Subgroups — C₂×#20246137 #26375118

Order 12032.

2526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246137	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689030	C₂	1	4	yes	maximal
3	C₂×#15689022	C₂	1	2	yes	maximal
4	C₂×#15689009	C₂	1	4	yes	maximal
5	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×Dic₄₇	C₂	1	2	yes	maximal
7	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15689030	C₂²	1	32	yes	—
15	#15689022	C₂²	1	16	yes	—
16	#15689009	C₂²	1	32	yes	—
17	#15688989	C₂²	1	8	yes	—
18	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
19	#15688969	C₂²	1	8	yes	—
20	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
21	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
22	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
23	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
24	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
25	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
26	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₄₇	C₄	1	16	yes	complemented
28	C₂×C₄⋊Q₈×C₄₇	C₂²	1	8	yes	—
29	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×Dic₄₇	C₂³	1	2	yes	—
33	C₂³×Dic₄₇	C₂³	1	4	yes	—
34	C₂³×Dic₄₇	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₄₇	—	2	8	no	—
36	C₂×C₄⋊Dic₄₇	—	4	8	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
39	C₂×C₄×Dic₄₇	—	2	4	no	—
40	C₂×C₄×Dic₄₇	—	4	8	no	—
41	C₂×C₄×Dic₄₇	Q₈	1	8	yes	complemented
42	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
45	C₄₇⋊#95	C₂³	1	8	yes	—
46	C₄₇⋊#95	C₂³	1	8	yes	—
47	C₄₇⋊#95	C₂³	1	16	yes	—
48	C₄₇⋊#95	C₂³	1	8	yes	—
49	C₄₇⋊#95	C₂³	1	8	yes	—
50	C₄₇⋊#95	C₂³	1	16	yes	—
51	C₂²×Q₈×C₄₇	—	4	8	no	—
52	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
53	C₂²×Q₈×C₄₇	C₂³	1	8	yes	—
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₄₇	C₂×C₄	1	64	yes	complemented
59	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
60	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
63	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
65	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂×Q₈×C₄₇	—	4	32	no	—
67	C₂×Q₈×C₄₇	—	4	16	no	—
68	C₂×Q₈×C₄₇	C₂²×C₄	1	32	yes	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	8	no	—
71	C₂²×C₄×C₄₇	—	2	8	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₄₇	#41	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×Dic₄₇	—	2	4	no	—
84	C₂²×Dic₄₇	—	2	4	no	—
85	C₂²×Dic₄₇	—	2	4	no	—
86	C₂²×Dic₄₇	—	2	8	no	—
87	C₂²×Dic₄₇	—	2	8	no	—
88	C₂²×Dic₄₇	—	2	4	no	—
89	C₂²×Dic₄₇	—	2	4	no	—
90	C₂²×Dic₄₇	—	2	8	no	—
91	C₂²×Dic₄₇	C₂×Q₈	1	8	yes	—
92	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
93	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
94	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
95	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
96	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
98	C₄⋊Dic₄₇	—	2	16	no	—
99	C₄⋊Dic₄₇	—	4	16	no	—
100	C₄×Dic₄₇	—	2	8	no	—
101	C₄×Dic₄₇	—	4	16	no	—
102	C₄×Dic₄₇	C₂×Q₈	1	16	yes	complemented
103	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
104	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
105	C₂³×C₄₇	C₂×#41	1	2	yes	—
106	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
107	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
108	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
109	Q₈×C₄₇	—	4	32	no	—
110	C₂×C₄×C₄₇	—	2	32	no	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	C₄×Q₈	1	16	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
124	C₂×Dic₄₇	—	2	16	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	2	8	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	8	no	—
133	C₂×Dic₄₇	—	4	8	no	—
134	C₂×Dic₄₇	—	4	16	no	—
135	C₂×Dic₄₇	C₂²×Q₈	1	8	yes	—
136	C₂×#2331	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
138	C₂²×C₄₇	#518	1	4	yes	—
139	C₂²×C₄₇	#519	1	4	yes	—
140	C₂²×C₄₇	#542	1	8	yes	—
141	C₂²×C₄₇	#544	1	8	yes	—
142	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
144	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
146	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
147	C₄×C₄₇	—	2	16	no	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	C₂×C₄×Q₈	1	8	yes	—
151	Dic₄₇	—	2	8	no	—
152	Dic₄₇	—	4	4	no	—
153	Dic₄₇	—	4	8	no	—
154	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
155	#2331	—	47	16	no	—
156	C₂×#391	—	47	4	no	—
157	C₂×#389	—	47	2	no	—
158	C₂×#387	—	47	4	no	—
159	C₂×#382	—	47	1	no	—
160	C₂×C₄×C₄⋊C₄	—	47	2	no	—
161	C₂×#376	—	47	1	no	—
162	C₂×C₄₇	#2331	1	8	yes	complemented
163	C₂×C₄₇	C₂²×C₄×Q₈	1	1	yes	characteristic
164	C₂×C₄₇	C₂×#518	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#542	1	2	yes	—
167	C₂×C₄₇	C₂×#544	1	2	yes	—
168	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
169	C₂×C₄⋊Q₈	Dic₄₇	1	16	yes	complemented
170	C₂×C₄⋊Q₈	C₂×D₄₇	1	8	yes	—
171	C₂²×C₄⋊C₄	—	47	2	no	—
172	C₂²×C₄⋊C₄	—	47	4	no	—
173	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
174	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
175	C₂²×C₄²	—	47	1	no	—
176	C₂²×C₄²	—	47	1	no	—
177	C₂²×C₄²	—	47	4	no	—
178	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
179	#391	—	47	32	no	—
180	#389	—	47	16	no	—
181	#387	—	47	32	no	—
182	#382	—	47	8	no	—
183	C₄×C₄⋊C₄	—	47	16	no	—
184	#376	—	47	8	no	—
185	C₂×#95	—	47	2	no	—
186	C₂×#95	—	47	2	no	—
187	C₂×#95	—	47	4	no	—
188	C₂×#95	—	47	2	no	—
189	C₂×#95	—	47	2	no	—
190	C₂×#95	—	47	4	no	—
191	C₄₇	C₂×#2331	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	4	8	no	—
193	C₂²×Q₈	C₂×Dic₄₇	1	16	yes	—
194	C₂²×Q₈	C₂²×D₄₇	1	8	yes	—
195	C₂³×C₄	—	47	2	no	—
196	C₂³×C₄	—	47	2	no	—
197	C₂³×C₄	—	47	4	no	—
198	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
199	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊Q₈	C₂×Dic₄₇	1	64	yes	complemented
203	C₂×C₄⋊C₄	—	47	8	no	—
204	C₂×C₄⋊C₄	—	47	16	no	—
205	C₂×C₄⋊C₄	—	94	8	no	—
206	C₂×C₄⋊C₄	—	188	8	no	—
207	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
208	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
209	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
210	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
211	C₂×C₄²	—	47	8	no	—
212	C₂×C₄²	—	47	4	no	—
213	C₂×C₄²	—	47	4	no	—
214	C₂×C₄²	—	47	16	no	—
215	C₂×C₄²	—	94	4	no	—
216	C₂×C₄²	—	188	8	no	—
217	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
218	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
219	#95	—	47	8	no	—
220	#95	—	47	8	no	—
221	#95	—	47	16	no	—
222	#95	—	47	8	no	—
223	#95	—	47	8	no	—
224	#95	—	47	16	no	—
225	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
226	C₂×Q₈	—	4	32	no	—
227	C₂×Q₈	—	4	16	no	—
228	C₂×Q₈	C₂²×Dic₄₇	1	32	yes	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	47	8	no	—
234	C₂²×C₄	—	47	4	no	—
235	C₂²×C₄	—	47	4	no	—
236	C₂²×C₄	—	47	8	no	—
237	C₂²×C₄	—	94	4	no	—
238	C₂²×C₄	—	94	4	no	—
239	C₂²×C₄	—	94	4	no	—
240	C₂²×C₄	—	94	8	no	—
241	C₂²×C₄	—	94	8	no	—
242	C₂²×C₄	—	94	4	no	—
243	C₂²×C₄	—	94	4	no	—
244	C₂²×C₄	—	94	8	no	—
245	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
246	C₂²×C₄	#10625354	1	4	yes	—
247	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
248	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
249	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
250	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
251	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
252	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
253	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
254	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
255	C₄⋊C₄	—	94	16	no	—
256	C₄⋊C₄	—	188	16	no	—
257	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
258	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
259	C₄²	—	47	16	no	—
260	C₄²	—	94	8	no	—
261	C₄²	—	188	16	no	—
262	C₄²	C₂²×Dic₄₇	1	16	yes	—
263	C₂³	C₂³×Dic₄₇	1	2	yes	—
264	C₂³	C₂×Q₈×D₄₇	1	2	yes	—
265	C₂³	C₂×#10625354	1	2	yes	—
266	C₂³	#13094069	1	4	yes	—
267	C₂³	#13094072	1	4	yes	—
268	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
269	Q₈	—	4	32	no	—
270	C₂×C₄	—	2	32	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	47	8	no	—
278	C₂×C₄	—	94	16	no	—
279	C₂×C₄	—	94	4	no	—
280	C₂×C₄	—	94	4	no	—
281	C₂×C₄	—	94	4	no	—
282	C₂×C₄	—	94	8	no	—
283	C₂×C₄	—	94	8	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	8	no	—
287	C₂×C₄	—	188	8	no	—
288	C₂×C₄	—	188	16	no	—
289	C₂×C₄	Q₈×Dic₄₇	1	16	yes	—
290	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
291	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
292	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
293	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
294	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
295	C₂×C₄	C₂×#10625354	1	4	yes	—
296	C₂²	C₂×Q₈×Dic₄₇	1	4	yes	—
297	C₂²	#15689598	1	4	yes	—
298	C₂²	#15689601	1	4	yes	—
299	C₂²	#15689747	1	8	yes	—
300	C₂²	#15689750	1	8	yes	—
301	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
302	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
303	C₂²	C₂²×#10625354	1	1	yes	characteristic
304	C₂²	C₂×#13094069	1	2	yes	—
305	C₂²	C₂×#13094072	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	2	16	no	—
308	C₄	—	2	4	no	—
309	C₄	—	94	8	no	—
310	C₄	—	188	4	no	—
311	C₄	—	188	8	no	—
312	C₄	C₂×Q₈×Dic₄₇	1	8	yes	—
313	C₂	#20246137	1	8	yes	direct product factor
314	C₂	C₂²×Q₈×Dic₄₇	1	1	yes	characteristic
315	C₂	C₂×#15689598	1	1	yes	characteristic
316	C₂	C₂×#15689601	1	1	yes	characteristic
317	C₂	C₂×#15689747	1	2	yes	—
318	C₂	C₂×#15689750	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#2331
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20246137	C₂	direct (N × H)
27. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
41. C₂×C₄×Dic₄₇	Q₈	semidirect
58. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
102. C₄×Dic₄₇	C₂×Q₈	semidirect
162. C₂×C₄₇	#2331	semidirect
169. C₂×C₄⋊Q₈	Dic₄₇	semidirect
191. C₄₇	C₂×#2331	semidirect
202. C₄⋊Q₈	C₂×Dic₄₇	semidirect
313. C₂	#20246137	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20246137	2
C₂×#15689030	2
C₂×#15689022	2
C₂×#15689009	2
C₂×#15688989	2
C₂×C₄⋊C₄×Dic₄₇	2
C₂×#15688969	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#2331	47

Subgroups — C2×#20246137 #26375118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246137 #26375118