Subgroups — C₂×C₄×#13094024 #26374800

Order 12032.

3358 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	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₉₄	C₂	1	2	yes	maximal
3	C₄×#13094024	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15689029	C₂	1	2	yes	maximal
5	C₂×#15689021	C₂	1	4	yes	maximal
6	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₄⋊Dic₄₇	C₂	1	4	yes	maximal
8	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₉₄	C₂²	1	2	yes	—
10	C₂×#13094024	C₄	1	16	yes	direct product factor
11	C₂×#13094024	C₂²	1	8	yes	—
12	C₂×C₄×Dic₉₄	—	2	16	no	—
13	C₂×C₄×Dic₉₄	C₂²	1	16	yes	—
14	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
16	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
18	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
19	#15689029	C₂²	1	16	yes	—
20	#15689021	C₂²	1	32	yes	—
21	#15688993	C₂²	1	8	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	32	yes	—
23	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
29	C₄³×C₄₇	C₂²	1	8	yes	—
30	C₂²×Dic₉₄	—	2	8	no	—
31	C₂²×Dic₉₄	—	2	8	no	—
32	C₂²×Dic₉₄	C₂×C₄	1	16	yes	—
33	C₂²×Dic₉₄	C₂³	1	8	yes	—
34	C₂³×Dic₄₇	C₂³	1	4	yes	—
35	C₂³×Dic₄₇	C₂³	1	4	yes	—
36	#13094024	C₂×C₄	1	64	yes	direct product factor
37	C₄×Dic₉₄	—	2	64	no	—
38	C₂×C₄⋊Dic₄₇	—	2	8	no	—
39	C₂×C₄⋊Dic₄₇	—	2	16	no	—
40	C₂×C₄⋊Dic₄₇	C₂×C₄	1	32	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
44	C₂×Dic₄₇⋊C₄	—	2	16	no	—
45	C₂×Dic₄₇⋊C₄	C₂³	1	16	yes	—
46	C₂×C₄×Dic₄₇	—	2	16	no	—
47	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
48	C₄₇⋊#95	C₂³	1	32	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	—	2	4	no	—
55	C₂×C₄²×C₄₇	—	2	8	no	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
57	C₂×C₄²×C₄₇	D₄	1	8	yes	—
58	C₂×C₄²×C₄₇	Q₈	1	16	yes	complemented
59	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
64	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
74	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
76	C₂²×C₄×C₄₇	#41	1	4	yes	—
77	C₂²×C₄×C₄₇	#41	1	8	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×Dic₄₇	—	2	8	no	—
84	C₂²×Dic₄₇	—	2	8	no	—
85	C₂²×Dic₄₇	—	2	8	no	—
86	C₂²×Dic₄₇	—	2	16	no	—
87	C₂²×Dic₄₇	C₂²×C₄	1	16	yes	—
88	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
89	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
90	C₂×Dic₉₄	—	2	64	no	—
91	C₂×Dic₉₄	—	2	16	no	—
92	C₂×Dic₉₄	—	2	16	no	—
93	C₂×Dic₉₄	C₂²×C₄	1	32	yes	—
94	C₄²×C₄₇	—	2	8	no	—
95	C₄²×C₄₇	—	2	16	no	—
96	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
97	C₄²×C₄₇	C₂×D₄	1	16	yes	—
98	C₄²×C₄₇	C₂×Q₈	1	32	yes	complemented
99	C₄⋊Dic₄₇	—	2	16	no	—
100	C₄⋊Dic₄₇	—	2	32	no	—
101	C₄⋊Dic₄₇	C₂²×C₄	1	64	yes	—
102	Dic₄₇⋊C₄	—	2	32	no	—
103	C₄×Dic₄₇	—	2	32	no	—
104	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
105	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
106	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
107	C₂³×C₄₇	C₂×#41	1	2	yes	—
108	C₂³×C₄₇	C₂×#41	1	4	yes	—
109	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	2	16	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	C₄×D₄	1	16	yes	—
116	C₂×C₄×C₄₇	C₄×Q₈	1	32	yes	complemented
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	16	yes	complemented
118	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
123	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
124	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
125	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
127	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
128	C₂×Dic₄₇	—	2	32	no	—
129	C₂×Dic₄₇	—	2	8	no	—
130	C₂×Dic₄₇	—	2	8	no	—
131	C₂×Dic₄₇	—	2	8	no	—
132	C₂×Dic₄₇	—	2	16	no	—
133	C₂×Dic₄₇	—	4	16	no	—
134	C₂×Dic₄₇	C₂³×C₄	1	16	yes	—
135	Dic₉₄	—	2	64	no	—
136	C₂×C₄×C₄⋊Q₈	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	C₂×C₄×D₄	1	4	yes	—
138	C₂²×C₄₇	C₂×C₄×Q₈	1	8	yes	—
139	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
140	C₂²×C₄₇	#532	1	4	yes	—
141	C₂²×C₄₇	#533	1	8	yes	—
142	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
144	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
145	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×#41	1	2	yes	—
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	—	2	8	no	—
149	C₄×C₄₇	C₂×C₄×D₄	1	8	yes	—
150	C₄×C₄₇	C₂×C₄×Q₈	1	16	yes	complemented
151	C₄×C₄₇	C₂×C₄⋊Q₈	1	8	yes	complemented
152	Dic₄₇	—	2	16	no	—
153	Dic₄₇	—	4	8	no	—
154	C₂²×C₄⋊Q₈	—	47	1	no	—
155	C₂²×C₄×Q₈	—	47	2	no	—
156	C₄×C₄⋊Q₈	—	47	16	no	—
157	C₂×#391	—	47	2	no	—
158	C₂×#389	—	47	4	no	—
159	C₂×#384	—	47	1	no	—
160	C₂×C₄×C₄⋊C₄	—	47	4	no	—
161	C₂×C₄³	D₄₇	1	1	yes	characteristic
162	C₂×C₄₇	C₄×C₄⋊Q₈	1	8	yes	complemented
163	C₂×C₄₇	C₂²×C₄×D₄	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×C₄×Q₈	1	2	yes	—
165	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#532	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#533	1	2	yes	—
168	C₂³×Q₈	—	47	2	no	—
169	C₂×C₄⋊Q₈	—	47	16	no	—
170	C₂×C₄⋊Q₈	—	47	8	no	—
171	C₂×C₄×Q₈	—	47	16	no	—
172	C₂×C₄×Q₈	—	94	16	no	—
173	C₂²×C₄⋊C₄	—	47	4	no	—
174	C₂²×C₄⋊C₄	—	47	4	no	—
175	C₂²×C₄⋊C₄	—	47	2	no	—
176	C₂²×C₄⋊C₄	—	47	4	no	—
177	C₂²×C₄²	—	47	4	no	—
178	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
179	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
181	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	#391	—	47	16	no	—
184	#389	—	47	32	no	—
185	#384	—	47	8	no	—
186	C₄×C₄⋊C₄	—	47	32	no	—
187	C₂×#95	—	47	8	no	—
188	C₄³	C₂×D₄₇	1	8	yes	—
189	C₄₇	C₂×C₄×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	47	16	no	—
191	C₂²×Q₈	—	47	8	no	—
192	C₂²×Q₈	—	94	8	no	—
193	C₂²×Q₈	—	94	8	no	—
194	C₂³×C₄	—	47	4	no	—
195	C₂³×C₄	—	47	4	no	—
196	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
199	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₄⋊Q₈	—	47	64	no	—
202	C₄×Q₈	—	94	64	no	—
203	C₂×C₄⋊C₄	—	47	32	no	—
204	C₂×C₄⋊C₄	—	47	16	no	—
205	C₂×C₄⋊C₄	—	47	16	no	—
206	C₂×C₄⋊C₄	—	47	8	no	—
207	C₂×C₄⋊C₄	—	47	16	no	—
208	C₂×C₄⋊C₄	—	94	16	no	—
209	C₂×C₄⋊C₄	—	94	8	no	—
210	C₂×C₄⋊C₄	—	94	16	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	—	2	8	no	—
213	C₂×C₄²	—	47	16	no	—
214	C₂×C₄²	—	94	16	no	—
215	C₂×C₄²	Dic₉₄	1	16	yes	complemented
216	C₂×C₄²	C₄×D₄₇	1	8	yes	—
217	C₂×C₄²	D₁₈₈	1	8	yes	—
218	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
219	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
220	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
221	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
222	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
223	#95	—	47	32	no	—
224	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
225	C₂×Q₈	—	47	32	no	—
226	C₂×Q₈	—	94	64	no	—
227	C₂×Q₈	—	94	16	no	—
228	C₂×Q₈	—	94	16	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	47	16	no	—
233	C₂²×C₄	—	47	8	no	—
234	C₂²×C₄	—	47	8	no	—
235	C₂²×C₄	—	94	8	no	—
236	C₂²×C₄	—	94	8	no	—
237	C₂²×C₄	—	94	8	no	—
238	C₂²×C₄	—	94	16	no	—
239	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
240	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
241	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
242	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
243	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
244	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
245	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
246	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
247	C₂²×C₄	#10625350	1	4	yes	—
248	C₂²×C₄	#10625350	1	8	yes	—
249	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
250	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
251	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
252	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
253	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
254	C₄⋊C₄	—	47	64	no	—
255	C₄⋊C₄	—	94	32	no	—
256	C₄⋊C₄	—	94	16	no	—
257	C₄⋊C₄	—	94	32	no	—
258	C₄²	—	2	8	no	—
259	C₄²	—	2	16	no	—
260	C₄²	—	94	32	no	—
261	C₄²	C₂×Dic₉₄	1	32	yes	complemented
262	C₄²	C₂×C₄×D₄₇	1	16	yes	—
263	C₄²	C₂×D₁₈₈	1	16	yes	—
264	C₂³	C₂²×C₄×D₄₇	1	2	yes	—
265	C₂³	C₂²×D₁₈₈	1	2	yes	—
266	C₂³	C₂²×Dic₉₄	1	4	yes	—
267	C₂³	C₂×#10625350	1	2	yes	—
268	C₂³	C₂×#10625350	1	4	yes	—
269	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
270	Q₈	—	94	64	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	16	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	47	16	no	—
277	C₂×C₄	—	94	32	no	—
278	C₂×C₄	—	94	8	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	—	94	8	no	—
281	C₂×C₄	—	94	16	no	—
282	C₂×C₄	—	188	16	no	—
283	C₂×C₄	C₄×D₁₈₈	1	16	yes	—
284	C₂×C₄	C₄×Dic₉₄	1	32	yes	complemented
285	C₂×C₄	#13094024	1	16	yes	direct product factor
286	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
287	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
288	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
289	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
290	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
291	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
292	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
293	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
294	C₂×C₄	C₂×#10625350	1	4	yes	—
295	C₂×C₄	C₂×#10625350	1	8	yes	—
296	C₂²	C₂×C₄×D₁₈₈	1	4	yes	—
297	C₂²	C₂×C₄×Dic₉₄	1	8	yes	—
298	C₂²	C₂×#13094024	1	4	yes	—
299	C₂²	#15689673	1	4	yes	—
300	C₂²	#15689680	1	8	yes	—
301	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
302	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
303	C₂²	C₂³×Dic₉₄	1	2	yes	—
304	C₂²	C₂²×#10625350	1	1	yes	characteristic
305	C₂²	C₂²×#10625350	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	2	8	no	—
308	C₄	—	94	16	no	—
309	C₄	—	188	8	no	—
310	C₄	C₂×C₄×D₁₈₈	1	8	yes	—
311	C₄	C₂×C₄×Dic₉₄	1	16	yes	complemented
312	C₄	C₂×#13094024	1	8	yes	direct product factor
313	C₂	C₄×#13094024	1	8	yes	direct product factor
314	C₂	C₂²×C₄×D₁₈₈	1	1	yes	characteristic
315	C₂	C₂²×C₄×Dic₉₄	1	2	yes	—
316	C₂	C₂²×#13094024	1	1	yes	characteristic
317	C₂	C₂×#15689673	1	1	yes	characteristic
318	C₂	C₂×#15689680	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	C₂×C₄×C₄⋊Q₈
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. C₄×#13094024	C₂	direct (N × H)
10. C₂×#13094024	C₄	direct (N × H)
36. #13094024	C₂×C₄	direct (N × H)
58. C₂×C₄²×C₄₇	Q₈	semidirect
98. C₄²×C₄₇	C₂×Q₈	semidirect
116. C₂×C₄×C₄₇	C₄×Q₈	semidirect
117. C₂×C₄×C₄₇	C₄⋊Q₈	semidirect
150. C₄×C₄₇	C₂×C₄×Q₈	semidirect
151. C₄×C₄₇	C₂×C₄⋊Q₈	semidirect
162. C₂×C₄₇	C₄×C₄⋊Q₈	semidirect
189. C₄₇	C₂×C₄×C₄⋊Q₈	semidirect
215. C₂×C₄²	Dic₉₄	semidirect
261. C₄²	C₂×Dic₉₄	semidirect
284. C₂×C₄	C₄×Dic₉₄	semidirect
285. C₂×C₄	#13094024	direct (N × H)
311. C₄	C₂×C₄×Dic₉₄	semidirect
312. C₄	C₂×#13094024	direct (N × H)
313. C₂	C₄×#13094024	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×C₄×Dic₉₄	2
C₄×#13094024	2
C₂×#15689029	2
C₂×#15689021	2
C₂×#15688993	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂×C₄³×C₄₇	2
C₂×C₄×C₄⋊Q₈	47

Subgroups — C2×C4×#13094024 #26374800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13094024 #26374800