Subgroups — C₂×#20241676 #26181876

Order 12032.

966 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	C₂×#15689174	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241676	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625324	C₄	1	2	yes	—
10	C₂²×C₈×D₄₇	C₄	1	2	yes	—
11	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
12	#15689174	C₄	1	4	yes	—
13	#15689174	C₂²	1	4	yes	—
14	C₂×Dic₄₇⋊C₈	C₄	1	2	yes	—
15	C₂×(C₂×D₄₇)⋊C₈	C₄	1	2	yes	—
16	C₄×#10625324	C₄	1	4	yes	—
17	C₄×#10625324	C₂²	1	4	yes	—
18	C₂×#13093934	C₄	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
23	C₄×C₈×D₄₇	C₄	1	4	yes	—
24	C₄×C₈×D₄₇	C₂²	1	4	yes	—
25	#15688845	C₂²	1	4	yes	complemented
26	#15688845	C₂²	1	4	yes	complemented
27	#15688845	C₂²	1	4	yes	complemented
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#322	C₂²	1	4	yes	complemented
32	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×#10625324	—	2	8	no	—
36	C₂×#10625324	C₂×C₄	1	8	yes	—
37	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₇	—	2	8	no	—
41	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
42	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄²×D₄₇	C₂×C₄	1	2	yes	—
46	C₄²×D₄₇	C₂×C₄	1	2	yes	—
47	C₄²×D₄₇	C₂×C₄	1	2	yes	—
48	C₄²×D₄₇	C₂³	1	2	yes	—
49	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
50	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
51	#13093934	C₂×C₄	1	8	yes	—
52	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
54	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
55	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
56	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
57	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
58	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
59	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
61	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₄₇	C₂³	1	2	yes	—
67	C₄×C₈×C₄₇	C₂³	1	2	yes	—
68	C₄×C₈×C₄₇	C₂³	1	2	yes	—
69	C₂³×D₄₇	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
75	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₄₇	—	2	4	no	—
77	C₂×C₄×D₄₇	—	2	4	no	—
78	C₂×C₄×D₄₇	C₄²	1	2	yes	—
79	C₂×C₄×D₄₇	C₄²	1	2	yes	—
80	C₂×C₄×D₄₇	C₄²	1	2	yes	—
81	C₂×C₄×D₄₇	M₁₆	1	8	yes	—
82	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄×Dic₄₇	C₄²	1	2	yes	—
96	C₄×Dic₄₇	C₄²	1	2	yes	—
97	C₄×Dic₄₇	C₄²	1	2	yes	—
98	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
99	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
100	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	M₁₆	1	8	yes	complemented
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
107	#10625324	—	2	16	no	—
108	C₈×D₄₇	—	2	16	no	—
109	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
110	C₂²×D₄₇	—	2	2	no	—
111	C₂²×D₄₇	—	2	2	no	—
112	C₂²×D₄₇	#97	1	4	yes	—
113	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
119	C₂×C₄×C₄₇	#131	1	4	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	#97	1	4	yes	—
126	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
127	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
128	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
129	C₄×D₄₇	—	2	4	no	—
130	C₄×D₄₇	—	2	4	no	—
131	C₄×D₄₇	—	2	4	no	—
132	C₄×D₄₇	—	2	4	no	—
133	C₄×D₄₇	#97	1	8	yes	—
134	C₄×D₄₇	C₂×M₁₆	1	8	yes	—
135	C₈×C₄₇	—	2	4	no	—
136	C₈×C₄₇	—	2	4	no	—
137	C₈×C₄₇	—	2	4	no	—
138	C₄₇⋊C₈	—	2	4	no	—
139	C₄₇⋊C₈	—	2	4	no	—
140	C₄₇⋊C₈	C₂×M₁₆	1	8	yes	complemented
141	C₂×#1451	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	C₂×#97	1	2	yes	—
143	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
144	C₂²×C₄₇	#405	1	2	yes	—
145	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
146	C₂×D₄₇	—	2	2	no	—
147	C₂×D₄₇	—	2	2	no	—
148	C₂×D₄₇	—	2	8	no	—
149	C₂×D₄₇	C₂×#97	1	4	yes	—
150	C₄×C₄₇	C₂×#97	1	2	yes	—
151	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
152	C₄×C₄₇	#405	1	2	yes	—
153	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
154	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
155	C₄×C₄₇	C₂×#131	1	2	yes	—
156	Dic₄₇	—	2	2	no	—
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	C₂×#97	1	4	yes	—
159	C₂×#431	—	47	1	no	—
160	C₂×C₄×M₁₆	—	47	1	no	—
161	C₂²×C₄×C₈	—	47	1	no	—
162	#1451	—	47	8	no	—
163	C₂×#322	—	47	1	no	—
164	C₂×#322	—	47	1	no	—
165	C₂×#322	—	47	1	no	—
166	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
167	C₂×C₄₇	#1451	1	4	yes	complemented
168	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
169	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
171	D₄₇	—	2	4	no	—
172	C₂²×M₁₆	—	47	2	no	—
173	C₂³×C₈	—	47	2	no	—
174	C₂²×C₄²	—	47	1	no	—
175	#431	—	47	4	no	—
176	#431	—	47	4	no	—
177	C₂×C₄⋊C₈	—	47	2	no	—
178	C₂×C₂²⋊C₈	—	47	2	no	—
179	C₄×M₁₆	—	47	4	no	—
180	C₄×M₁₆	—	47	4	no	—
181	C₂×#97	—	47	2	no	—
182	C₂×C₄×C₈	—	47	2	no	—
183	C₂×C₄×C₈	—	47	1	no	—
184	C₂×C₄×C₈	—	47	1	no	—
185	C₂×C₄×C₈	—	47	1	no	—
186	C₂×C₄×C₈	—	47	4	no	—
187	C₂×C₄×C₈	—	47	4	no	—
188	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
191	#322	—	47	4	no	—
192	#322	—	47	4	no	—
193	#322	—	47	4	no	—
194	#322	C₂×D₄₇	1	4	yes	complemented
195	C₄₇	C₂×#1451	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	47	1	no	—
197	C₂³×C₄	—	47	1	no	—
198	C₂³×C₄	—	47	1	no	—
199	C₂×M₁₆	—	47	8	no	—
200	C₂×M₁₆	—	94	8	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	—	47	2	no	—
203	C₂²×C₈	—	47	2	no	—
204	C₂²×C₈	—	47	8	no	—
205	C₂²×C₈	—	94	8	no	—
206	C₂²×C₈	C₄×D₄₇	1	2	yes	—
207	C₂²×C₈	C₄×D₄₇	1	2	yes	—
208	C₂²×C₈	C₄×D₄₇	1	2	yes	—
209	C₂×C₄²	—	47	1	no	—
210	C₂×C₄²	—	47	1	no	—
211	C₂×C₄²	—	47	1	no	—
212	C₂×C₄²	—	47	2	no	—
213	C₂×C₄²	—	47	2	no	—
214	C₂×C₄²	—	47	2	no	—
215	C₂×C₄²	—	47	2	no	—
216	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
217	C₄⋊C₈	—	47	8	no	—
218	C₂²⋊C₈	—	47	8	no	—
219	#97	—	47	8	no	—
220	C₄×C₈	—	47	8	no	—
221	C₄×C₈	—	47	2	no	—
222	C₄×C₈	—	47	2	no	—
223	C₄×C₈	—	47	2	no	—
224	C₄×C₈	—	47	2	no	—
225	C₄×C₈	—	47	2	no	—
226	C₄×C₈	—	47	2	no	—
227	C₄×C₈	C₄×D₄₇	1	2	yes	—
228	C₄×C₈	C₄×D₄₇	1	2	yes	—
229	C₄×C₈	C₄×D₄₇	1	2	yes	—
230	C₄×C₈	C₂²×D₄₇	1	2	yes	—
231	C₄×C₈	C₂²×D₄₇	1	2	yes	—
232	C₄×C₈	C₂²×D₄₇	1	2	yes	—
233	C₂⁴	—	47	1	no	—
234	C₂²×C₄	—	47	1	no	—
235	C₂²×C₄	—	47	2	no	—
236	C₂²×C₄	—	47	8	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	47	1	no	—
239	C₂²×C₄	—	47	1	no	—
240	C₂²×C₄	—	47	2	no	—
241	C₂²×C₄	—	47	2	no	—
242	C₂²×C₄	—	47	2	no	—
243	C₂²×C₄	—	47	2	no	—
244	C₂²×C₄	—	94	4	no	—
245	C₂²×C₄	—	94	4	no	—
246	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
249	M₁₆	—	94	16	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	47	8	no	—
254	C₂×C₈	—	47	4	no	—
255	C₂×C₈	—	47	4	no	—
256	C₂×C₈	—	47	4	no	—
257	C₂×C₈	—	94	16	no	—
258	C₂×C₈	—	94	4	no	—
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
261	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
262	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
263	C₄²	—	47	2	no	—
264	C₄²	—	47	2	no	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	2	no	—
267	C₄²	—	47	2	no	—
268	C₄²	—	47	2	no	—
269	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₄²	C₂³×D₄₇	1	1	yes	characteristic
273	C₂³	—	47	2	no	—
274	C₂³	—	47	4	no	—
275	C₂³	—	94	2	no	—
276	C₂³	—	94	2	no	—
277	C₂³	C₄²×D₄₇	1	1	yes	characteristic
278	C₂×C₄	—	47	2	no	—
279	C₂×C₄	—	47	4	no	—
280	C₂×C₄	—	47	8	no	—
281	C₂×C₄	—	47	8	no	—
282	C₂×C₄	—	47	2	no	—
283	C₂×C₄	—	47	2	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	94	4	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
293	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
294	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
295	C₂×C₄	#13094030	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
299	C₈	—	2	4	no	—
300	C₈	—	2	4	no	—
301	C₈	—	2	4	no	—
302	C₈	—	47	8	no	—
303	C₈	—	94	4	no	—
304	C₈	—	94	4	no	—
305	C₂²	—	47	4	no	—
306	C₂²	—	94	8	no	—
307	C₂²	—	94	2	no	—
308	C₂²	—	94	2	no	—
309	C₂²	#97×D₄₇	1	2	yes	—
310	C₂²	#15689075	1	2	yes	—
311	C₂²	#15689077	1	2	yes	—
312	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
313	C₄	—	47	4	no	—
314	C₄	—	94	2	no	—
315	C₄	—	94	2	no	—
316	C₄	#97×D₄₇	1	2	yes	—
317	C₄	#15689075	1	2	yes	—
318	C₄	#15689077	1	2	yes	—
319	C₄	C₂×M₁₆×D₄₇	1	2	yes	—
320	C₄	C₂×M₁₆×D₄₇	1	2	yes	—
321	C₄	C₂×#13094030	1	2	yes	—
322	C₂	—	94	4	no	—
323	C₂	#20241676	1	4	yes	direct product factor
324	C₂	C₂×#97×D₄₇	1	1	yes	characteristic
325	C₂	C₂×#15689075	1	1	yes	characteristic
326	C₂	C₂×#15689077	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂³×C₄₇	C₂×C₄²
prime radical	C₂³×D₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	C₂×#1451
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #20241676	C₂	direct (N × H)
5. C₂×#15688845	C₂	semidirect
6. C₂×#15688845	C₂	semidirect
7. C₂×#15688845	C₂	semidirect
8. C₂×C₄₇×#322	C₂	semidirect
25. #15688845	C₂²	semidirect
26. #15688845	C₂²	semidirect
27. #15688845	C₂²	semidirect
31. C₄₇×#322	C₂²	semidirect
103. C₂×C₄₇⋊C₈	M₁₆	semidirect
140. C₄₇⋊C₈	C₂×M₁₆	semidirect
166. C₂×#322	D₄₇	semidirect
167. C₂×C₄₇	#1451	semidirect
194. #322	C₂×D₄₇	semidirect
195. C₄₇	C₂×#1451	semidirect
323. C₂	#20241676	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×#15689174	2
C₂×C₄×#10625324	2
C₂×C₄×C₈×D₄₇	2
#20241676	2
C₂×#15688845	2
C₂×C₄₇×#322	2
C₂×#1451	47

Subgroups — C2×#20241676 #26181876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241676 #26181876