₄₇	1	2	yes	—
266	C₂×C₄⋊C₄	—	94	1	no	—
267	C₂×C₄⋊C₄	—	94	1	no	—
268	C₂×C₄⋊C₄	—	94	1	no	—
269	C₂×C₄⋊C₄	—	94	1	no	—
270	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
271	#103	—	94	8	no	—
272	#102	—	94	8	no	—
273	#102	—	94	8	no	—
274	#102	—	94	8	no	—
275	C₄×C₈	—	47	2	no	—
276	C₄×C₈	—	47	2	no	—
277	C₄×C₈	—	47	2	no	—
278	C₄×C₈	—	47	2	no	—
279	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
280	C₄×C₈	C₂²×D₄₇	1	2	yes	—
281	C₂⁴	—	4	1	no	—
282	C₂⁴	—	4	1	no	—
283	C₂×Q₈	—	94	2	no	—
284	C₂×Q₈	—	94	2	no	—
285	C₂×Q₈	—	188	4	no	—
286	C₂×Q₈	—	188	4	no	—
287	C₂×Q₈	—	188	1	no	—
288	C₂×Q₈	—	188	1	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	4	no	—
302	C₂×D₄	—	4	4	no	—
303	C₂×D₄	—	4	4	no	—
304	C₂×D₄	—	4	4	no	—
305	C₂×D₄	—	4	4	no	—
306	C₂×D₄	—	4	4	no	—
307	C₂²×C₄	—	188	1	no	—
308	C₂²×C₄	—	188	1	no	—
309	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
310	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
311	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
312	SD₁₆	—	188	16	no	—
313	D₈	—	4	16	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	47	4	no	—
316	C₂×C₈	—	47	4	no	—
317	C₂×C₈	—	94	4	no	—
318	C₂×C₈	—	94	4	no	—
319	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
320	C₄⋊C₄	—	94	2	no	—
321	C₄⋊C₄	—	94	2	no	—
322	C₄⋊C₄	—	94	2	no	—
323	C₄⋊C₄	—	94	2	no	—
324	C₄⋊C₄	—	188	1	no	—
325	C₄⋊C₄	—	188	1	no	—
326	C₄⋊C₄	—	188	1	no	—
327	C₄⋊C₄	—	188	1	no	—
328	C₄²	D₄×D₄₇	1	1	yes	characteristic
329	C₄²	D₄×D₄₇	1	1	yes	characteristic
330	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
331	C₄²	C₂³×D₄₇	1	1	yes	characteristic
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	—	8	2	no	—
335	C₂³	—	8	2	no	—
336	C₂³	—	8	2	no	—
337	C₂³	—	8	2	no	—
338	C₂³	—	8	2	no	—
339	C₂³	—	8	2	no	—
340	C₂³	#13094020	1	1	yes	characteristic
341	Q₈	—	376	2	no	—
342	Q₈	—	376	2	no	—
343	Q₈	—	188	4	no	—
344	Q₈	—	188	4	no	—
345	D₄	—	4	4	no	—
346	D₄	—	4	4	no	—
347	D₄	—	4	4	no	—
348	D₄	—	4	4	no	—
349	D₄	—	4	4	no	—
350	D₄	—	4	4	no	—
351	D₄	—	8	2	no	—
352	D₄	—	8	2	no	—
353	D₄	—	8	2	no	—
354	D₄	—	8	2	no	—
355	D₄	—	8	2	no	—
356	D₄	—	8	2	no	—
357	C₂×C₄	—	376	2	no	—
358	C₂×C₄	—	376	2	no	—
359	C₂×C₄	—	188	2	no	—
360	C₂×C₄	—	188	2	no	—
361	C₂×C₄	#13094020	1	1	yes	characteristic
362	C₂×C₄	#13094020	1	1	yes	characteristic
363	C₂×C₄	#13094020	1	1	yes	characteristic
364	C₂×C₄	#13094043	1	4	yes	—
365	C₂×C₄	#13094048	1	4	yes	—
366	C₂×C₄	#13094050	1	4	yes	—
367	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
368	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
369	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
370	C₈	—	2	4	no	—
371	C₈	—	94	4	no	—
372	C₈	—	94	4	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	8	2	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	16	4	no	—
380	C₂²	—	16	4	no	—
381	C₂²	#15689506	1	2	yes	—
382	C₂²	#15689510	1	2	yes	—
383	C₂²	#15689511	1	2	yes	—
384	C₂²	C₂×#13094020	1	1	yes	characteristic
385	C₄	—	376	2	no	—
386	C₄	—	376	2	no	—
387	C₄	#15689506	1	2	yes	—
388	C₄	#15689510	1	2	yes	—
389	C₄	#15689511	1	2	yes	—
390	C₄	C₂×#13094043	1	2	yes	—
391	C₄	C₂×#13094048	1	2	yes	—
392	C₄	C₂×#13094050	1	2	yes	—
393	C₂	—	16	2	no	—
394	C₂	—	16	2	no	—
395	C₂	#20243026	1	4	yes	direct product factor
396	C₂	C₂×#15689506	1	1	yes	characteristic
397	C₂	C₂×#15689510	1	1	yes	characteristic
398	C₂	C₂×#15689511	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting, prime radical	C₂×C₄₇×#493	C₂
socle	C₂³×C₄₇	#127
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	C₂×#1717
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#15689495	C₂	semidirect
2. C₂×#15689474	C₂	semidirect
3. C₂×#15689469	C₂	semidirect
4. C₂×#15689469	C₂	semidirect
5. C₂×#15689469	C₂	semidirect
6. #20243026	C₂	direct (N × H)
7. C₂×#15688845	C₂	semidirect
13. #15689495	C₂²	semidirect
15. #15689474	C₂²	semidirect
19. #15689469	C₂²	semidirect
20. #15689469	C₂²	semidirect
21. #15689469	C₂²	semidirect
28. #15688845	C₂²	semidirect
204. C₂×C₄₇	#1717	semidirect
234. C₄₇	C₂×#1717	semidirect
395. C₂	#20243026	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×#15689495	2
C₂×#15689474	2
C₂×#15689469	2
#20243026	2
C₂×#15688845	2
C₂×C₄₇×#493	2
C₂×#1717	47

Subgroups — C2×#20243026 #26204017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243026 #26204017