Subgroups — C₂×#20256019 #27095830

Order 12032.

2402 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20256019	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689469	C₂	1	3	yes	maximal, complemented
4	C₂×#15689403	C₂	1	3	yes	maximal, complemented
5	C₂×#15689315	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#10625334	C₂	1	3	yes	maximal, complemented
8	C₂×C₄₇×#550	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₄₇	—	2	1	no	—
10	C₂²×#10625334	C₂²	1	3	yes	—
11	Q₈×D₁₈₈	—	2	4	no	—
12	Q₈×D₁₈₈	C₂²	1	8	yes	complemented
13	C₂×#13094024	C₂²	1	3	yes	—
14	C₂×#13094004	—	2	3	no	—
15	C₂×C₄×D₁₈₈	C₂²	1	3	yes	—
16	#15689469	C₂²	1	24	yes	complemented
17	#15689403	C₂²	1	24	yes	complemented
18	#15689315	C₂²	1	24	yes	complemented
19	Q₈×C₄₇⋊C₈	C₂²	1	8	yes	complemented
20	C₄×#10625334	C₂²	1	24	yes	complemented
21	C₂×#13093957	C₂²	1	3	yes	—
22	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
23	C₂×#13093945	C₂²	1	6	yes	—
24	C₂×#13093945	C₂²	1	3	yes	—
25	C₂×#13093944	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
27	C₂²×#41×C₄₇	—	2	1	no	—
28	C₄₇×#550	—	2	4	no	—
29	C₄₇×#550	C₂²	1	8	yes	complemented
30	C₂×#127×C₄₇	C₂²	1	3	yes	—
31	C₂×#121×C₄₇	—	2	3	no	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₄₇	C₂²	1	3	yes	—
34	C₂²×Dic₉₄	—	2	3	no	—
35	C₂×Q₈×D₄₇	—	2	8	no	—
36	C₂×Q₈×D₄₇	—	4	8	no	—
37	C₂²×D₁₈₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄×D₄₇	—	2	3	no	—
39	C₂×#10625334	—	2	12	no	—
40	C₂×#10625334	C₂³	1	12	yes	—
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
43	#13094024	—	2	6	no	—
44	#13094024	C₂³	1	12	yes	—
45	#13094004	—	2	24	no	—
46	C₄×D₁₈₈	—	2	6	no	—
47	C₄×D₁₈₈	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₄₇	—	2	3	no	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	3	yes	—
50	C₂×(C₂×D₄₇)⋊C₄	—	2	3	no	—
51	#13093957	C₂³	1	12	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	12	yes	—
53	#13093945	C₂³	1	24	yes	—
54	#13093945	C₂³	1	12	yes	—
55	#13093944	C₂³	1	4	yes	—
56	C₄×C₄₇⋊C₈	C₂³	1	12	yes	—
57	C₂×#41×C₄₇	—	2	8	no	—
58	C₂×#41×C₄₇	—	4	8	no	—
59	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₄₇	—	2	3	no	—
61	C₂²×D₄×C₄₇	—	2	3	no	—
62	C₂²×D₄×C₄₇	C₂³	1	3	yes	—
63	C₂³×C₄×C₄₇	—	2	3	no	—
64	#127×C₄₇	—	2	6	no	—
65	#127×C₄₇	C₂³	1	12	yes	—
66	#121×C₄₇	—	2	24	no	—
67	C₄×Q₈×C₄₇	D₄	1	4	yes	—
68	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
69	C₄×D₄×C₄₇	—	2	6	no	—
70	C₄×D₄×C₄₇	C₂³	1	12	yes	—
71	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
72	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₄₇	—	2	3	no	—
74	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
75	C₂⁴×C₄₇	—	2	3	no	—
76	C₂³×D₄₇	—	2	1	no	—
77	#41×C₄₇	—	4	32	no	—
78	C₂×Q₈×C₄₇	D₈	1	8	yes	complemented
79	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
80	C₂×D₄×C₄₇	—	2	3	no	—
81	C₂×D₄×C₄₇	—	2	12	no	—
82	C₂×D₄×C₄₇	—	2	12	no	—
83	C₂×D₄×C₄₇	—	2	6	no	—
84	C₂×D₄×C₄₇	—	2	6	no	—
85	C₂×D₄×C₄₇	—	4	12	no	—
86	C₂×D₄×C₄₇	—	8	6	no	—
87	C₂×D₄×C₄₇	C₂⁴	1	6	yes	—
88	C₂²×C₄×C₄₇	—	2	1	no	—
89	C₂²×C₄×C₄₇	—	2	12	no	—
90	C₂²×C₄×C₄₇	—	4	12	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
92	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₇	—	2	3	no	—
95	Q₈×D₄₇	—	4	32	no	—
96	C₂×D₁₈₈	—	2	1	no	—
97	C₂×D₁₈₈	—	2	2	no	—
98	C₂×D₁₈₈	#41	1	4	yes	complemented
99	C₂×D₁₈₈	C₂⁴	1	2	yes	—
100	C₂×C₄×D₄₇	—	2	12	no	—
101	C₂×C₄×D₄₇	—	4	12	no	—
102	C₂×Dic₉₄	—	2	12	no	—
103	C₂×Dic₉₄	—	4	12	no	—
104	C₄⋊C₄×C₄₇	—	2	1	no	—
105	C₄⋊C₄×C₄₇	C₂×D₄	1	12	yes	—
106	C₄⋊C₄×C₄₇	C₂⁴	1	2	yes	—
107	C₂²⋊C₄×C₄₇	—	2	12	no	—
108	C₄²×C₄₇	C₂×D₄	1	6	yes	—
109	C₄²×C₄₇	C₂⁴	1	6	yes	—
110	#10625334	—	2	24	no	—
111	(C₂×D₄₇)⋊C₄	—	2	12	no	—
112	C₄⋊Dic₄₇	—	2	12	no	—
113	C₄⋊Dic₄₇	—	2	3	no	—
114	C₄⋊Dic₄₇	C₂⁴	1	6	yes	—
115	C₂×C₄₇⋊C₈	—	2	6	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
118	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
119	C₂³×C₄₇	—	2	6	no	—
120	C₂³×C₄₇	—	4	6	no	—
121	C₂³×C₄₇	—	4	6	no	—
122	C₂³×C₄₇	—	4	6	no	—
123	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂²×D₄₇	—	2	4	no	—
125	C₂²×D₄₇	—	2	2	no	—
126	C₂²×D₄₇	—	4	2	no	—
127	C₂²×D₄₇	—	4	2	no	—
128	Q₈×C₄₇	C₂×D₈	1	16	yes	complemented
129	D₄×C₄₇	—	2	6	no	—
130	D₄×C₄₇	—	2	6	no	—
131	D₄×C₄₇	—	4	6	no	—
132	D₄×C₄₇	—	4	24	no	—
133	D₄×C₄₇	—	8	12	no	—
134	C₂×C₄×C₄₇	—	2	6	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	24	no	—
138	C₂×C₄×C₄₇	C₂×D₈	1	12	yes	—
139	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
140	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
142	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
143	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
144	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
145	C₂×Dic₄₇	—	2	6	no	—
146	C₂×Dic₄₇	—	4	6	no	—
147	D₁₈₈	—	2	4	no	—
148	D₁₈₈	—	2	2	no	—
149	D₁₈₈	C₂×#41	1	4	yes	complemented
150	C₄×D₄₇	—	4	24	no	—
151	Dic₉₄	—	4	24	no	—
152	C₄₇⋊C₈	—	2	6	no	—
153	C₄₇⋊C₈	—	2	2	no	—
154	C₂×#3358	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	4	6	no	—
156	C₂²×C₄₇	—	4	6	no	—
157	C₂²×C₄₇	—	4	6	no	—
158	C₂²×C₄₇	—	8	12	no	—
159	C₂²×C₄₇	#548	1	2	yes	—
160	C₂²×C₄₇	C₂²×D₈	1	2	yes	—
161	C₂²×C₄₇	#577	1	2	yes	—
162	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
163	C₂×D₄₇	—	2	4	no	—
164	C₂×D₄₇	—	4	8	no	—
165	C₂×D₄₇	—	4	2	no	—
166	C₂×D₄₇	—	4	2	no	—
167	C₄×C₄₇	—	2	6	no	—
168	C₄×C₄₇	—	4	2	no	—
169	C₄×C₄₇	#548	1	2	yes	—
170	C₄×C₄₇	C₂²×D₈	1	12	yes	—
171	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
172	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
173	Dic₄₇	—	4	6	no	—
174	C₂×#550	D₄₇	1	1	yes	characteristic, complemented
175	C₂×D₄×Q₈	—	47	1	no	—
176	#3358	—	47	16	no	—
177	C₂×#486	—	47	3	no	—
178	C₂×#474	—	47	3	no	—
179	C₂×#459	—	47	3	no	—
180	C₂×C₈×Q₈	—	47	1	no	—
181	C₂×C₄×D₈	—	47	3	no	—
182	C₂×C₄₇	—	8	6	no	—
183	C₂×C₄₇	#3358	1	4	yes	complemented
184	C₂×C₄₇	C₂×#548	1	1	yes	characteristic
185	C₂×C₄₇	C₂³×D₈	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#577	1	1	yes	characteristic
187	D₄₇	—	4	4	no	—
188	C₂²×#41	—	2	1	no	—
189	C₂³×Q₈	—	94	1	no	—
190	C₂²×D₈	—	47	3	no	—
191	#550	—	2	4	no	—
192	#550	C₂×D₄₇	1	8	yes	complemented
193	D₄×Q₈	—	47	8	no	—
194	D₄×Q₈	—	94	4	no	—
195	C₂×C₄⋊Q₈	—	47	3	no	—
196	C₂×#127	C₂×D₄₇	1	3	yes	—
197	C₂×C₂²⋊Q₈	—	94	3	no	—
198	C₂×#121	—	2	3	no	—
199	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
200	C₂×C₄×D₄	—	47	3	no	—
201	C₂×C₄×D₄	C₂×D₄₇	1	3	yes	—
202	#486	—	47	24	no	—
203	#474	—	47	24	no	—
204	#459	—	47	24	no	—
205	C₈×Q₈	—	47	8	no	—
206	C₄×D₈	—	47	24	no	—
207	C₂×#107	—	47	3	no	—
208	C₂×C₄⋊C₈	—	47	3	no	—
209	C₂×#102	—	47	3	no	—
210	C₂×#102	—	47	6	no	—
211	C₂×#102	—	47	1	no	—
212	C₂×C₄×C₈	—	47	3	no	—
213	C₄₇	C₂×#3358	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	2	8	no	—
215	C₂×#41	—	4	8	no	—
216	C₂²×Q₈	—	94	8	no	—
217	C₂²×Q₈	—	94	3	no	—
218	C₂²×Q₈	—	188	8	no	—
219	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×D₄	—	2	3	no	—
221	C₂²×D₄	—	2	3	no	—
222	C₂²×D₄	—	47	1	no	—
223	C₂²×D₄	C₂²×D₄₇	1	3	yes	—
224	C₂³×C₄	—	2	3	no	—
225	C₂³×C₄	—	94	3	no	—
226	C₂×D₈	—	47	12	no	—
227	C₂×D₈	—	94	12	no	—
228	C₂²×C₈	—	47	3	no	—
229	C₂²×C₈	—	47	1	no	—
230	C₄⋊Q₈	—	47	12	no	—
231	C₄⋊Q₈	—	94	6	no	—
232	#127	—	2	6	no	—
233	#127	C₂²×D₄₇	1	12	yes	—
234	C₂²⋊Q₈	—	94	24	no	—
235	#121	—	2	24	no	—
236	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
237	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
238	C₄×D₄	—	2	6	no	—
239	C₄×D₄	—	47	12	no	—
240	C₄×D₄	—	94	6	no	—
241	C₄×D₄	C₂²×D₄₇	1	12	yes	—
242	C₂×C₄⋊C₄	—	47	3	no	—
243	C₂×C₄⋊C₄	—	94	3	no	—
244	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
245	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
246	C₂×C₂²⋊C₄	—	2	3	no	—
247	C₂×C₂²⋊C₄	—	94	3	no	—
248	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
249	#107	—	47	12	no	—
250	C₄⋊C₈	—	47	12	no	—
251	#102	—	47	12	no	—
252	#102	—	47	24	no	—
253	#102	—	47	4	no	—
254	C₄×C₈	—	47	12	no	—
255	C₂⁴	—	2	3	no	—
256	C₂⁴	—	94	1	no	—
257	#41	—	4	32	no	—
258	C₂×Q₈	—	94	12	no	—
259	C₂×Q₈	—	188	32	no	—
260	C₂×Q₈	—	188	12	no	—
261	C₂×Q₈	#10625334	1	8	yes	complemented
262	C₂×Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
263	C₂×D₄	—	2	3	no	—
264	C₂×D₄	—	2	12	no	—
265	C₂×D₄	—	2	12	no	—
266	C₂×D₄	—	2	6	no	—
267	C₂×D₄	—	2	6	no	—
268	C₂×D₄	—	4	12	no	—
269	C₂×D₄	—	8	6	no	—
270	C₂×D₄	—	47	4	no	—
271	C₂×D₄	—	47	2	no	—
272	C₂×D₄	—	94	1	no	—
273	C₂×D₄	—	94	2	no	—
274	C₂×D₄	C₂³×D₄₇	1	6	yes	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	2	12	no	—
277	C₂²×C₄	—	4	12	no	—
278	C₂²×C₄	—	94	3	no	—
279	C₂²×C₄	—	94	12	no	—
280	C₂²×C₄	—	188	12	no	—
281	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
282	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
283	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
284	D₈	—	94	24	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	47	6	no	—
287	C₂×C₈	—	94	6	no	—
288	C₂×C₈	—	94	2	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	47	6	no	—
291	C₄⋊C₄	—	94	3	no	—
292	C₄⋊C₄	—	94	12	no	—
293	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
294	C₄⋊C₄	C₂³×D₄₇	1	2	yes	—
295	C₂²⋊C₄	—	2	12	no	—
296	C₂²⋊C₄	—	94	12	no	—
297	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
298	C₄²	C₂³×D₄₇	1	6	yes	—
299	C₂³	—	2	6	no	—
300	C₂³	—	4	6	no	—
301	C₂³	—	4	6	no	—
302	C₂³	—	4	6	no	—
303	C₂³	—	94	4	no	—
304	C₂³	—	94	2	no	—
305	C₂³	—	188	2	no	—
306	C₂³	—	188	2	no	—
307	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
308	Q₈	—	188	24	no	—
309	Q₈	C₂×#10625334	1	16	yes	complemented
310	D₄	—	2	6	no	—
311	D₄	—	2	6	no	—
312	D₄	—	4	6	no	—
313	D₄	—	4	24	no	—
314	D₄	—	8	12	no	—
315	D₄	—	47	4	no	—
316	D₄	—	94	4	no	—
317	D₄	—	94	2	no	—
318	C₂×C₄	—	2	6	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	24	no	—
322	C₂×C₄	—	94	6	no	—
323	C₂×C₄	—	188	6	no	—
324	C₂×C₄	—	188	24	no	—
325	C₂×C₄	C₂×#10625334	1	12	yes	—
326	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
327	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
329	C₂×C₄	#41×D₄₇	1	2	yes	—
330	C₂×C₄	#13094071	1	2	yes	—
331	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
332	C₈	—	94	6	no	—
333	C₈	—	94	2	no	—
334	C₂²	—	4	6	no	—
335	C₂²	—	4	6	no	—
336	C₂²	—	4	6	no	—
337	C₂²	—	8	12	no	—
338	C₂²	—	94	4	no	—
339	C₂²	—	188	8	no	—
340	C₂²	—	188	2	no	—
341	C₂²	—	188	2	no	—
342	C₂²	#15689780	1	2	yes	—
343	C₂²	C₂²×#10625334	1	2	yes	—
344	C₂²	#15689934	1	2	yes	—
345	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
346	C₄	—	2	6	no	—
347	C₄	—	4	2	no	—
348	C₄	—	188	6	no	—
349	C₄	#15689780	1	2	yes	—
350	C₄	C₂²×#10625334	1	12	yes	—
351	C₄	C₂×#41×D₄₇	1	1	yes	characteristic
352	C₄	C₂×#13094071	1	1	yes	characteristic
353	C₂	—	8	6	no	—
354	C₂	—	188	4	no	—
355	C₂	#20256019	1	4	yes	direct product factor
356	C₂	C₂×#15689780	1	1	yes	characteristic
357	C₂	C₂³×#10625334	1	1	yes	characteristic
358	C₂	C₂×#15689934	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#550	C₂
socle	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#550	D₄₇
hypocenter	C₄₇	C₂×#3358
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#550

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₁₈₈	C₂	semidirect
2. #20256019	C₂	direct (N × H)
3. C₂×#15689469	C₂	semidirect
4. C₂×#15689403	C₂	semidirect
5. C₂×#15689315	C₂	semidirect
6. C₂×Q₈×C₄₇⋊C₈	C₂	semidirect
7. C₂×C₄×#10625334	C₂	semidirect
8. C₂×C₄₇×#550	C₂	semidirect
12. Q₈×D₁₈₈	C₂²	semidirect
16. #15689469	C₂²	semidirect
17. #15689403	C₂²	semidirect
18. #15689315	C₂²	semidirect
19. Q₈×C₄₇⋊C₈	C₂²	semidirect
20. C₄×#10625334	C₂²	semidirect
29. C₄₇×#550	C₂²	semidirect
78. C₂×Q₈×C₄₇	D₈	semidirect
98. C₂×D₁₈₈	#41	semidirect
128. Q₈×C₄₇	C₂×D₈	semidirect
149. D₁₈₈	C₂×#41	semidirect
174. C₂×#550	D₄₇	semidirect
183. C₂×C₄₇	#3358	semidirect
192. #550	C₂×D₄₇	semidirect
213. C₄₇	C₂×#3358	semidirect
261. C₂×Q₈	#10625334	semidirect
309. Q₈	C₂×#10625334	semidirect
355. C₂	#20256019	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×Q₈×D₁₈₈	2
#20256019	2
C₂×#15689469	2
C₂×#15689403	2
C₂×#15689315	2
C₂×Q₈×C₄₇⋊C₈	2
C₂×C₄×#10625334	2
C₂×C₄₇×#550	2
C₂×#3358	47

Subgroups — C2×#20256019 #27095830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256019 #27095830