Subgroups — C₂²×#15689697 #27123086

Order 12032.

11406 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10625350	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689697	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13094022	C₂	1	2	yes	maximal, complemented
4	C₂²×#13094013	C₂	1	2	yes	maximal, complemented
5	C₂²×#13094007	C₂	1	4	yes	maximal, complemented
6	C₂²×#13094000	C₂	1	4	yes	maximal, complemented
7	C₂²×#13093983	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625350	—	2	16	no	—
10	C₂²×#10625350	C₂²	1	24	yes	—
11	C₂³×Dic₉₄	C₂²	1	1	yes	characteristic
12	C₂³×D₁₈₈	C₂²	1	1	yes	characteristic
13	C₂³×C₂²⋊D₄₇	C₂²	1	2	yes	—
14	C₂³×C₄×D₄₇	C₂²	1	2	yes	—
15	#15689697	C₂²	1	256	yes	direct product factor
16	C₂×#13094022	C₂²	1	48	yes	complemented
17	C₂×#13094013	C₂²	1	48	yes	complemented
18	C₂×#13094007	C₂²	1	96	yes	complemented
19	C₂×#13094000	C₂²	1	96	yes	complemented
20	C₂×#13093983	C₂²	1	24	yes	complemented
21	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂²×Dic₄₇⋊C₄	C₂²	1	8	yes	—
23	C₂²×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
24	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
25	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
26	C₂³×Q₈×C₄₇	—	2	16	no	—
27	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
28	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
29	C₂³×Q₈×C₄₇	C₂²	1	24	yes	complemented
30	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
31	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂⁴×D₄₇	C₂³	1	2	yes	—
33	C₂×#10625350	—	2	192	no	—
34	C₂×#10625350	C₂³	1	64	yes	—
35	C₂²×Dic₉₄	—	2	8	no	—
36	C₂²×Dic₉₄	C₂³	1	12	yes	—
37	C₂²×D₁₈₈	—	2	8	no	—
38	C₂²×D₁₈₈	C₂³	1	12	yes	—
39	C₂²×C₂²⋊D₄₇	—	2	16	no	—
40	C₂²×C₂²⋊D₄₇	C₂³	1	24	yes	—
41	C₂³×Dic₄₇	C₂³	1	4	yes	—
42	C₂³×Dic₄₇	C₂³	1	2	yes	—
43	C₂²×C₄×D₄₇	—	2	16	no	—
44	C₂²×C₄×D₄₇	C₂³	1	24	yes	—
45	#13094022	C₂³	1	128	yes	complemented
46	#13094013	C₂³	1	128	yes	complemented
47	#13094007	C₂³	1	256	yes	complemented
48	#13094000	C₂³	1	256	yes	complemented
49	#13093983	C₂³	1	64	yes	complemented
50	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
51	C₂×Dic₄₇⋊C₄	C₂³	1	96	yes	—
52	C₂×C₂²⋊Dic₄₇	C₂³	1	24	yes	—
53	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	96	yes	—
54	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
55	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₄₇	—	2	16	no	—
57	C₂²×Q₈×C₄₇	—	2	16	no	—
58	C₂²×Q₈×C₄₇	—	2	192	no	—
59	C₂²×Q₈×C₄₇	C₂³	1	24	yes	—
60	C₂²×Q₈×C₄₇	C₂³	1	24	yes	—
61	C₂²×Q₈×C₄₇	C₂³	1	64	yes	complemented
62	C₂³×C₄×C₄₇	—	2	16	no	—
63	C₂³×C₄×C₄₇	D₄	1	16	yes	—
64	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
65	C₂³×C₄×C₄₇	C₂³	1	24	yes	—
66	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
69	C₂⁴×C₄₇	—	2	8	no	—
70	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
71	C₂⁴×C₄₇	C₂⁴	1	6	yes	—
72	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂³×D₄₇	—	2	8	no	—
74	C₂³×D₄₇	—	2	16	no	—
75	C₂³×D₄₇	C₂⁴	1	12	yes	—
76	C₂×Q₈×C₄₇	—	2	96	no	—
77	C₂×Q₈×C₄₇	—	2	96	no	—
78	C₂×Q₈×C₄₇	—	2	256	no	—
79	C₂×Q₈×C₄₇	C₂⁴	1	32	yes	—
80	C₂×Q₈×C₄₇	C₂⁴	1	32	yes	—
81	C₂²×C₄×C₄₇	—	2	96	no	—
82	C₂²×C₄×C₄₇	—	2	16	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	96	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	24	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	32	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
90	C₂²×C₄×C₄₇	C₂⁴	1	16	yes	—
91	C₂×C₂²⋊D₄₇	—	2	96	no	—
92	C₂×C₂²⋊D₄₇	C₂⁴	1	32	yes	—
93	C₂²×Dic₄₇	—	2	8	no	—
94	C₂²×Dic₄₇	—	2	16	no	—
95	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
96	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
97	#10625350	—	2	256	no	—
98	C₂×D₁₈₈	—	2	48	no	—
99	C₂×D₁₈₈	C₂⁴	1	16	yes	—
100	C₂×C₄×D₄₇	—	2	96	no	—
101	C₂×C₄×D₄₇	C₂⁴	1	32	yes	—
102	C₂×Dic₉₄	—	2	48	no	—
103	C₂×Dic₉₄	C₂⁴	1	16	yes	—
104	C₂²⋊Dic₄₇	C₂⁴	1	32	yes	—
105	(C₂×D₄₇)⋊C₄	C₂⁴	1	128	yes	—
106	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
107	Dic₄₇⋊C₄	C₂⁴	1	128	yes	—
108	C₄×Dic₄₇	C₂⁴	1	32	yes	—
109	C₂³×C₄₇	—	2	32	no	—
110	C₂³×C₄₇	—	2	24	no	—
111	C₂³×C₄₇	C₂²×D₄	1	24	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
113	C₂³×C₄₇	2₋¹⁺⁴	1	8	yes	—
114	C₂³×C₄₇	C₂⁵	1	4	yes	—
115	C₂³×C₄₇	C₂⁵	1	3	yes	—
116	C₂²×D₄₇	—	2	24	no	—
117	C₂²×D₄₇	—	2	48	no	—
118	C₂²×D₄₇	—	4	32	no	—
119	C₂²×D₄₇	C₂⁵	1	8	yes	—
120	Q₈×C₄₇	—	2	64	no	—
121	Q₈×C₄₇	—	2	64	no	—
122	C₂×C₄×C₄₇	—	2	64	no	—
123	C₂×C₄×C₄₇	—	2	48	no	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	24	yes	—
125	C₂×C₄×C₄₇	C₂²×D₄	1	24	yes	—
126	C₂×C₄×C₄₇	C₂²×D₄	1	64	yes	—
127	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
128	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
129	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
130	C₂²⋊D₄₇	—	2	64	no	—
131	C₂×Dic₄₇	—	2	24	no	—
132	C₂×Dic₄₇	—	2	48	no	—
133	C₂×Dic₄₇	C₂⁵	1	16	yes	—
134	C₂×Dic₄₇	C₂⁵	1	8	yes	—
135	D₁₈₈	—	2	32	no	—
136	C₄×D₄₇	—	2	64	no	—
137	Dic₉₄	—	2	32	no	—
138	C₂²×#536	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	8	no	—
140	C₂²×C₄₇	—	2	48	no	—
141	C₂²×C₄₇	#536	1	16	yes	complemented
142	C₂²×C₄₇	C₂³×D₄	1	8	yes	—
143	C₂²×C₄₇	C₂³×D₄	1	6	yes	—
144	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	12	yes	—
145	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
146	C₂×D₄₇	—	2	8	no	—
147	C₂×D₄₇	—	2	16	no	—
148	C₂×D₄₇	—	4	48	no	—
149	C₄×C₄₇	—	2	16	no	—
150	C₄×C₄₇	C₂³×D₄	1	8	yes	—
151	C₄×C₄₇	C₂³×D₄	1	8	yes	—
152	Dic₄₇	—	2	8	no	—
153	Dic₄₇	—	2	16	no	—
154	C₂³×#41	—	47	1	no	—
155	C₂⁴×Q₈	D₄₇	1	1	yes	characteristic, complemented
156	C₂×#536	—	47	48	no	—
157	C₂²×C₄⋊Q₈	—	47	2	no	—
158	C₂²×#124	—	47	2	no	—
159	C₂²×#123	—	47	4	no	—
160	C₂²×C₂²⋊Q₈	—	47	4	no	—
161	C₂²×#117	—	47	1	no	—
162	C₂×C₄₇	—	2	8	no	—
163	C₂×C₄₇	C₂×#536	1	12	yes	complemented
164	C₂×C₄₇	C₂⁴×D₄	1	1	yes	characteristic
165	C₂×C₄₇	C₂²×2₋¹⁺⁴	1	2	yes	—
166	D₄₇	—	4	8	no	—
167	C₂²×#41	—	47	24	no	—
168	C₂²×#41	—	94	16	no	—
169	C₂³×Q₈	—	2	16	no	—
170	C₂³×Q₈	—	47	1	no	—
171	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
172	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
173	C₂³×Q₈	C₂×D₄₇	1	24	yes	complemented
174	C₂³×D₄	—	47	2	no	—
175	C₂³×D₄	—	47	1	no	—
176	C₂⁴×C₄	—	47	2	no	—
177	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
178	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
179	#536	—	47	256	no	—
180	C₂×C₄⋊Q₈	—	47	48	no	—
181	C₂×#124	—	47	48	no	—
182	C₂×#123	—	47	96	no	—
183	C₂×C₂²⋊Q₈	—	47	96	no	—
184	C₂×#117	—	47	24	no	—
185	C₂²×C₄⋊C₄	—	47	2	no	—
186	C₂²×C₄⋊C₄	—	47	8	no	—
187	C₂²×C₂²⋊C₄	—	47	2	no	—
188	C₂²×C₂²⋊C₄	—	47	8	no	—
189	C₂²×C₄²	—	47	2	no	—
190	C₄₇	C₂²×#536	1	1	yes	characteristic, complemented, Sylow
191	C₂⁵	—	47	2	no	—
192	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×#41	—	47	64	no	—
194	C₂×#41	—	94	192	no	—
195	C₂²×Q₈	—	2	16	no	—
196	C₂²×Q₈	—	2	16	no	—
197	C₂²×Q₈	—	2	192	no	—
198	C₂²×Q₈	—	47	12	no	—
199	C₂²×Q₈	—	94	8	no	—
200	C₂²×Q₈	C₂²×D₄₇	1	24	yes	—
201	C₂²×Q₈	C₂²×D₄₇	1	24	yes	—
202	C₂²×Q₈	C₂²×D₄₇	1	64	yes	complemented
203	C₂²×D₄	—	47	24	no	—
204	C₂²×D₄	—	47	12	no	—
205	C₂²×D₄	—	94	16	no	—
206	C₂²×D₄	—	94	8	no	—
207	C₂³×C₄	—	2	16	no	—
208	C₂³×C₄	—	47	4	no	—
209	C₂³×C₄	—	47	2	no	—
210	C₂³×C₄	—	47	24	no	—
211	C₂³×C₄	—	94	16	no	—
212	C₂³×C₄	C₂²⋊D₄₇	1	16	yes	—
213	C₂³×C₄	C₂²×D₄₇	1	24	yes	—
214	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
215	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
216	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
217	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊Q₈	—	47	128	no	—
219	#124	—	47	128	no	—
220	#123	—	47	256	no	—
221	C₂²⋊Q₈	—	47	256	no	—
222	#117	—	47	64	no	—
223	C₂×C₄⋊C₄	—	47	24	no	—
224	C₂×C₄⋊C₄	—	47	96	no	—
225	C₂×C₂²⋊C₄	—	47	24	no	—
226	C₂×C₂²⋊C₄	—	47	96	no	—
227	C₂×C₄²	—	47	24	no	—
228	C₂⁴	—	2	8	no	—
229	C₂⁴	—	47	12	no	—
230	C₂⁴	—	94	8	no	—
231	C₂⁴	—	94	16	no	—
232	C₂⁴	C₂×C₂²⋊D₄₇	1	8	yes	—
233	C₂⁴	C₂³×D₄₇	1	6	yes	—
234	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
235	#41	—	94	256	no	—
236	C₂×Q₈	—	2	96	no	—
237	C₂×Q₈	—	2	96	no	—
238	C₂×Q₈	—	2	256	no	—
239	C₂×Q₈	—	47	16	no	—
240	C₂×Q₈	—	94	48	no	—
241	C₂×Q₈	C₂³×D₄₇	1	32	yes	—
242	C₂×Q₈	C₂³×D₄₇	1	32	yes	—
243	C₂×D₄	—	47	32	no	—
244	C₂×D₄	—	47	16	no	—
245	C₂×D₄	—	94	96	no	—
246	C₂×D₄	—	94	48	no	—
247	C₂²×C₄	—	2	96	no	—
248	C₂²×C₄	—	2	16	no	—
249	C₂²×C₄	—	47	24	no	—
250	C₂²×C₄	—	47	12	no	—
251	C₂²×C₄	—	47	32	no	—
252	C₂²×C₄	—	94	8	no	—
253	C₂²×C₄	—	94	16	no	—
254	C₂²×C₄	—	94	96	no	—
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
257	C₂²×C₄	C₂×C₂²⋊D₄₇	1	96	yes	—
258	C₂²×C₄	C₂³×D₄₇	1	24	yes	—
259	C₂²×C₄	C₂³×D₄₇	1	32	yes	—
260	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
261	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
262	C₂²×C₄	C₂³×D₄₇	1	16	yes	—
263	C₄⋊C₄	—	47	128	no	—
264	C₄⋊C₄	—	47	32	no	—
265	C₂²⋊C₄	—	47	32	no	—
266	C₂²⋊C₄	—	47	128	no	—
267	C₄²	—	47	32	no	—
268	C₂³	—	2	32	no	—
269	C₂³	—	2	24	no	—
270	C₂³	—	47	8	no	—
271	C₂³	—	94	24	no	—
272	C₂³	—	94	48	no	—
273	C₂³	—	188	32	no	—
274	C₂³	C₂²×C₂²⋊D₄₇	1	24	yes	—
275	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
276	C₂³	#13094072	1	8	yes	—
277	C₂³	C₂⁴×D₄₇	1	4	yes	—
278	C₂³	C₂⁴×D₄₇	1	3	yes	—
279	Q₈	—	2	64	no	—
280	Q₈	—	2	64	no	—
281	Q₈	—	94	32	no	—
282	D₄	—	94	64	no	—
283	D₄	—	94	32	no	—
284	C₂×C₄	—	2	64	no	—
285	C₂×C₄	—	2	48	no	—
286	C₂×C₄	—	47	16	no	—
287	C₂×C₄	—	47	8	no	—
288	C₂×C₄	—	94	24	no	—
289	C₂×C₄	—	94	48	no	—
290	C₂×C₄	—	94	64	no	—
291	C₂×C₄	C₂²×C₂²⋊D₄₇	1	24	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₄₇	1	24	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₄₇	1	64	yes	—
294	C₂×C₄	C₂⁴×D₄₇	1	16	yes	—
295	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
296	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	48	no	—
299	C₂²	—	94	8	no	—
300	C₂²	—	94	16	no	—
301	C₂²	—	188	48	no	—
302	C₂²	#15689697	1	16	yes	direct product factor
303	C₂²	C₂³×C₂²⋊D₄₇	1	8	yes	—
304	C₂²	C₂³×C₂²⋊D₄₇	1	6	yes	—
305	C₂²	C₂×#13094072	1	12	yes	—
306	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
307	C₄	—	2	16	no	—
308	C₄	—	94	8	no	—
309	C₄	—	94	16	no	—
310	C₄	C₂³×C₂²⋊D₄₇	1	8	yes	—
311	C₄	C₂³×C₂²⋊D₄₇	1	8	yes	—
312	C₂	—	2	8	no	—
313	C₂	—	188	8	no	—
314	C₂	C₂×#15689697	1	12	yes	direct product factor
315	C₂	C₂⁴×C₂²⋊D₄₇	1	1	yes	characteristic
316	C₂	C₂²×#13094072	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂⁴×Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂³×#10625350	C₂
hypercenter	C₂⁴×Q₈	D₄₇
hypocenter	C₄₇	C₂²×#536
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15689697	C₂	direct (N × H)
3. C₂²×#13094022	C₂	semidirect
4. C₂²×#13094013	C₂	semidirect
5. C₂²×#13094007	C₂	semidirect
6. C₂²×#13094000	C₂	semidirect
7. C₂²×#13093983	C₂	semidirect
8. C₂⁴×Q₈×C₄₇	C₂	semidirect
15. #15689697	C₂²	direct (N × H)
16. C₂×#13094022	C₂²	semidirect
17. C₂×#13094013	C₂²	semidirect
18. C₂×#13094007	C₂²	semidirect
19. C₂×#13094000	C₂²	semidirect
20. C₂×#13093983	C₂²	semidirect
29. C₂³×Q₈×C₄₇	C₂²	semidirect
45. #13094022	C₂³	semidirect
46. #13094013	C₂³	semidirect
47. #13094007	C₂³	semidirect
48. #13094000	C₂³	semidirect
49. #13093983	C₂³	semidirect
61. C₂²×Q₈×C₄₇	C₂³	semidirect
141. C₂²×C₄₇	#536	semidirect
155. C₂⁴×Q₈	D₄₇	semidirect
163. C₂×C₄₇	C₂×#536	semidirect
173. C₂³×Q₈	C₂×D₄₇	semidirect
190. C₄₇	C₂²×#536	semidirect
202. C₂²×Q₈	C₂²×D₄₇	semidirect
302. C₂²	#15689697	direct (N × H)
314. C₂	C₂×#15689697	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#10625350	2
C₂×#15689697	2
C₂²×#13094022	2
C₂²×#13094013	2
C₂²×#13094007	2
C₂²×#13094000	2
C₂²×#13093983	2
C₂⁴×Q₈×C₄₇	2
C₂²×#536	47

Subgroups — C22×#15689697 #27123086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689697 #27123086