Subgroups — C₂×#20245512 #26370840

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093970	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689541	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689399	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689395	C₂	1	1	yes	maximal, characteristic
5	#20245512	C₂	1	8	yes	maximal, direct product factor
6	C₂×#15688952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688940	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#362	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094057	—	2	1	no	—
10	C₂×#13094055	—	2	1	no	—
11	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
12	C₂×#13094030	C₂²	1	1	yes	characteristic
13	C₂×#13093970	—	2	1	no	—
14	C₂×#13093970	—	2	1	no	—
15	C₂×#13093970	—	2	2	no	—
16	C₂×#13093970	C₂²	1	4	yes	complemented
17	#15689541	—	2	2	no	—
18	#15689541	C₂²	1	4	yes	—
19	C₂×#13093963	—	2	1	no	—
20	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₄₇	—	2	1	no	—
22	#15689399	—	2	2	no	—
23	#15689399	C₂²	1	4	yes	—
24	#15689395	—	2	2	no	—
25	#15689395	C₂²	1	4	yes	—
26	C₂×#13093960	C₂²	1	1	yes	characteristic
27	C₂×#13093943	—	2	1	no	—
28	C₂×#13093942	—	2	1	no	—
29	#15688952	C₂²	1	4	yes	complemented
30	#15688940	C₂²	1	4	yes	complemented
31	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#362	C₂²	1	4	yes	complemented
35	C₂×#10625350	D₄	1	1	yes	characteristic
36	C₂²×Dic₉₄	—	2	1	no	—
37	#13094057	—	2	4	no	—
38	#13094057	—	4	2	no	—
39	#13094055	—	2	4	no	—
40	#13094055	—	4	2	no	—
41	C₂×Dic₁₈₈	—	2	1	no	—
42	C₂×Dic₁₈₈	—	2	1	no	—
43	C₂×Dic₁₈₈	—	2	1	no	—
44	C₂×Dic₁₈₈	—	2	1	no	—
45	C₂×Dic₁₈₈	—	2	1	no	—
46	C₂×Dic₁₈₈	—	4	1	no	—
47	C₂×Dic₁₈₈	C₂³	1	2	yes	—
48	C₂×#10625337	—	4	1	no	—
49	C₂×#10625325	—	4	1	no	—
50	C₂×#10625336	—	4	1	no	—
51	#13094030	—	2	1	no	—
52	#13094030	—	2	1	no	—
53	#13094030	D₄	1	2	yes	—
54	#13094030	C₂³	1	2	yes	—
55	C₂×#10625329	—	2	1	no	—
56	C₂×#10625329	D₄	1	1	yes	characteristic
57	C₂×#10625324	—	2	1	no	—
58	C₂×C₈×D₄₇	—	2	1	no	—
59	#13093970	—	2	4	no	—
60	#13093970	—	2	4	no	—
61	#13093963	—	2	4	no	—
62	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
63	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
64	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
65	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
66	C₁₆×D₄₇	—	2	4	no	—
67	#13093960	—	2	1	no	—
68	#13093960	C₂³	1	2	yes	—
69	#13093943	—	2	4	no	—
70	#13093942	—	2	4	no	—
71	C₂²×Q₈×C₄₇	—	2	1	no	—
72	C₂×Q₁₆×C₄₇	—	2	1	no	—
73	C₂×Q₁₆×C₄₇	—	2	1	no	—
74	C₂×Q₁₆×C₄₇	—	2	1	no	—
75	C₂×Q₁₆×C₄₇	—	2	1	no	—
76	C₂×Q₁₆×C₄₇	—	2	1	no	—
77	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
78	C₂×M₁₆×C₄₇	—	2	1	no	—
79	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	M₃₂×C₄₇	—	2	1	no	—
81	M₃₂×C₄₇	C₂³	1	2	yes	—
82	C₂×C₁₆×C₄₇	—	2	1	no	—
83	#108×C₄₇	—	2	1	no	—
84	#108×C₄₇	C₂³	1	2	yes	—
85	C₂×Q₈×C₄₇	—	2	2	no	—
86	C₂×Q₈×C₄₇	—	4	1	no	—
87	C₂×Q₈×C₄₇	—	4	1	no	—
88	C₂×Q₈×C₄₇	—	4	1	no	—
89	C₂×Q₈×C₄₇	—	4	1	no	—
90	C₂×Q₈×C₄₇	—	4	1	no	—
91	C₂²×C₄×C₄₇	—	4	1	no	—
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₄₇	—	2	1	no	—
94	C₂²×Dic₄₇	—	4	1	no	—
95	#10625350	—	2	1	no	—
96	#10625350	—	2	1	no	—
97	#10625350	D₈	1	2	yes	—
98	#10625350	C₂×D₄	1	2	yes	—
99	C₂×D₁₈₈	D₈	1	1	yes	characteristic
100	C₂×C₄×D₄₇	—	2	1	no	—
101	C₂×Dic₉₄	—	2	2	no	—
102	C₂×Dic₉₄	—	4	1	no	—
103	C₂×Dic₉₄	—	4	1	no	—
104	C₂×Dic₉₄	—	4	1	no	—
105	C₂×Dic₉₄	—	4	1	no	—
106	C₂×Dic₉₄	—	4	1	no	—
107	C₂×Dic₉₄	D₈	1	1	yes	characteristic
108	Q₁₆×C₄₇	—	2	2	no	—
109	Q₁₆×C₄₇	—	2	2	no	—
110	Q₁₆×C₄₇	—	4	1	no	—
111	Q₁₆×C₄₇	—	4	1	no	—
112	M₁₆×C₄₇	—	2	2	no	—
113	M₁₆×C₄₇	—	4	1	no	—
114	C₂×C₈×C₄₇	—	4	1	no	—
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
119	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
120	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
121	#10625337	—	4	4	no	—
122	#10625336	—	4	4	no	—
123	#10625329	—	2	1	no	—
124	#10625329	—	2	2	no	—
125	#10625329	—	4	1	no	—
126	#10625329	C₂×D₄	1	2	yes	—
127	C₂×C₄₇⋊C₈	—	2	1	no	—
128	C₂×C₄₇⋊C₈	—	4	1	no	—
129	Dic₁₈₈	—	2	2	no	—
130	Dic₁₈₈	—	2	2	no	—
131	Dic₁₈₈	—	4	4	no	—
132	Dic₁₈₈	—	4	1	no	—
133	Dic₁₈₈	—	4	1	no	—
134	#10625325	—	4	4	no	—
135	#10625324	—	2	2	no	—
136	#10625324	—	2	2	no	—
137	C₈×D₄₇	—	2	2	no	—
138	C₈×D₄₇	—	2	2	no	—
139	C₁₆×C₄₇	—	2	2	no	—
140	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
141	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
142	C₂³×C₄₇	#135	1	1	yes	characteristic
143	C₂²×D₄₇	—	2	1	no	—
144	Q₈×C₄₇	—	4	2	no	—
145	Q₈×C₄₇	—	4	2	no	—
146	Q₈×C₄₇	—	8	1	no	—
147	Q₈×C₄₇	—	8	1	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	8	1	no	—
150	C₂×C₄×C₄₇	—	8	1	no	—
151	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
156	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₄₇	—	2	2	no	—
158	C₂²⋊D₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	2	1	no	—
160	C₂×Dic₄₇	—	4	2	no	—
161	C₂×Dic₄₇	—	8	1	no	—
162	C₂×Dic₄₇	—	8	1	no	—
163	D₁₈₈	—	2	1	no	—
164	D₁₈₈	C₂×D₈	1	2	yes	—
165	C₄×D₄₇	—	2	2	no	—
166	C₄×D₄₇	—	2	2	no	—
167	Dic₉₄	—	2	1	no	—
168	Dic₉₄	—	4	2	no	—
169	Dic₉₄	—	4	2	no	—
170	Dic₉₄	—	8	1	no	—
171	Dic₉₄	—	8	1	no	—
172	Dic₉₄	C₂×D₈	1	2	yes	—
173	C₈×C₄₇	—	4	2	no	—
174	C₈×C₄₇	#121	1	1	yes	characteristic
175	C₈×C₄₇	#121	1	1	yes	characteristic
176	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
177	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
178	C₄₇⋊C₈	—	2	2	no	—
179	C₄₇⋊C₈	—	4	2	no	—
180	C₂×#2222	—	47	1	no	maximal, Sylow
181	C₂²×C₄₇	—	2	2	no	—
182	C₂²×C₄₇	#466	1	1	yes	characteristic
183	C₂²×C₄₇	#466	1	1	yes	characteristic
184	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
185	C₂×D₄₇	—	2	2	no	—
186	C₂×D₄₇	—	4	2	no	—
187	C₄×C₄₇	—	8	2	no	—
188	C₄×C₄₇	#466	1	1	yes	characteristic
189	C₄×C₄₇	#466	1	1	yes	characteristic
190	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
191	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
192	Dic₄₇	—	2	2	no	—
193	Dic₄₇	—	8	2	no	—
194	C₂²×Q₃₂	—	47	1	no	—
195	C₂×#504	—	47	1	no	—
196	C₂×#473	—	47	1	no	—
197	C₂×#473	—	47	1	no	—
198	#2222	—	47	8	no	—
199	C₂×#368	—	47	1	no	—
200	C₂×#362	—	47	1	no	—
201	C₂×#362	D₄₇	1	1	yes	characteristic, complemented
202	C₂×C₄₇	—	2	1	no	—
203	C₂×C₄₇	—	2	1	no	—
204	C₂×C₄₇	#2222	1	2	yes	complemented
205	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
206	D₄₇	—	4	2	no	—
207	C₂×#137	—	94	1	no	—
208	C₂×#137	—	94	1	no	—
209	C₂²×Q₁₆	—	47	1	no	—
210	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
211	C₂×#131	—	47	1	no	—
212	C₂×Q₃₂	—	47	4	no	—
213	C₂×Q₃₂	—	94	1	no	—
214	C₂×Q₃₂	—	94	1	no	—
215	C₂×Q₃₂	—	94	2	no	—
216	#504	—	47	4	no	—
217	#504	—	94	2	no	—
218	C₂×M₃₂	—	94	1	no	—
219	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
220	C₂²×C₁₆	—	47	1	no	—
221	C₂²×C₁₆	—	94	1	no	—
222	#473	—	47	4	no	—
223	#473	—	47	4	no	—
224	#473	—	94	2	no	—
225	#473	—	94	2	no	—
226	C₂×#108	—	47	1	no	—
227	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
228	C₂×#101	—	94	1	no	—
229	C₂×#101	—	94	1	no	—
230	#368	—	47	4	no	—
231	#362	—	47	4	no	—
232	#362	C₂×D₄₇	1	4	yes	complemented
233	C₄₇	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
234	C₂×#41	—	47	1	no	—
235	C₂²×Q₈	—	2	1	no	—
236	C₂²×Q₈	—	94	1	no	—
237	#137	—	94	4	no	—
238	#137	—	94	4	no	—
239	#137	—	188	2	no	—
240	#137	—	188	2	no	—
241	C₂×Q₁₆	—	2	1	no	—
242	C₂×Q₁₆	—	2	1	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	2	1	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	47	2	no	—
247	C₂×Q₁₆	—	94	1	no	—
248	C₂×Q₁₆	—	94	1	no	—
249	C₂×Q₁₆	—	94	1	no	—
250	C₂×Q₁₆	—	94	1	no	—
251	C₂×Q₁₆	—	94	1	no	—
252	C₂×Q₁₆	—	188	1	no	—
253	C₂×Q₁₆	—	188	1	no	—
254	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
255	C₂×SD₁₆	—	188	1	no	—
256	C₂×SD₁₆	—	188	1	no	—
257	#131	—	47	2	no	—
258	#131	—	47	2	no	—
259	#131	—	94	1	no	—
260	#131	—	94	1	no	—
261	C₂×M₁₆	—	2	1	no	—
262	C₂×M₁₆	—	47	1	no	—
263	C₂×M₁₆	—	94	1	no	—
264	C₂×M₁₆	—	94	1	no	—
265	C₂²×C₈	—	94	1	no	—
266	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
267	Q₃₂	—	94	4	no	—
268	Q₃₂	—	94	4	no	—
269	M₃₂	—	2	1	no	—
270	M₃₂	—	94	4	no	—
271	M₃₂	C₂²×D₄₇	1	2	yes	—
272	C₂×C₁₆	—	2	1	no	—
273	C₂×C₁₆	—	47	1	no	—
274	C₂×C₁₆	—	47	1	no	—
275	C₂×C₁₆	—	47	2	no	—
276	C₂×C₁₆	—	47	2	no	—
277	C₂×C₁₆	—	94	4	no	—
278	#108	—	2	1	no	—
279	#108	—	47	2	no	—
280	#108	—	94	1	no	—
281	#108	C₂²×D₄₇	1	2	yes	—
282	#101	—	94	4	no	—
283	#101	—	94	4	no	—
284	#41	—	47	2	no	—
285	#41	—	47	2	no	—
286	#41	—	94	1	no	—
287	#41	—	94	1	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	4	1	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×Q₈	—	4	1	no	—
292	C₂×Q₈	—	4	1	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	47	1	no	—
295	C₂×Q₈	—	94	2	no	—
296	C₂×Q₈	—	188	1	no	—
297	C₂×Q₈	—	188	1	no	—
298	C₂×Q₈	—	188	1	no	—
299	C₂×Q₈	—	188	1	no	—
300	C₂×Q₈	—	188	1	no	—
301	C₂×D₄	—	47	1	no	—
302	C₂×D₄	—	94	1	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	94	1	no	—
305	C₂²×C₄	—	188	1	no	—
306	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
307	Q₁₆	—	2	2	no	—
308	Q₁₆	—	2	2	no	—
309	Q₁₆	—	4	1	no	—
310	Q₁₆	—	4	1	no	—
311	Q₁₆	—	94	2	no	—
312	Q₁₆	—	94	2	no	—
313	Q₁₆	—	188	4	no	—
314	Q₁₆	—	188	4	no	—
315	Q₁₆	—	188	1	no	—
316	Q₁₆	—	188	1	no	—
317	SD₁₆	—	188	4	no	—
318	SD₁₆	—	188	4	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	4	1	no	—
321	M₁₆	—	47	2	no	—
322	M₁₆	—	94	1	no	—
323	M₁₆	—	94	2	no	—
324	M₁₆	—	94	2	no	—
325	M₁₆	—	94	2	no	—
326	M₁₆	—	188	1	no	—
327	C₂×C₈	—	4	1	no	—
328	C₂×C₈	—	94	1	no	—
329	C₂×C₈	—	94	2	no	—
330	C₂×C₈	—	94	2	no	—
331	C₂×C₈	—	188	1	no	—
332	C₂×C₈	#10625350	1	1	yes	characteristic
333	C₂×C₈	#10625350	1	1	yes	characteristic
334	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
335	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
336	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
337	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
338	C₁₆	—	2	2	no	—
339	C₁₆	—	47	2	no	—
340	C₁₆	—	47	2	no	—
341	C₂³	—	94	1	no	—
342	C₂³	#13094045	1	1	yes	characteristic
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	8	1	no	—
346	Q₈	—	8	1	no	—
347	Q₈	—	47	2	no	—
348	Q₈	—	94	1	no	—
349	Q₈	—	376	1	no	—
350	Q₈	—	376	1	no	—
351	Q₈	—	188	2	no	—
352	Q₈	—	188	2	no	—
353	D₄	—	47	2	no	—
354	D₄	—	94	2	no	—
355	D₄	—	94	2	no	—
356	D₄	—	94	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	1	no	—
359	C₂×C₄	—	8	1	no	—
360	C₂×C₄	—	94	1	no	—
361	C₂×C₄	—	94	2	no	—
362	C₂×C₄	—	94	2	no	—
363	C₂×C₄	—	376	1	no	—
364	C₂×C₄	—	376	1	no	—
365	C₂×C₄	—	188	2	no	—
366	C₂×C₄	#13093997	1	1	yes	characteristic
367	C₂×C₄	#13093997	1	1	yes	characteristic
368	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
369	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
370	C₂×C₄	#13094045	1	1	yes	characteristic
371	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
372	C₈	—	4	2	no	—
373	C₈	—	94	2	no	—
374	C₈	—	188	2	no	—
375	C₈	#13093997	1	1	yes	characteristic
376	C₈	#13093997	1	1	yes	characteristic
377	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
378	C₈	C₂×#10625350	1	1	yes	characteristic
379	C₂²	—	2	2	no	—
380	C₂²	—	94	2	no	—
381	C₂²	—	188	2	no	—
382	C₂²	#15689358	1	1	yes	characteristic
383	C₂²	#15689358	1	1	yes	characteristic
384	C₂²	C₂×#13094045	1	1	yes	characteristic
385	C₄	—	8	2	no	—
386	C₄	—	94	2	no	—
387	C₄	—	376	2	no	—
388	C₄	#15689358	1	1	yes	characteristic
389	C₄	#15689358	1	1	yes	characteristic
390	C₄	C₂×#13093997	1	1	yes	characteristic
391	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	188	2	no	—
395	C₂	#20245512	1	2	yes	direct product factor
396	C₂	C₂×#15689358	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689358
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#362	C₂
socle	C₂²×C₄₇	#466
prime radical	C₂×#10625350	D₄
hypercenter	C₂×#362	D₄₇
hypocenter	C₄₇	C₂×#2222
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093970	C₂	semidirect
5. #20245512	C₂	direct (N × H)
6. C₂×#15688952	C₂	semidirect
7. C₂×#15688940	C₂	semidirect
8. C₂×C₄₇×#362	C₂	semidirect
16. C₂×#13093970	C₂²	semidirect
29. #15688952	C₂²	semidirect
30. #15688940	C₂²	semidirect
34. C₄₇×#362	C₂²	semidirect
201. C₂×#362	D₄₇	semidirect
204. C₂×C₄₇	#2222	semidirect
232. #362	C₂×D₄₇	semidirect
233. C₄₇	C₂×#2222	semidirect
395. C₂	#20245512	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×#13093970	2
C₂×#15689541	2
C₂×#15689399	2
C₂×#15689395	2
#20245512	2
C₂×#15688952	2
C₂×#15688940	2
C₂×C₄₇×#362	2
C₂×#2222	47

Subgroups — C2×#20245512 #26370840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245512 #26370840