Subgroups — C₂×#20244365 #26220895

Order 12032.

1422 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	C₂×#15689680	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	#20244365	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688883	C₂	1	4	yes	maximal
5	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	#15689680	—	2	2	no	—
8	#15689680	—	2	2	no	—
9	#15689680	C₄	1	4	yes	complemented
10	#15689680	C₂²	1	4	yes	—
11	C₂×#13094024	C₄	1	2	yes	complemented
12	C₂×#13094016	C₄	1	2	yes	complemented
13	C₂×#13094003	—	2	2	no	—
14	C₂×C₄×Dic₉₄	—	2	2	no	—
15	C₂×#13093983	C₂²	1	2	yes	—
16	#15689147	—	2	2	no	—
17	#15689147	C₂²	1	4	yes	—
18	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	#15688883	C₂²	1	16	yes	—
20	C₂²×C₄²×C₄₇	—	2	2	no	—
21	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄³×C₄₇	—	2	2	no	—
24	C₄³×C₄₇	C₂²	1	4	yes	—
25	C₂²×Dic₉₄	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	8	no	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	#13094024	—	2	4	no	—
31	#13094024	C₂×C₄	1	8	yes	complemented
32	#13094016	—	2	4	no	—
33	#13094016	C₂×C₄	1	8	yes	complemented
34	#13094003	—	2	8	no	—
35	#13094003	—	2	8	no	—
36	C₄×Dic₉₄	—	2	8	no	—
37	C₄×Dic₉₄	—	2	8	no	—
38	#13093983	—	2	2	no	—
39	#13093983	—	2	2	no	—
40	#13093983	C₂×C₄	1	4	yes	—
41	#13093983	C₂³	1	4	yes	—
42	C₂×C₄⋊Dic₄₇	—	2	2	no	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
44	C₂×Dic₄₇⋊C₄	—	2	4	no	—
45	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
48	C₂³×C₄×C₄₇	—	2	2	no	—
49	C₂³×C₄×C₄₇	D₄	1	2	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	—	2	16	no	—
52	C₂×C₄²×C₄₇	—	2	8	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	—	2	4	no	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	D₄	1	2	yes	—
59	C₂×C₄²×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	D₄	1	2	yes	—
61	C₂×C₄²×C₄₇	Q₈	1	8	yes	complemented
62	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
64	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	16	no	—
66	C₂²×C₄×C₄₇	—	2	8	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	8	no	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₄₇	#41	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	2	4	no	—
82	C₂×Dic₉₄	—	2	8	no	—
83	C₂×Dic₉₄	—	4	8	no	—
84	C₄²×C₄₇	—	2	32	no	—
85	C₄²×C₄₇	—	2	8	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
88	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂×D₄	1	4	yes	—
93	C₄²×C₄₇	C₂×Q₈	1	8	yes	complemented
94	C₂²⋊Dic₄₇	—	2	4	no	—
95	C₂²⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	2	8	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
99	Dic₄₇⋊C₄	—	2	16	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	—	2	4	no	—
102	#10625329	—	2	8	no	—
103	#10625329	—	4	4	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	4	4	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
110	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	16	no	—
112	C₂×C₄×C₄₇	—	2	32	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	#104	1	8	yes	—
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
123	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	#124	1	2	yes	—
126	C₂×C₄×C₄₇	#124	1	2	yes	—
127	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×Dic₄₇	—	2	8	no	—
134	C₂×Dic₄₇	—	4	8	no	—
135	Dic₉₄	—	4	16	no	—
136	C₄₇⋊C₈	—	4	4	no	—
137	C₂×#1994	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	8	no	—
141	C₂²×C₄₇	#391	1	2	yes	—
142	C₂²×C₄₇	#391	1	2	yes	—
143	C₂²×C₄₇	C₂×#104	1	4	yes	—
144	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
146	C₄×C₄₇	—	2	16	no	—
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	#391	1	2	yes	—
149	C₄×C₄₇	#391	1	2	yes	—
150	C₄×C₄₇	C₂×#104	1	8	yes	—
151	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
152	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
153	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	Dic₄₇	—	4	8	no	—
155	C₂×#533	—	47	1	no	—
156	C₂×#423	—	47	1	no	—
157	C₂×C₄³	D₄₇	1	1	yes	characteristic
158	#1994	—	47	8	no	—
159	C₂×#339	—	47	4	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	#1994	1	4	yes	complemented
163	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×#104	1	2	yes	—
165	C₂²×M₁₆	—	47	2	no	—
166	#533	—	47	4	no	—
167	#533	—	47	4	no	—
168	#533	—	94	2	no	—
169	#533	—	94	2	no	—
170	C₂×C₄⋊Q₈	—	47	2	no	—
171	C₂×#125	—	47	2	no	—
172	C₂×C₂²⋊Q₈	—	94	2	no	—
173	C₂×C₄×Q₈	—	94	2	no	—
174	C₂×#117	—	47	2	no	—
175	C₂²×C₄²	—	2	2	no	—
176	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
177	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
178	#423	—	47	4	no	—
179	#423	—	94	2	no	—
180	C₂×C₄⋊C₈	—	94	2	no	—
181	C₄³	—	2	2	no	—
182	C₄³	C₂×D₄₇	1	4	yes	—
183	#339	—	47	16	no	—
184	C₄₇	C₂×#1994	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	94	2	no	—
186	C₂³×C₄	—	2	2	no	—
187	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×M₁₆	—	47	4	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	8	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₄⋊Q₈	—	47	8	no	—
194	C₄⋊Q₈	—	94	4	no	—
195	#125	—	47	8	no	—
196	#125	—	94	4	no	—
197	C₂²⋊Q₈	—	94	8	no	—
198	C₂²⋊Q₈	—	94	8	no	—
199	C₄×Q₈	—	94	8	no	—
200	C₄×Q₈	—	94	8	no	—
201	#117	—	47	4	no	—
202	#117	—	47	4	no	—
203	#117	—	94	2	no	—
204	#117	—	94	2	no	—
205	C₂×C₄⋊C₄	—	47	4	no	—
206	C₂×C₄⋊C₄	—	94	4	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₂²⋊C₄	—	94	2	no	—
209	C₂×C₄²	—	2	16	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	8	no	—
213	C₂×C₄²	—	94	2	no	—
214	C₂×C₄²	Dic₉₄	1	8	yes	complemented
215	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₄₇	1	2	yes	—
218	C₂×C₄²	D₁₈₈	1	2	yes	—
219	C₂×C₄²	D₁₈₈	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
221	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
222	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
223	C₄⋊C₈	—	94	8	no	—
224	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×Q₈	—	94	8	no	—
226	C₂×Q₈	—	188	8	no	—
227	C₂²×C₄	—	2	16	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	94	4	no	—
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
234	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
235	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	#10625350	1	4	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	M₁₆	—	94	8	no	—
245	M₁₆	—	188	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	188	4	no	—
248	C₄⋊C₄	—	47	8	no	—
249	C₄⋊C₄	—	94	16	no	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	—	94	8	no	—
252	C₂²⋊C₄	—	94	4	no	—
253	C₂²⋊C₄	—	94	4	no	—
254	C₄²	—	2	32	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	94	4	no	—
258	C₄²	—	94	4	no	—
259	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
260	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
261	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
262	C₄²	C₂×Dic₉₄	1	8	yes	complemented
263	C₄²	C₂×C₄×D₄₇	1	2	yes	—
264	C₄²	C₂×C₄×D₄₇	1	2	yes	—
265	C₄²	C₂×D₁₈₈	1	4	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
268	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂³	#13094003	1	4	yes	—
270	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	Q₈	—	188	16	no	—
272	C₂×C₄	—	2	16	no	—
273	C₂×C₄	—	2	32	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	94	8	no	—
279	C₂×C₄	—	188	8	no	—
280	C₂×C₄	#13093952	1	8	yes	—
281	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
282	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
285	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
286	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
287	C₂×C₄	#13094003	1	4	yes	—
288	C₂×C₄	#13094012	1	2	yes	—
289	C₂×C₄	#13094012	1	2	yes	—
290	C₂×C₄	#13094024	1	2	yes	—
291	C₂×C₄	#13094024	1	2	yes	—
292	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
295	C₂×C₄	C₂×#10625350	1	2	yes	—
296	C₈	—	188	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	8	no	—
300	C₂²	#15689029	1	2	yes	—
301	C₂²	#15689029	1	2	yes	—
302	C₂²	C₂×#13093952	1	4	yes	—
303	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
304	C₂²	C₂×#13094003	1	2	yes	—
305	C₄	—	2	16	no	—
306	C₄	—	2	4	no	—
307	C₄	—	188	8	no	—
308	C₄	#15689029	1	2	yes	—
309	C₄	#15689029	1	2	yes	—
310	C₄	C₂×#13093952	1	8	yes	—
311	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
312	C₄	C₂×#13094012	1	1	yes	characteristic
313	C₄	C₂×#13094024	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#20244365	1	4	yes	direct product factor
317	C₂	C₂×#15689029	1	1	yes	characteristic
318	C₂	C₂²×#13093952	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	C₂×#1994
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. #20244365	C₂	direct (N × H)
9. #15689680	C₄	semidirect
11. C₂×#13094024	C₄	semidirect
12. C₂×#13094016	C₄	semidirect
31. #13094024	C₂×C₄	semidirect
33. #13094016	C₂×C₄	semidirect
61. C₂×C₄²×C₄₇	Q₈	semidirect
93. C₄²×C₄₇	C₂×Q₈	semidirect
162. C₂×C₄₇	#1994	semidirect
184. C₄₇	C₂×#1994	semidirect
214. C₂×C₄²	Dic₉₄	semidirect
262. C₄²	C₂×Dic₉₄	semidirect
316. C₂	#20244365	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×#15689680	2
C₂×#15689147	2
#20244365	2
C₂×#15688883	2
C₂×C₄³×C₄₇	2
C₂×#1994	47

Subgroups — C2×#20244365 #26220895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244365 #26220895