Subgroups — C₂×#20244722 #26244306

Order 12032.

1454 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093956	C₂	1	2	yes	maximal
3	C₂×#15689021	C₂	1	1	yes	maximal, characteristic
4	#20244722	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688886	C₂	1	2	yes	maximal
6	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	—	2	1	no	—
8	C₂×#13094024	—	2	8	no	—
9	C₂×#13094024	—	2	2	no	—
10	C₂×#13094024	C₂²	1	4	yes	—
11	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₄₇⋊C₄	—	2	1	no	—
16	C₂×#13093956	—	2	8	no	—
17	C₂×#13093956	C₂²	1	8	yes	—
18	#15689021	—	2	2	no	—
19	#15689021	C₂²	1	4	yes	—
20	C₂×C₄₇⋊#95	—	2	1	no	—
21	#15688886	C₂²	1	8	yes	—
22	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#336	C₂²	1	4	yes	—
25	C₂²×Dic₉₄	—	2	8	no	—
26	C₂²×Dic₉₄	—	2	4	no	—
27	C₂²×Dic₉₄	—	4	4	no	—
28	C₂²×Dic₉₄	—	4	4	no	—
29	C₂³×Dic₄₇	—	2	1	no	—
30	C₂³×Dic₄₇	—	2	2	no	—
31	C₂³×Dic₄₇	—	2	1	no	—
32	#13094024	—	2	16	no	—
33	#13094024	—	2	16	no	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	—	2	1	no	—
36	C₂×C₄⋊Dic₄₇	—	2	4	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	4	no	—
39	C₂×C₄⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄⋊Dic₄₇	—	2	4	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×C₄⋊Dic₄₇	D₄	1	8	yes	—
43	C₂×C₄⋊Dic₄₇	Q₈	1	4	yes	complemented
44	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×Dic₄₇⋊C₄	—	2	4	no	—
48	C₂×Dic₄₇⋊C₄	—	4	4	no	—
49	#13093956	—	2	16	no	—
50	C₄₇⋊#95	—	2	4	no	—
51	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	—	2	4	no	—
55	C₂²×C₈×C₄₇	—	2	4	no	—
56	C₂²×C₈×C₄₇	Q₈	1	8	yes	complemented
57	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₇	—	2	4	no	—
59	C₂×C₄²×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
67	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	8	no	—
77	C₂²×Dic₄₇	—	2	4	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	2	2	no	—
81	C₂²×Dic₄₇	—	4	2	no	—
82	C₂²×Dic₄₇	—	4	4	no	—
83	C₂²×Dic₄₇	—	4	2	no	—
84	C₂²×Dic₄₇	—	4	4	no	—
85	C₂²×Dic₄₇	—	4	2	no	—
86	C₂²×Dic₄₇	—	4	2	no	—
87	C₂²×Dic₄₇	—	4	2	no	—
88	C₂×Dic₉₄	—	2	16	no	—
89	C₂×Dic₉₄	—	4	8	no	—
90	C₂×Dic₉₄	—	4	32	no	—
91	C₂×Dic₉₄	—	4	8	no	—
92	C₂×C₈×C₄₇	—	2	16	no	—
93	C₂×C₈×C₄₇	—	2	4	no	—
94	C₂×C₈×C₄₇	—	2	4	no	—
95	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	complemented
96	C₄²×C₄₇	—	2	4	no	—
97	C₄²×C₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	16	no	—
99	C₄⋊Dic₄₇	—	2	8	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	C₄⋊Dic₄₇	C₂×D₄	1	8	yes	—
106	C₄⋊Dic₄₇	C₂×Q₈	1	4	yes	complemented
107	Dic₄₇⋊C₄	—	4	8	no	—
108	C₂³×C₄₇	#120	1	2	yes	—
109	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
110	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
111	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
112	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
113	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
114	C₂³×C₄₇	#137	1	2	yes	—
115	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	#120	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
126	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
128	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₄₇	—	2	8	no	—
134	C₂×Dic₄₇	—	2	4	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	16	no	—
140	C₂×Dic₄₇	—	4	8	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	8	4	no	—
145	Dic₉₄	—	4	32	no	—
146	C₈×C₄₇	—	2	8	no	—
147	C₂×#2058	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	#395	1	4	yes	—
149	C₂²×C₄₇	#448	1	4	yes	—
150	C₂²×C₄₇	#450	1	4	yes	—
151	C₂²×C₄₇	#475	1	4	yes	—
152	C₂²×C₄₇	#476	1	4	yes	—
153	C₂²×C₄₇	#498	1	4	yes	—
154	C₂²×C₄₇	#501	1	4	yes	—
155	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
159	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	#395	1	4	yes	—
165	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
166	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
167	Dic₄₇	—	4	8	no	—
168	Dic₄₇	—	4	4	no	—
169	Dic₄₇	—	8	2	no	—
170	C₂²×C₄⋊Q₈	—	47	1	no	—
171	C₂²×#106	—	47	2	no	—
172	C₂×#389	—	47	1	no	—
173	#2058	—	47	8	no	—
174	C₂×#340	—	47	2	no	—
175	C₂×#336	D₄₇	1	1	yes	characteristic
176	C₂×C₄₇	#2058	1	8	yes	complemented
177	C₂×C₄₇	C₂×#395	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#450	1	1	yes	characteristic
179	C₂×C₄₇	C₂×#448	1	1	yes	characteristic
180	C₂×C₄₇	C₂×#476	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#475	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#498	1	1	yes	characteristic
183	C₂×C₄₇	C₂×#501	1	1	yes	characteristic
184	C₂³×Q₈	—	94	1	no	—
185	C₂³×C₈	C₂×D₄₇	1	2	yes	—
186	C₂×C₄⋊Q₈	—	47	4	no	—
187	C₂×C₄⋊Q₈	—	94	8	no	—
188	C₂×C₄⋊Q₈	—	94	2	no	—
189	C₂²×C₄⋊C₄	—	47	2	no	—
190	C₂²×C₄⋊C₄	—	47	1	no	—
191	C₂²×C₄⋊C₄	—	47	1	no	—
192	C₂²×C₄⋊C₄	—	94	1	no	—
193	C₂²×C₄⋊C₄	—	94	1	no	—
194	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
195	C₂×#106	—	47	8	no	—
196	C₂×#106	—	94	8	no	—
197	#389	—	47	4	no	—
198	#389	—	94	2	no	—
199	C₂×#95	—	94	1	no	—
200	#340	—	47	8	no	—
201	#336	C₂×D₄₇	1	4	yes	—
202	C₄₇	C₂×#2058	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	94	8	no	—
204	C₂²×Q₈	—	94	4	no	—
205	C₂²×Q₈	—	188	4	no	—
206	C₂²×Q₈	—	188	4	no	—
207	C₂³×C₄	—	94	2	no	—
208	C₂³×C₄	—	94	1	no	—
209	C₂³×C₄	—	94	1	no	—
210	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
211	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
212	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	Dic₉₄	1	8	yes	complemented
216	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
217	C₄⋊Q₈	—	94	16	no	—
218	C₄⋊Q₈	—	94	16	no	—
219	C₂×C₄⋊C₄	—	47	8	no	—
220	C₂×C₄⋊C₄	—	47	4	no	—
221	C₂×C₄⋊C₄	—	47	4	no	—
222	C₂×C₄⋊C₄	—	47	2	no	—
223	C₂×C₄⋊C₄	—	47	2	no	—
224	C₂×C₄⋊C₄	—	94	8	no	—
225	C₂×C₄⋊C₄	—	94	2	no	—
226	C₂×C₄⋊C₄	—	94	1	no	—
227	C₂×C₄⋊C₄	—	94	4	no	—
228	C₂×C₄⋊C₄	—	94	4	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	94	4	no	—
231	C₂×C₄⋊C₄	—	94	1	no	—
232	C₂×C₄⋊C₄	—	94	4	no	—
233	C₂×C₄⋊C₄	—	188	4	no	—
234	C₂×C₄²	—	2	4	no	—
235	C₂×C₄²	D₁₈₈	1	2	yes	—
236	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
237	#106	—	94	16	no	—
238	#95	—	94	4	no	—
239	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂×Q₈	—	94	16	no	—
241	C₂×Q₈	—	188	32	no	—
242	C₂×Q₈	—	188	8	no	—
243	C₂×Q₈	—	188	8	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	94	8	no	—
249	C₂²×C₄	—	94	4	no	—
250	C₂²×C₄	—	94	2	no	—
251	C₂²×C₄	—	94	4	no	—
252	C₂²×C₄	—	94	2	no	—
253	C₂²×C₄	—	188	2	no	—
254	C₂²×C₄	—	188	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	C₂²×C₄	—	188	4	no	—
258	C₂²×C₄	—	188	2	no	—
259	C₂²×C₄	—	188	4	no	—
260	C₂²×C₄	#10625325	1	4	yes	—
261	C₂²×C₄	#10625325	1	4	yes	—
262	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
263	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
264	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
265	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₄₇	1	4	yes	—
267	C₂²×C₄	#10625352	1	2	yes	—
268	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
269	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
270	C₂×C₈	—	2	16	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	C₂×Dic₉₄	1	8	yes	complemented
274	C₄⋊C₄	—	47	8	no	—
275	C₄⋊C₄	—	47	4	no	—
276	C₄⋊C₄	—	94	16	no	—
277	C₄⋊C₄	—	94	8	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	94	4	no	—
280	C₄⋊C₄	—	94	2	no	—
281	C₄⋊C₄	—	94	4	no	—
282	C₄⋊C₄	—	188	2	no	—
283	C₄⋊C₄	—	188	8	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	4	no	—
286	C₂³	#13093994	1	2	yes	—
287	C₂³	#13094004	1	2	yes	—
288	C₂³	#13094024	1	2	yes	—
289	C₂³	C₂×#10625325	1	2	yes	—
290	C₂³	C₂×#10625325	1	2	yes	—
291	C₂³	#13094052	1	2	yes	—
292	C₂³	#13094057	1	2	yes	—
293	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
294	Q₈	—	188	32	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	94	8	no	—
302	C₂×C₄	—	94	4	no	—
303	C₂×C₄	—	376	4	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	16	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	4	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	4	no	—
312	C₂×C₄	—	188	8	no	—
313	C₂×C₄	#13093994	1	2	yes	—
314	C₂×C₄	#13094001	1	8	yes	—
315	C₂×C₄	#13094004	1	2	yes	—
316	C₂×C₄	#13094023	1	8	yes	—
317	C₂×C₄	#13094024	1	2	yes	—
318	C₂×C₄	C₂×#10625325	1	4	yes	—
319	C₂×C₄	C₂×#10625325	1	4	yes	—
320	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
321	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
322	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
323	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
324	C₈	—	2	8	no	—
325	C₂²	#15689044	1	4	yes	—
326	C₂²	#15689264	1	4	yes	—
327	C₂²	#15689276	1	4	yes	—
328	C₂²	#15689409	1	4	yes	—
329	C₂²	#15689414	1	4	yes	—
330	C₂²	#15689521	1	4	yes	—
331	C₂²	#15689532	1	4	yes	—
332	C₂²	C₂×#13093994	1	1	yes	characteristic
333	C₂²	C₂×#13094004	1	1	yes	characteristic
334	C₂²	C₂×#13094024	1	1	yes	characteristic
335	C₂²	C₂²×#10625325	1	1	yes	characteristic
336	C₂²	C₂²×#10625325	1	1	yes	characteristic
337	C₂²	C₂×#13094052	1	1	yes	characteristic
338	C₂²	C₂×#13094057	1	1	yes	characteristic
339	C₄	—	2	4	no	—
340	C₄	—	2	4	no	—
341	C₄	—	376	2	no	—
342	C₄	—	188	8	no	—
343	C₄	—	188	4	no	—
344	C₄	#15689044	1	4	yes	—
345	C₄	C₂×#13094001	1	2	yes	—
346	C₄	C₂×#13094023	1	2	yes	—
347	C₂	#20244722	1	8	yes	direct product factor
348	C₂	C₂×#15689044	1	1	yes	characteristic
349	C₂	C₂×#15689276	1	1	yes	characteristic
350	C₂	C₂×#15689264	1	1	yes	characteristic
351	C₂	C₂×#15689414	1	1	yes	characteristic
352	C₂	C₂×#15689409	1	1	yes	characteristic
353	C₂	C₂×#15689521	1	1	yes	characteristic
354	C₂	C₂×#15689532	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	C₂×#2058
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
4. #20244722	C₂	direct (N × H)
43. C₂×C₄⋊Dic₄₇	Q₈	semidirect
56. C₂²×C₈×C₄₇	Q₈	semidirect
95. C₂×C₈×C₄₇	C₂×Q₈	semidirect
106. C₄⋊Dic₄₇	C₂×Q₈	semidirect
176. C₂×C₄₇	#2058	semidirect
202. C₄₇	C₂×#2058	semidirect
215. C₂²×C₈	Dic₉₄	semidirect
273. C₂×C₈	C₂×Dic₉₄	semidirect
347. C₂	#20244722	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×#13094024	2
C₂²×#13093956	2
C₂×#15689021	2
#20244722	2
C₂×#15688886	2
C₂×C₄₇×#336	2
C₂×#2058	47

Subgroups — C2×#20244722 #26244306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244722 #26244306