Subgroups — C₂²×#15689858 #27123327

Order 12032.

8710 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689858	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13094021	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13094013	C₂	1	2	yes	maximal, complemented
4	C₂²×#13094012	C₂	1	2	yes	maximal, complemented
5	C₂²×#13094009	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₈₈	C₂	1	2	yes	maximal, complemented
8	C₂²×#13093986	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Dic₉₄	C₂²	1	1	yes	characteristic
11	C₂³×D₁₈₈	C₂²	1	2	yes	—
12	C₂³×D₁₈₈	C₂²	1	2	yes	—
13	C₂³×D₁₈₈	C₂²	1	2	yes	—
14	C₂³×C₄×D₄₇	C₂²	1	4	yes	—
15	#15689858	C₂²	1	256	yes	direct product factor
16	C₂×#13094021	C₂²	1	24	yes	—
17	C₂×#13094013	C₂²	1	48	yes	complemented
18	C₂×#13094012	C₂²	1	48	yes	complemented
19	C₂×#13094009	C₂²	1	96	yes	complemented
20	C₂×C₄×Dic₉₄	C₂²	1	24	yes	complemented
21	C₂×C₄×D₁₈₈	C₂²	1	48	yes	complemented
22	C₂×#13093986	C₂²	1	48	yes	complemented
23	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
26	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
27	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
28	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
29	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	C₂²	1	24	yes	complemented
31	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
33	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
34	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	C₂⁴×D₄₇	C₂³	1	4	yes	—
36	C₂²×Dic₉₄	—	2	8	no	—
37	C₂²×Dic₉₄	C₂³	1	12	yes	—
38	C₂²×D₁₈₈	—	2	16	no	—
39	C₂²×D₁₈₈	—	2	16	no	—
40	C₂²×D₁₈₈	—	4	8	no	—
41	C₂²×D₁₈₈	C₂³	1	24	yes	—
42	C₂²×D₁₈₈	C₂³	1	24	yes	—
43	C₂²×D₁₈₈	C₂³	1	24	yes	—
44	C₂³×Dic₄₇	C₂³	1	2	yes	—
45	C₂³×Dic₄₇	C₂³	1	2	yes	—
46	C₂²×C₄×D₄₇	—	4	16	no	—
47	C₂²×C₄×D₄₇	C₂³	1	48	yes	—
48	#13094021	C₂³	1	64	yes	—
49	#13094013	C₂³	1	128	yes	complemented
50	#13094012	C₂³	1	128	yes	complemented
51	#13094009	C₂³	1	256	yes	complemented
52	C₄×Dic₉₄	C₂³	1	64	yes	complemented
53	C₄×D₁₈₈	C₂³	1	128	yes	complemented
54	#13093986	C₂³	1	128	yes	complemented
55	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
56	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	—
57	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	48	yes	—
58	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	48	yes	—
59	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	48	yes	—
60	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
61	C₂²×Q₈×C₄₇	—	2	8	no	—
62	C₂²×Q₈×C₄₇	C₂³	1	12	yes	—
63	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
68	C₄×Q₈×C₄₇	C₂³	1	64	yes	complemented
69	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
70	C₂×C₄⋊C₄×C₄₇	C₂³	1	24	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	24	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
73	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂³×D₄₇	—	2	16	no	—
75	C₂³×D₄₇	—	2	16	no	—
76	C₂³×D₄₇	—	2	16	no	—
77	C₂³×D₄₇	C₂⁴	1	24	yes	—
78	C₂×Q₈×C₄₇	—	2	48	no	—
79	C₂×Q₈×C₄₇	C₂⁴	1	16	yes	—
80	C₂²×C₄×C₄₇	—	2	8	no	—
81	C₂²×C₄×C₄₇	—	2	8	no	—
82	C₂²×C₄×C₄₇	—	2	4	no	—
83	C₂²×C₄×C₄₇	#41	1	8	yes	—
84	C₂²×C₄×C₄₇	#41	1	8	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
90	C₂²×Dic₄₇	—	2	8	no	—
91	C₂²×Dic₄₇	—	2	8	no	—
92	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
93	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
94	C₂×D₁₈₈	—	2	96	no	—
95	C₂×D₁₈₈	—	2	96	no	—
96	C₂×D₁₈₈	—	4	48	no	—
97	C₂×D₁₈₈	C₂⁴	1	32	yes	—
98	C₂×D₁₈₈	C₂⁴	1	32	yes	—
99	C₂×D₁₈₈	C₂⁴	1	32	yes	—
100	C₂×C₄×D₄₇	—	4	96	no	—
101	C₂×C₄×D₄₇	C₂⁴	1	64	yes	—
102	C₂×Dic₉₄	—	2	48	no	—
103	C₂×Dic₉₄	C₂⁴	1	16	yes	—
104	C₄⋊C₄×C₄₇	C₂⁴	1	16	yes	—
105	C₄⋊C₄×C₄₇	C₂⁴	1	32	yes	—
106	C₄²×C₄₇	C₂⁴	1	32	yes	—
107	C₄²×C₄₇	C₂⁴	1	16	yes	—
108	(C₂×D₄₇)⋊C₄	C₂⁴	1	64	yes	—
109	(C₂×D₄₇)⋊C₄	C₂⁴	1	64	yes	—
110	(C₂×D₄₇)⋊C₄	C₂⁴	1	64	yes	—
111	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
112	Dic₄₇⋊C₄	C₂⁴	1	32	yes	—
113	C₄×Dic₄₇	C₂⁴	1	32	yes	—
114	C₂³×C₄₇	C₂×#41	1	4	yes	—
115	C₂³×C₄₇	C₂×#41	1	4	yes	—
116	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
117	C₂³×C₄₇	C₂⁵	1	3	yes	—
118	C₂²×D₄₇	—	2	48	no	—
119	C₂²×D₄₇	—	2	48	no	—
120	C₂²×D₄₇	—	2	48	no	—
121	C₂²×D₄₇	—	4	64	no	—
122	C₂²×D₄₇	C₂⁵	1	16	yes	—
123	Q₈×C₄₇	—	2	32	no	—
124	C₂×C₄×C₄₇	—	2	24	no	—
125	C₂×C₄×C₄₇	—	2	24	no	—
126	C₂×C₄×C₄₇	—	2	12	no	—
127	C₂×C₄×C₄₇	C₂×#41	1	24	yes	—
128	C₂×C₄×C₄₇	C₂×#41	1	24	yes	—
129	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
130	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
131	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
132	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
133	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
134	C₂×Dic₄₇	—	2	24	no	—
135	C₂×Dic₄₇	—	2	24	no	—
136	C₂×Dic₄₇	C₂⁵	1	8	yes	—
137	C₂×Dic₄₇	C₂⁵	1	8	yes	—
138	D₁₈₈	—	2	64	no	—
139	D₁₈₈	—	2	64	no	—
140	D₁₈₈	—	4	32	no	—
141	C₄×D₄₇	—	4	64	no	—
142	Dic₉₄	—	2	32	no	—
143	C₂²×#559	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	#559	1	16	yes	complemented
145	C₂²×C₄₇	C₂²×#41	1	6	yes	—
146	C₂²×C₄₇	C₂²×#41	1	6	yes	—
147	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	6	yes	—
148	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
149	C₂×D₄₇	—	2	16	no	—
150	C₂×D₄₇	—	2	16	no	—
151	C₂×D₄₇	—	2	16	no	—
152	C₂×D₄₇	—	4	96	no	—
153	C₄×C₄₇	—	2	8	no	—
154	C₄×C₄₇	—	2	8	no	—
155	C₄×C₄₇	—	2	4	no	—
156	C₄×C₄₇	C₂²×#41	1	8	yes	—
157	C₄×C₄₇	C₂²×#41	1	8	yes	—
158	Dic₄₇	—	2	8	no	—
159	Dic₄₇	—	2	8	no	—
160	C₂×#559	—	47	48	no	—
161	C₂²×#127	—	47	1	no	—
162	C₂²×#124	—	47	2	no	—
163	C₂²×#124	—	47	2	no	—
164	C₂²×#123	—	47	4	no	—
165	C₂²×C₄×Q₈	—	47	1	no	—
166	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
167	C₂²×C₄×D₄	—	47	2	no	—
168	C₂²×C₄×D₄	—	47	2	no	—
169	C₂×C₄₇	C₂×#559	1	12	yes	complemented
170	C₂×C₄₇	C₂³×#41	1	1	yes	characteristic
171	C₂×C₄₇	C₂³×#41	1	1	yes	characteristic
172	C₂×C₄₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
173	D₄₇	—	4	16	no	—
174	C₂³×Q₈	—	47	1	no	—
175	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂³×D₄	—	47	2	no	—
177	C₂³×D₄	—	47	2	no	—
178	C₂³×D₄	—	47	2	no	—
179	C₂⁴×C₄	—	47	4	no	—
180	#559	—	47	256	no	—
181	C₂×#127	—	47	24	no	—
182	C₂×#124	—	47	48	no	—
183	C₂×#124	—	47	48	no	—
184	C₂×#123	—	47	96	no	—
185	C₂×C₄×Q₈	—	47	24	no	—
186	C₂×C₄×Q₈	C₂×D₄₇	1	24	yes	complemented
187	C₂×C₄×D₄	—	47	48	no	—
188	C₂×C₄×D₄	—	47	48	no	—
189	C₂²×C₄⋊C₄	—	47	1	no	—
190	C₂²×C₄⋊C₄	—	47	2	no	—
191	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
192	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
193	C₂²×C₂²⋊C₄	—	47	4	no	—
194	C₂²×C₂²⋊C₄	—	47	4	no	—
195	C₂²×C₂²⋊C₄	—	47	4	no	—
196	C₂²×C₄²	—	47	2	no	—
197	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
198	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
199	C₄₇	C₂²×#559	1	1	yes	characteristic, complemented, Sylow
200	C₂⁵	—	47	4	no	—
201	C₂²×Q₈	—	2	8	no	—
202	C₂²×Q₈	—	47	12	no	—
203	C₂²×Q₈	—	94	8	no	—
204	C₂²×Q₈	C₂²×D₄₇	1	12	yes	—
205	C₂²×D₄	—	47	24	no	—
206	C₂²×D₄	—	47	24	no	—
207	C₂²×D₄	—	47	24	no	—
208	C₂²×D₄	—	94	16	no	—
209	C₂²×D₄	—	94	16	no	—
210	C₂²×D₄	—	188	8	no	—
211	C₂³×C₄	—	47	2	no	—
212	C₂³×C₄	—	47	2	no	—
213	C₂³×C₄	—	47	48	no	—
214	C₂³×C₄	—	188	16	no	—
215	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
216	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
217	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
219	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	#127	—	47	64	no	—
221	#124	—	47	128	no	—
222	#124	—	47	128	no	—
223	#123	—	47	256	no	—
224	C₄×Q₈	—	47	64	no	—
225	C₄×Q₈	C₂²×D₄₇	1	64	yes	complemented
226	C₄×D₄	—	47	128	no	—
227	C₄×D₄	—	47	128	no	—
228	C₂×C₄⋊C₄	—	47	12	no	—
229	C₂×C₄⋊C₄	—	47	24	no	—
230	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
231	C₂×C₄⋊C₄	C₂²×D₄₇	1	24	yes	—
232	C₂×C₂²⋊C₄	—	47	48	no	—
233	C₂×C₂²⋊C₄	—	47	48	no	—
234	C₂×C₂²⋊C₄	—	47	48	no	—
235	C₂×C₄²	—	47	24	no	—
236	C₂×C₄²	C₂²×D₄₇	1	24	yes	—
237	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
238	C₂⁴	—	47	24	no	—
239	C₂⁴	—	94	16	no	—
240	C₂⁴	—	94	16	no	—
241	C₂⁴	—	94	16	no	—
242	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
243	C₂×Q₈	—	2	48	no	—
244	C₂×Q₈	—	47	16	no	—
245	C₂×Q₈	—	94	48	no	—
246	C₂×Q₈	C₂³×D₄₇	1	16	yes	—
247	C₂×D₄	—	47	32	no	—
248	C₂×D₄	—	47	32	no	—
249	C₂×D₄	—	47	32	no	—
250	C₂×D₄	—	94	96	no	—
251	C₂×D₄	—	94	96	no	—
252	C₂×D₄	—	188	48	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	2	8	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	47	12	no	—
257	C₂²×C₄	—	47	12	no	—
258	C₂²×C₄	—	47	64	no	—
259	C₂²×C₄	—	94	8	no	—
260	C₂²×C₄	—	94	8	no	—
261	C₂²×C₄	—	188	96	no	—
262	C₂²×C₄	#10625350	1	8	yes	—
263	C₂²×C₄	#10625354	1	8	yes	—
264	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
265	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
266	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
267	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
268	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
269	C₄⋊C₄	—	47	32	no	—
270	C₄⋊C₄	—	47	16	no	—
271	C₄⋊C₄	C₂³×D₄₇	1	16	yes	—
272	C₄⋊C₄	C₂³×D₄₇	1	32	yes	—
273	C₂²⋊C₄	—	47	64	no	—
274	C₂²⋊C₄	—	47	64	no	—
275	C₂²⋊C₄	—	47	64	no	—
276	C₄²	—	47	32	no	—
277	C₄²	C₂³×D₄₇	1	32	yes	—
278	C₄²	C₂³×D₄₇	1	16	yes	—
279	C₂³	—	47	16	no	—
280	C₂³	—	94	48	no	—
281	C₂³	—	94	48	no	—
282	C₂³	—	94	48	no	—
283	C₂³	—	188	64	no	—
284	C₂³	C₂×#10625350	1	4	yes	—
285	C₂³	C₂×#10625354	1	4	yes	—
286	C₂³	#13094070	1	4	yes	—
287	C₂³	C₂⁴×D₄₇	1	3	yes	—
288	Q₈	—	2	32	no	—
289	Q₈	—	94	32	no	—
290	D₄	—	94	64	no	—
291	D₄	—	94	64	no	—
292	D₄	—	188	32	no	—
293	C₂×C₄	—	2	24	no	—
294	C₂×C₄	—	2	24	no	—
295	C₂×C₄	—	2	12	no	—
296	C₂×C₄	—	47	8	no	—
297	C₂×C₄	—	47	8	no	—
298	C₂×C₄	—	94	24	no	—
299	C₂×C₄	—	94	24	no	—
300	C₂×C₄	—	188	64	no	—
301	C₂×C₄	C₂×#10625350	1	24	yes	—
302	C₂×C₄	C₂×#10625354	1	24	yes	—
303	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
304	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
305	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
306	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
307	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
308	C₂²	—	94	16	no	—
309	C₂²	—	94	16	no	—
310	C₂²	—	94	16	no	—
311	C₂²	—	188	96	no	—
312	C₂²	#15689858	1	16	yes	direct product factor
313	C₂²	C₂²×#10625350	1	6	yes	—
314	C₂²	C₂²×#10625354	1	6	yes	—
315	C₂²	C₂×#13094070	1	6	yes	—
316	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
317	C₄	—	2	8	no	—
318	C₄	—	2	8	no	—
319	C₄	—	2	4	no	—
320	C₄	—	94	8	no	—
321	C₄	—	94	8	no	—
322	C₄	C₂²×#10625350	1	8	yes	—
323	C₄	C₂²×#10625354	1	8	yes	—
324	C₂	—	188	16	no	—
325	C₂	C₂×#15689858	1	12	yes	direct product factor
326	C₂	C₂³×#10625350	1	1	yes	characteristic
327	C₂	C₂³×#10625354	1	1	yes	characteristic
328	C₂	C₂²×#13094070	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂²×#13094021	C₂
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂²×#559
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. C₂×#15689858	C₂	direct (N × H)
3. C₂²×#13094013	C₂	semidirect
4. C₂²×#13094012	C₂	semidirect
5. C₂²×#13094009	C₂	semidirect
6. C₂²×C₄×Dic₉₄	C₂	semidirect
7. C₂²×C₄×D₁₈₈	C₂	semidirect
8. C₂²×#13093986	C₂	semidirect
9. C₂²×C₄×Q₈×C₄₇	C₂	semidirect
15. #15689858	C₂²	direct (N × H)
17. C₂×#13094013	C₂²	semidirect
18. C₂×#13094012	C₂²	semidirect
19. C₂×#13094009	C₂²	semidirect
20. C₂×C₄×Dic₉₄	C₂²	semidirect
21. C₂×C₄×D₁₈₈	C₂²	semidirect
22. C₂×#13093986	C₂²	semidirect
30. C₂×C₄×Q₈×C₄₇	C₂²	semidirect
49. #13094013	C₂³	semidirect
50. #13094012	C₂³	semidirect
51. #13094009	C₂³	semidirect
52. C₄×Dic₉₄	C₂³	semidirect
53. C₄×D₁₈₈	C₂³	semidirect
54. #13093986	C₂³	semidirect
68. C₄×Q₈×C₄₇	C₂³	semidirect
144. C₂²×C₄₇	#559	semidirect
166. C₂²×C₄×Q₈	D₄₇	semidirect
169. C₂×C₄₇	C₂×#559	semidirect
186. C₂×C₄×Q₈	C₂×D₄₇	semidirect
199. C₄₇	C₂²×#559	semidirect
225. C₄×Q₈	C₂²×D₄₇	semidirect
312. C₂²	#15689858	direct (N × H)
325. C₂	C₂×#15689858	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×#15689858	2
C₂²×#13094021	2
C₂²×#13094013	2
C₂²×#13094012	2
C₂²×#13094009	2
C₂²×C₄×Dic₉₄	2
C₂²×C₄×D₁₈₈	2
C₂²×#13093986	2
C₂²×C₄×Q₈×C₄₇	2
C₂²×#559	47

Subgroups — C22×#15689858 #27123327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689858 #27123327