Subgroups — C₂×#20242927 #26198925

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689469	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689419	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689320	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689250	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242927	C₂	1	8	yes	maximal, direct product factor
6	C₂×#15688875	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688871	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#331	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625335	—	2	1	no	—
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
12	#15689469	—	2	2	no	—
13	#15689469	C₂²	1	4	yes	—
14	#15689419	—	2	2	no	—
15	#15689419	C₂²	1	4	yes	complemented
16	#15689320	—	2	2	no	—
17	#15689320	C₂²	1	4	yes	—
18	#15689250	—	2	2	no	—
19	#15689250	C₂²	1	4	yes	complemented
20	C₂×#13093958	—	2	1	no	—
21	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×Dic₄₇⋊C₈	—	2	1	no	—
23	C₂×#13093949	—	2	1	no	—
24	C₂×#13093945	—	2	1	no	—
25	C₂×#13093945	—	2	1	no	—
26	C₂×#13093945	—	2	1	no	—
27	C₂×#13093934	—	2	1	no	—
28	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
29	#15688875	C₂²	1	4	yes	complemented
30	#15688871	C₂²	1	4	yes	complemented
31	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#331	C₂²	1	4	yes	—
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
37	C₂×#10625335	—	2	4	no	—
38	C₂×#10625335	—	4	4	no	—
39	C₂²×C₄₇⋊C₈	—	2	1	no	—
40	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	#13094024	—	2	1	no	—
43	#13094024	—	2	1	no	—
44	#13094024	—	2	1	no	—
45	#13094024	C₂³	1	2	yes	—
46	C₄×Dic₉₄	—	2	1	no	—
47	C₄×Dic₉₄	—	2	1	no	—
48	C₄×Dic₉₄	D₄	1	2	yes	—
49	C₄×Dic₉₄	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
53	C₂×Dic₄₇⋊C₄	—	2	1	no	—
54	C₂×C₄×Dic₄₇	—	2	1	no	—
55	#13093958	—	2	4	no	—
56	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
57	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
58	Dic₄₇⋊C₈	—	2	4	no	—
59	#13093949	—	2	4	no	—
60	#13093945	—	2	4	no	—
61	#13093945	—	2	4	no	—
62	#13093945	—	2	4	no	—
63	#13093934	—	2	4	no	—
64	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
65	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
66	C₂²×D₄×C₄₇	—	2	1	no	—
67	C₂²×D₄×C₄₇	—	2	1	no	—
68	C₂²×D₄×C₄₇	—	2	1	no	—
69	C₂²×C₈×C₄₇	—	2	1	no	—
70	C₂²×C₈×C₄₇	—	2	1	no	—
71	#127×C₄₇	—	2	1	no	—
72	#127×C₄₇	—	2	1	no	—
73	#127×C₄₇	—	2	1	no	—
74	#127×C₄₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₇	—	2	1	no	—
77	C₄⋊C₈×C₄₇	—	2	1	no	—
78	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
80	C₂⁴×C₄₇	—	4	1	no	—
81	C₂×D₄×C₄₇	—	2	2	no	—
82	C₂×D₄×C₄₇	—	2	2	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	4	1	no	—
85	C₂×D₄×C₄₇	—	4	1	no	—
86	C₂×D₄×C₄₇	—	4	1	no	—
87	C₂×D₄×C₄₇	—	4	2	no	—
88	C₂×D₄×C₄₇	—	4	2	no	—
89	C₂×D₄×C₄₇	—	4	2	no	—
90	C₂×D₄×C₄₇	—	4	2	no	—
91	C₂×D₄×C₄₇	—	8	2	no	—
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
95	C₂²×Dic₄₇	—	2	1	no	—
96	C₂²×Dic₄₇	—	2	1	no	—
97	C₂²×Dic₄₇	—	4	1	no	—
98	C₂×Dic₉₄	—	2	2	no	—
99	C₂×Dic₉₄	—	2	2	no	—
100	C₂×Dic₉₄	—	2	1	no	—
101	C₂×Dic₉₄	—	2	2	no	—
102	C₂×Dic₉₄	—	4	1	no	—
103	C₂×Dic₉₄	—	8	2	no	—
104	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	—	2	2	no	—
107	C₂×C₈×C₄₇	—	4	2	no	—
108	C₂×C₈×C₄₇	—	4	2	no	—
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₇	#41	1	1	yes	characteristic
112	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
113	#10625335	—	4	8	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	4	1	no	—
118	C₄⋊Dic₄₇	—	4	1	no	—
119	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
120	Dic₄₇⋊C₄	—	2	2	no	—
121	Dic₄₇⋊C₄	—	2	2	no	—
122	C₄×Dic₄₇	—	2	2	no	—
123	C₄×Dic₄₇	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	4	2	no	—
128	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
129	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
130	C₂³×C₄₇	—	4	2	no	—
131	C₂³×C₄₇	—	8	2	no	—
132	C₂³×C₄₇	—	8	2	no	—
133	C₂³×C₄₇	—	8	2	no	—
134	C₂³×C₄₇	#121	1	1	yes	characteristic
135	D₄×C₄₇	—	4	2	no	—
136	D₄×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	2	no	—
138	D₄×C₄₇	—	4	2	no	—
139	D₄×C₄₇	—	8	2	no	—
140	D₄×C₄₇	—	8	2	no	—
141	D₄×C₄₇	—	8	2	no	—
142	D₄×C₄₇	—	8	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#135	1	2	yes	—
148	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₄₇	#137	1	2	yes	—
150	C₂×C₄×C₄₇	#137	1	2	yes	—
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₄₇	—	2	2	no	—
155	C₂×Dic₄₇	—	2	2	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₂×Dic₄₇	—	4	2	no	—
158	C₂×Dic₄₇	—	4	2	no	—
159	C₂×Dic₄₇	—	8	2	no	—
160	Dic₉₄	—	2	2	no	—
161	Dic₉₄	—	2	2	no	—
162	Dic₉₄	—	4	2	no	—
163	Dic₉₄	—	8	2	no	—
164	Dic₉₄	—	8	2	no	—
165	C₈×C₄₇	—	4	2	no	—
166	C₈×C₄₇	—	4	2	no	—
167	C₄₇⋊C₈	—	2	2	no	—
168	C₄₇⋊C₈	—	2	2	no	—
169	C₄₇⋊C₈	—	4	2	no	—
170	C₂×#1698	—	47	1	no	maximal, Sylow
171	C₂²×C₄₇	—	8	2	no	—
172	C₂²×C₄₇	—	8	2	no	—
173	C₂²×C₄₇	—	8	2	no	—
174	C₂²×C₄₇	—	16	4	no	—
175	C₂²×C₄₇	#464	1	2	yes	—
176	C₂²×C₄₇	#468	1	2	yes	—
177	C₂²×C₄₇	#472	1	2	yes	—
178	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	#464	1	2	yes	—
182	C₄×C₄₇	#468	1	2	yes	—
183	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
184	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
186	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
187	Dic₄₇	—	4	2	no	—
188	Dic₄₇	—	4	2	no	—
189	Dic₄₇	—	8	2	no	—
190	C₂×#486	—	47	1	no	—
191	C₂×#477	—	47	1	no	—
192	C₂×#460	—	47	1	no	—
193	C₂×#446	—	47	1	no	—
194	#1698	—	47	8	no	—
195	C₂×#335	—	47	1	no	—
196	C₂×#333	—	47	1	no	—
197	C₂×#331	D₄₇	1	1	yes	characteristic
198	C₂×C₄₇	—	16	2	no	—
199	C₂×C₄₇	#1698	1	4	yes	complemented
200	C₂×C₄₇	C₂×#464	1	1	yes	characteristic
201	C₂×C₄₇	C₂×#468	1	1	yes	characteristic
202	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
203	C₂²×SD₁₆	—	94	1	no	—
204	C₂×C₄⋊Q₈	—	47	1	no	—
205	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
206	C₂×C₄×Q₈	—	47	1	no	—
207	#486	—	47	4	no	—
208	#486	—	94	2	no	—
209	#477	—	47	4	no	—
210	#477	—	94	2	no	—
211	#460	—	47	4	no	—
212	#460	—	94	2	no	—
213	#446	—	47	4	no	—
214	#446	—	94	2	no	—
215	C₂×#107	—	94	1	no	—
216	C₂×C₄⋊C₈	—	47	1	no	—
217	C₂×C₄⋊C₈	—	94	1	no	—
218	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
220	C₂×#103	—	94	1	no	—
221	C₂×#102	—	94	1	no	—
222	C₂×#102	—	94	1	no	—
223	C₂×#102	—	94	1	no	—
224	C₂×#97	—	94	1	no	—
225	C₂×C₄×C₈	—	47	1	no	—
226	#335	—	47	4	no	—
227	#333	—	47	4	no	—
228	#331	C₂×D₄₇	1	4	yes	—
229	C₄₇	C₂×#1698	1	1	yes	characteristic, complemented, Sylow
230	C₂²×Q₈	—	47	1	no	—
231	C₂²×Q₈	—	94	1	no	—
232	C₂²×D₄	—	2	1	no	—
233	C₂²×D₄	—	2	1	no	—
234	C₂²×D₄	—	2	1	no	—
235	C₂×SD₁₆	—	94	4	no	—
236	C₂×SD₁₆	—	188	4	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	47	1	no	—
240	C₂²×C₈	—	47	1	no	—
241	C₂²×C₈	—	94	1	no	—
242	C₄⋊Q₈	—	47	2	no	—
243	C₄⋊Q₈	—	94	1	no	—
244	C₄⋊Q₈	—	94	1	no	—
245	C₄⋊Q₈	—	94	1	no	—
246	#127	—	2	1	no	—
247	#127	—	2	1	no	—
248	#127	—	2	1	no	—
249	#127	C₂²×D₄₇	1	2	yes	—
250	C₄×Q₈	—	47	2	no	—
251	C₄×Q₈	—	47	2	no	—
252	C₄×Q₈	—	94	1	no	—
253	C₄×Q₈	—	94	1	no	—
254	C₂×C₄⋊C₄	—	47	1	no	—
255	C₂×C₄⋊C₄	—	94	1	no	—
256	C₂×C₄⋊C₄	—	94	1	no	—
257	C₂×C₄⋊C₄	—	94	1	no	—
258	C₂×C₄²	—	94	1	no	—
259	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
260	#107	—	94	4	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	47	2	no	—
264	C₄⋊C₈	—	94	4	no	—
265	C₄⋊C₈	—	94	1	no	—
266	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
267	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
268	#103	—	94	4	no	—
269	#102	—	94	4	no	—
270	#102	—	94	4	no	—
271	#102	—	94	4	no	—
272	#97	—	94	4	no	—
273	C₄×C₈	—	47	2	no	—
274	C₄×C₈	—	47	2	no	—
275	C₂⁴	—	4	1	no	—
276	C₂×Q₈	—	47	2	no	—
277	C₂×Q₈	—	94	1	no	—
278	C₂×Q₈	—	94	2	no	—
279	C₂×Q₈	—	94	2	no	—
280	C₂×Q₈	—	94	2	no	—
281	C₂×Q₈	—	376	2	no	—
282	C₂×Q₈	—	188	1	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	4	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂×D₄	—	4	2	no	—
292	C₂×D₄	—	4	2	no	—
293	C₂×D₄	—	8	2	no	—
294	C₂²×C₄	—	94	1	no	—
295	C₂²×C₄	—	94	1	no	—
296	C₂²×C₄	—	188	1	no	—
297	C₂²×C₄	#10625350	1	1	yes	characteristic
298	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
299	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
300	SD₁₆	—	188	8	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	4	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	—	47	2	no	—
306	C₂×C₈	—	47	2	no	—
307	C₂×C₈	—	94	2	no	—
308	C₂×C₈	—	94	2	no	—
309	C₂×C₈	—	94	2	no	—
310	C₂×C₈	—	188	2	no	—
311	C₄⋊C₄	—	47	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	—	94	1	no	—
314	C₄⋊C₄	—	94	2	no	—
315	C₄⋊C₄	—	94	2	no	—
316	C₄⋊C₄	—	94	2	no	—
317	C₄⋊C₄	—	188	1	no	—
318	C₄⋊C₄	—	188	1	no	—
319	C₄²	—	94	2	no	—
320	C₄²	—	94	2	no	—
321	C₄²	#10625350	1	1	yes	characteristic
322	C₄²	D₄×D₄₇	1	1	yes	characteristic
323	C₄²	D₄×D₄₇	1	1	yes	characteristic
324	C₄²	C₂³×D₄₇	1	1	yes	characteristic
325	C₂³	—	4	2	no	—
326	C₂³	—	8	2	no	—
327	C₂³	—	8	2	no	—
328	C₂³	—	8	2	no	—
329	C₂³	#13093997	1	1	yes	characteristic
330	Q₈	—	94	2	no	—
331	Q₈	—	94	2	no	—
332	Q₈	—	376	2	no	—
333	Q₈	—	376	2	no	—
334	Q₈	—	188	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	D₄	—	8	2	no	—
340	D₄	—	8	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	94	2	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	376	2	no	—
347	C₂×C₄	—	188	2	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	#13093997	1	1	yes	characteristic
351	C₂×C₄	#13093997	1	1	yes	characteristic
352	C₂×C₄	#13093997	1	1	yes	characteristic
353	C₂×C₄	#13094043	1	2	yes	—
354	C₂×C₄	#13094048	1	2	yes	—
355	C₂×C₄	#13094053	1	2	yes	—
356	C₂×C₄	#13094053	1	2	yes	—
357	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
359	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
360	C₈	—	4	2	no	—
361	C₈	—	4	2	no	—
362	C₈	—	94	2	no	—
363	C₈	—	94	2	no	—
364	C₈	—	188	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	16	4	no	—
369	C₂²	#15689344	1	2	yes	—
370	C₂²	#15689368	1	2	yes	—
371	C₂²	#15689392	1	2	yes	—
372	C₂²	C₂×#13093997	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	376	2	no	—
376	C₄	—	188	2	no	—
377	C₄	—	188	2	no	—
378	C₄	#15689344	1	2	yes	—
379	C₄	#15689368	1	2	yes	—
380	C₄	C₂×#13094043	1	1	yes	characteristic
381	C₄	C₂×#13094048	1	1	yes	characteristic
382	C₄	C₂×#13094053	1	1	yes	characteristic
383	C₄	C₂×#13094053	1	1	yes	characteristic
384	C₂	—	16	2	no	—
385	C₂	#20242927	1	4	yes	direct product factor
386	C₂	C₂×#15689344	1	1	yes	characteristic
387	C₂	C₂×#15689368	1	1	yes	characteristic
388	C₂	C₂×#15689392	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093997
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#331	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂×#127×C₄₇	C₂²
hypercenter	C₂×#331	D₄₇
hypocenter	C₄₇	C₂×#1698
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#331

Complements

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

N	H ≅ G/N	type
2. C₂×#15689419	C₂	semidirect
4. C₂×#15689250	C₂	semidirect
5. #20242927	C₂	direct (N × H)
6. C₂×#15688875	C₂	semidirect
7. C₂×#15688871	C₂	semidirect
15. #15689419	C₂²	semidirect
19. #15689250	C₂²	semidirect
29. #15688875	C₂²	semidirect
30. #15688871	C₂²	semidirect
199. C₂×C₄₇	#1698	semidirect
229. C₄₇	C₂×#1698	semidirect
385. C₂	#20242927	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×#15689469	2
C₂×#15689419	2
C₂×#15689320	2
C₂×#15689250	2
#20242927	2
C₂×#15688875	2
C₂×#15688871	2
C₂×C₄₇×#331	2
C₂×#1698	47

Subgroups — C2×#20242927 #26198925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242927 #26198925