Subgroups — C₂×#20242740 #26198504

Order 12032.

958 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689315	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242740	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688867	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625334	—	2	2	no	—
8	C₂²×C₈×D₄₇	—	2	1	no	—
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	#15689315	—	2	4	no	—
11	#15689315	C₂²	1	8	yes	complemented
12	C₈×D₁₈₈	—	2	2	no	—
13	C₈×D₁₈₈	C₂²	1	4	yes	complemented
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093945	—	2	2	no	—
17	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
18	#15688876	C₂²	1	4	yes	complemented
19	#15688867	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₄₇	—	2	1	no	—
21	C₂²×D₈×C₄₇	—	2	1	no	—
22	C₂×#127×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#493	—	2	2	no	—
24	C₄₇×#493	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	—	2	1	no	—
28	C₂×#10625334	—	2	8	no	—
29	C₂×#10625334	—	4	8	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂×C₈×D₄₇	—	2	4	no	—
33	C₂×C₈×D₄₇	—	4	4	no	—
34	C₄×D₁₈₈	—	2	1	no	—
35	C₄×D₁₈₈	—	2	1	no	—
36	C₄×D₁₈₈	D₄	1	2	yes	—
37	C₄×D₁₈₈	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
40	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
43	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
44	#13093945	—	2	8	no	—
45	(C₂×D₄₇)⋊C₈	—	2	4	no	—
46	C₂²×D₄×C₄₇	—	2	2	no	—
47	C₂²×D₄×C₄₇	—	2	2	no	—
48	C₂²×D₄×C₄₇	—	2	2	no	—
49	C₂×D₈×C₄₇	—	2	4	no	—
50	C₂×D₈×C₄₇	—	2	4	no	—
51	C₂×D₈×C₄₇	—	4	4	no	—
52	C₂×D₈×C₄₇	—	4	4	no	—
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	#127×C₄₇	—	2	2	no	—
56	#127×C₄₇	—	2	2	no	—
57	#127×C₄₇	—	2	2	no	—
58	#127×C₄₇	C₂³	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₇	D₄	1	2	yes	—
61	C₄×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂⁴×C₄₇	—	4	2	no	—
63	C₂³×D₄₇	—	2	1	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	—	2	4	no	—
67	C₂×D₄×C₄₇	—	4	2	no	—
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂×D₄×C₄₇	—	4	2	no	—
70	C₂×D₄×C₄₇	—	4	4	no	—
71	C₂×D₄×C₄₇	—	4	4	no	—
72	C₂×D₄×C₄₇	—	4	4	no	—
73	C₂×D₄×C₄₇	—	4	4	no	—
74	C₂×D₄×C₄₇	—	8	4	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	1	no	—
79	C₂×D₁₈₈	—	2	1	no	—
80	C₂×D₁₈₈	—	2	2	no	—
81	C₂×D₁₈₈	D₈	1	4	yes	complemented
82	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
83	C₂×C₄×D₄₇	—	2	2	no	—
84	C₂×C₄×D₄₇	—	2	2	no	—
85	C₂×C₄×D₄₇	—	4	2	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	D₈×C₄₇	—	4	8	no	—
88	D₈×C₄₇	—	4	8	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	D₈	1	4	yes	complemented
91	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₇	#41	1	1	yes	characteristic
96	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
97	#10625334	—	4	16	no	—
98	(C₂×D₄₇)⋊C₄	—	2	2	no	—
99	(C₂×D₄₇)⋊C₄	—	2	2	no	—
100	C₄⋊Dic₄₇	—	2	1	no	—
101	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	—	4	2	no	—
105	C₂×C₄₇⋊C₈	—	4	4	no	—
106	C₈×D₄₇	—	4	8	no	—
107	C₂³×C₄₇	—	4	4	no	—
108	C₂³×C₄₇	—	8	4	no	—
109	C₂³×C₄₇	—	8	4	no	—
110	C₂³×C₄₇	—	8	4	no	—
111	C₂³×C₄₇	#121	1	1	yes	characteristic
112	C₂²×D₄₇	—	2	2	no	—
113	C₂²×D₄₇	—	2	4	no	—
114	C₂²×D₄₇	—	4	2	no	—
115	C₂²×D₄₇	—	4	2	no	—
116	D₄×C₄₇	—	4	4	no	—
117	D₄×C₄₇	—	4	4	no	—
118	D₄×C₄₇	—	4	4	no	—
119	D₄×C₄₇	—	4	4	no	—
120	D₄×C₄₇	—	8	4	no	—
121	D₄×C₄₇	—	8	4	no	—
122	D₄×C₄₇	—	8	4	no	—
123	D₄×C₄₇	—	8	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₄₇	#135	1	2	yes	—
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	D₁₈₈	—	2	2	no	—
138	D₁₈₈	—	2	4	no	—
139	D₁₈₈	C₂×D₈	1	4	yes	complemented
140	C₄×D₄₇	—	4	2	no	—
141	C₄×D₄₇	—	4	2	no	—
142	C₄×D₄₇	—	4	4	no	—
143	C₈×C₄₇	—	2	2	no	—
144	C₈×C₄₇	C₂×D₈	1	4	yes	complemented
145	C₄₇⋊C₈	—	4	2	no	—
146	C₄₇⋊C₈	—	4	4	no	—
147	C₂×#1667	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	8	4	no	—
149	C₂²×C₄₇	—	8	4	no	—
150	C₂²×C₄₇	—	8	4	no	—
151	C₂²×C₄₇	—	16	8	no	—
152	C₂²×C₄₇	#459	1	2	yes	—
153	C₂²×C₄₇	#466	1	2	yes	—
154	C₂²×C₄₇	#472	1	2	yes	—
155	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
156	C₂×D₄₇	—	2	4	no	—
157	C₂×D₄₇	—	4	8	no	—
158	C₂×D₄₇	—	4	2	no	—
159	C₂×D₄₇	—	4	2	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	#459	1	2	yes	—
163	C₄×C₄₇	#466	1	2	yes	—
164	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
167	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	Dic₄₇	—	4	2	no	—
169	C₂×#493	D₄₇	1	1	yes	characteristic, complemented
170	C₂×#459	—	47	2	no	—
171	C₂×C₈×D₄	—	47	1	no	—
172	#1667	—	47	8	no	—
173	C₂×#335	—	47	1	no	—
174	C₂×#331	—	47	2	no	—
175	C₂×C₄₇	—	16	4	no	—
176	C₂×C₄₇	#1667	1	4	yes	complemented
177	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
179	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
180	D₄₇	—	4	4	no	—
181	C₂²×D₈	—	2	1	no	—
182	C₂²×D₈	—	2	1	no	—
183	C₂²×D₈	—	94	2	no	—
184	C₂³×C₈	—	94	1	no	—
185	C₂×#127	C₂×D₄₇	1	2	yes	—
186	C₂×C₄×D₄	—	47	1	no	—
187	#493	—	2	2	no	—
188	#493	C₂×D₄₇	1	4	yes	complemented
189	#459	—	47	8	no	—
190	#459	—	94	4	no	—
191	C₈×D₄	—	47	4	no	—
192	C₈×D₄	—	94	2	no	—
193	C₂×C₄⋊C₈	—	47	1	no	—
194	C₂×C₄⋊C₈	—	47	2	no	—
195	C₂×#102	—	94	2	no	—
196	C₂×C₂²⋊C₈	—	94	1	no	—
197	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
198	#335	—	47	4	no	—
199	#331	—	47	8	no	—
200	C₄₇	C₂×#1667	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	—	2	2	no	—
203	C₂²×D₄	—	2	2	no	—
204	C₂²×D₄	—	47	1	no	—
205	C₂³×C₄	—	94	1	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	2	4	no	—
208	C₂×D₈	—	4	4	no	—
209	C₂×D₈	—	4	4	no	—
210	C₂×D₈	—	94	8	no	—
211	C₂×D₈	—	188	8	no	—
212	C₂²×C₈	—	94	2	no	—
213	C₂²×C₈	—	94	1	no	—
214	C₂²×C₈	—	94	4	no	—
215	C₂²×C₈	—	188	4	no	—
216	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
218	#127	—	2	2	no	—
219	#127	—	2	2	no	—
220	#127	—	2	2	no	—
221	#127	C₂²×D₄₇	1	4	yes	—
222	C₄×D₄	—	47	2	no	—
223	C₄×D₄	—	47	2	no	—
224	C₄×D₄	—	94	1	no	—
225	C₄×D₄	—	94	1	no	—
226	C₂×C₄⋊C₄	—	47	1	no	—
227	C₂×C₂²⋊C₄	—	94	1	no	—
228	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
229	C₄⋊C₈	—	47	2	no	—
230	C₄⋊C₈	—	47	4	no	—
231	C₄⋊C₈	—	94	2	no	—
232	C₄⋊C₈	—	94	1	no	—
233	#102	—	94	8	no	—
234	C₂²⋊C₈	—	94	4	no	—
235	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
236	C₄×C₈	C₂²×D₄₇	1	2	yes	—
237	C₂⁴	—	4	2	no	—
238	C₂⁴	—	94	1	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	4	no	—
248	C₂×D₄	—	4	4	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂×D₄	—	47	4	no	—
251	C₂×D₄	—	47	2	no	—
252	C₂×D₄	—	94	1	no	—
253	C₂×D₄	—	94	2	no	—
254	C₂²×C₄	—	94	1	no	—
255	C₂²×C₄	—	94	2	no	—
256	C₂²×C₄	—	94	2	no	—
257	C₂²×C₄	—	188	2	no	—
258	C₂²×C₄	—	188	2	no	—
259	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	#10625352	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	D₈	—	4	8	no	—
263	D₈	—	4	8	no	—
264	D₈	—	188	16	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	94	4	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	—	188	4	no	—
269	C₂×C₈	—	188	2	no	—
270	C₂×C₈	—	188	8	no	—
271	C₂×C₈	#10625334	1	4	yes	complemented
272	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
274	C₄⋊C₄	—	47	2	no	—
275	C₄⋊C₄	—	94	1	no	—
276	C₂²⋊C₄	—	94	2	no	—
277	C₂²⋊C₄	—	94	2	no	—
278	C₄²	D₄×D₄₇	1	1	yes	characteristic
279	C₄²	#10625352	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
281	C₄²	C₂³×D₄₇	1	1	yes	characteristic
282	C₂³	—	4	4	no	—
283	C₂³	—	8	4	no	—
284	C₂³	—	8	4	no	—
285	C₂³	—	8	4	no	—
286	C₂³	—	94	4	no	—
287	C₂³	—	94	2	no	—
288	C₂³	—	188	2	no	—
289	C₂³	—	188	2	no	—
290	C₂³	#13093995	1	1	yes	characteristic
291	D₄	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	4	no	—
296	D₄	—	8	4	no	—
297	D₄	—	8	4	no	—
298	D₄	—	8	4	no	—
299	D₄	—	47	4	no	—
300	D₄	—	94	4	no	—
301	D₄	—	94	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	94	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	4	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	#13093995	1	1	yes	characteristic
309	C₂×C₄	#13093995	1	1	yes	characteristic
310	C₂×C₄	#13093995	1	1	yes	characteristic
311	C₂×C₄	D₈×D₄₇	1	2	yes	—
312	C₂×C₄	C₂×#10625334	1	2	yes	—
313	C₂×C₄	#13094043	1	2	yes	—
314	C₂×C₄	#13094050	1	2	yes	—
315	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	188	4	no	—
320	C₈	—	188	2	no	—
321	C₈	C₂×#10625334	1	4	yes	complemented
322	C₂²	—	8	4	no	—
323	C₂²	—	8	4	no	—
324	C₂²	—	8	4	no	—
325	C₂²	—	16	8	no	—
326	C₂²	—	94	4	no	—
327	C₂²	—	188	8	no	—
328	C₂²	—	188	2	no	—
329	C₂²	—	188	2	no	—
330	C₂²	#15689315	1	2	yes	—
331	C₂²	#15689356	1	2	yes	—
332	C₂²	#15689390	1	2	yes	—
333	C₂²	C₂×#13093995	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	188	2	no	—
337	C₄	#15689315	1	2	yes	—
338	C₄	#15689356	1	2	yes	—
339	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
340	C₄	C₂²×#10625334	1	1	yes	characteristic
341	C₄	C₂×#13094043	1	1	yes	characteristic
342	C₄	C₂×#13094050	1	1	yes	characteristic
343	C₂	—	16	4	no	—
344	C₂	—	188	4	no	—
345	C₂	#20242740	1	4	yes	direct product factor
346	C₂	C₂×#15689315	1	1	yes	characteristic
347	C₂	C₂×#15689356	1	1	yes	characteristic
348	C₂	C₂×#15689390	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#493	C₂
socle	C₂³×C₄₇	#121
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	C₂×#1667
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×#15689315	C₂	semidirect
2. C₂×C₈×D₁₈₈	C₂	semidirect
3. #20242740	C₂	direct (N × H)
4. C₂×#15688876	C₂	semidirect
5. C₂×#15688867	C₂	semidirect
6. C₂×C₄₇×#493	C₂	semidirect
11. #15689315	C₂²	semidirect
13. C₈×D₁₈₈	C₂²	semidirect
18. #15688876	C₂²	semidirect
19. #15688867	C₂²	semidirect
24. C₄₇×#493	C₂²	semidirect
81. C₂×D₁₈₈	D₈	semidirect
90. C₂×C₈×C₄₇	D₈	semidirect
139. D₁₈₈	C₂×D₈	semidirect
144. C₈×C₄₇	C₂×D₈	semidirect
169. C₂×#493	D₄₇	semidirect
176. C₂×C₄₇	#1667	semidirect
188. #493	C₂×D₄₇	semidirect
200. C₄₇	C₂×#1667	semidirect
271. C₂×C₈	#10625334	semidirect
321. C₈	C₂×#10625334	semidirect
345. C₂	#20242740	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×#15689315	2
C₂×C₈×D₁₈₈	2
#20242740	2
C₂×#15688876	2
C₂×#15688867	2
C₂×C₄₇×#493	2
C₂×#1667	47

Subgroups — C2×#20242740 #26198504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242740 #26198504