Subgroups — C₄×#15689205 #26214741

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689205	C₂	1	2	yes	maximal
3	C₄×#13093986	C₂	1	2	yes	maximal, complemented
4	#107×Dic₄₇	C₂	1	2	yes	maximal, complemented
5	C₄×#13093944	C₂	1	2	yes	maximal, complemented
6	#20243202	C₂	1	4	yes	maximal, complemented
7	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#107×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
11	C₂×#13093986	C₂²	1	4	yes	—
12	C₂×C₄²×D₄₇	—	2	2	no	—
13	C₄×D₃₇₆	C₄	1	8	yes	complemented
14	#15689205	C₄	1	16	yes	direct product factor
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093946	C₂²	1	2	yes	—
17	C₂×#13093944	C₂²	1	4	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	C₄×(C₂×D₄₇)⋊C₄	—	2	2	no	—
23	#15688970	—	2	4	no	—
24	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688885	C₂²	1	4	yes	—
26	C₂×#107×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#340	C₂²	1	2	yes	—
30	C₂²×D₁₈₈	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₇	—	2	2	no	—
32	C₂²×C₄×D₄₇	—	2	4	no	—
33	C₂×D₃₇₆	C₂×C₄	1	8	yes	—
34	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₄×D₁₈₈	—	2	4	no	—
37	C₄×D₁₈₈	C₂×C₄	1	8	yes	complemented
38	#13093986	—	2	8	no	—
39	#13093986	C₂×C₄	1	16	yes	complemented
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
42	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
43	C₂×C₄×Dic₄₇	—	2	4	no	—
44	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
48	C₄²×D₄₇	—	4	8	no	—
49	#13093958	C₂×C₄	1	4	yes	complemented
50	#13093946	C₂×C₄	1	8	yes	complemented
51	#13093944	C₂×C₄	1	16	yes	complemented
52	C₄×C₄₇⋊C₈	—	2	2	no	—
53	C₈×Dic₄₇	—	2	4	no	—
54	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
55	C₄₇⋊#95	—	2	4	no	—
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
59	C₂×C₄²×C₄₇	—	2	2	no	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#107×C₄₇	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
63	#95×C₄₇	—	2	2	no	—
64	C₂³×D₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
71	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
72	C₂×D₁₈₈	—	2	4	no	—
73	C₂×D₁₈₈	—	2	2	no	—
74	C₂×D₁₈₈	C₂²×C₄	1	8	yes	—
75	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
76	C₂×C₄×D₄₇	—	2	8	no	—
77	C₂×C₄×D₄₇	—	2	16	no	—
78	C₂×C₄×D₄₇	—	4	4	no	—
79	C₂×C₄×D₄₇	—	4	4	no	—
80	C₂×C₄×D₄₇	—	4	8	no	—
81	C₂×C₄×D₄₇	—	4	8	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
88	C₄²×C₄₇	—	4	4	no	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	#41	1	2	yes	—
91	(C₂×D₄₇)⋊C₄	—	2	8	no	—
92	(C₂×D₄₇)⋊C₄	—	2	16	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
95	C₄×Dic₄₇	—	2	2	no	—
96	C₄×Dic₄₇	—	2	4	no	—
97	C₄×Dic₄₇	—	4	8	no	—
98	C₄×Dic₄₇	D₈	1	4	yes	complemented
99	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
100	C₄×Dic₄₇	#41	1	4	yes	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
104	D₃₇₆	C₄²	1	16	yes	complemented
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂²×D₄₇	—	2	8	no	—
107	C₂²×D₄₇	—	2	4	no	—
108	C₂²×D₄₇	—	4	2	no	—
109	C₂²×D₄₇	—	4	4	no	—
110	C₂²×D₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₄₇	#135	1	2	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
129	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
130	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
131	C₂×Dic₄₇	#135	1	4	yes	—
132	D₁₈₈	—	2	4	no	—
133	D₁₈₈	—	2	8	no	—
134	D₁₈₈	C₂×C₄²	1	8	yes	complemented
135	C₄×D₄₇	—	4	16	no	—
136	C₄×D₄₇	—	4	32	no	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	C₂×C₄²	1	4	yes	complemented
139	C₄₇⋊C₈	—	2	4	no	—
140	C₄²×D₈	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
142	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#135	1	2	yes	—
145	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
146	C₂×D₄₇	—	2	8	no	—
147	C₂×D₄₇	—	4	8	no	—
148	C₂×D₄₇	—	4	4	no	—
149	C₂×D₄₇	—	4	8	no	—
150	C₂×D₄₇	—	4	4	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	4	8	no	—
153	C₄×C₄₇	C₄×D₈	1	4	yes	complemented
154	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
155	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
156	C₄×C₄₇	C₄×#41	1	2	yes	—
157	Dic₄₇	—	2	4	no	—
158	Dic₄₇	—	4	4	no	—
159	Dic₄₇	C₄×D₈	1	8	yes	complemented
160	C₂×C₄×D₈	—	47	2	no	—
161	C₂×C₄×D₈	—	47	1	no	—
162	C₄²×D₄	—	47	2	no	—
163	C₄×#107	—	47	2	no	—
164	C₄×#107	D₄₇	1	1	yes	characteristic, complemented
165	C₄×#102	—	47	2	no	—
166	C₄×#102	—	47	4	no	—
167	C₄²×C₈	—	47	1	no	—
168	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
169	C₂×C₄₇	C₂×C₄×D₈	1	2	yes	—
170	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
171	C₂×C₄₇	C₄×#135	1	2	yes	—
172	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
173	D₄₇	—	4	8	no	—
174	C₂²×D₈	—	47	1	no	—
175	C₂×C₄×D₄	—	47	4	no	—
176	C₂×C₄×D₄	—	47	2	no	—
177	C₂²×C₄²	—	94	2	no	—
178	C₄×D₈	—	47	16	no	—
179	C₄×D₈	—	47	8	no	—
180	C₂×#107	—	47	1	no	—
181	C₂×#107	C₂×D₄₇	1	2	yes	—
182	C₂×#102	—	47	2	no	—
183	C₂×#102	—	47	4	no	—
184	C₂×C₄×C₈	—	47	2	no	—
185	C₂×C₄×C₈	—	47	2	no	—
186	C₂×C₄×C₈	—	47	1	no	—
187	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
188	C₄×C₄⋊C₄	—	47	4	no	—
189	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
190	C₄×C₂²⋊C₄	—	94	4	no	—
191	C₄×C₂²⋊C₄	—	94	2	no	—
192	C₄³	—	47	1	no	—
193	#340	—	47	4	no	—
194	#340	C₂×D₄₇	1	2	yes	—
195	C₄₇	C₄²×D₈	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	47	2	no	—
197	C₂³×C₄	—	94	4	no	—
198	C₂³×C₄	—	94	2	no	—
199	C₂×D₈	—	47	8	no	—
200	C₂×D₈	—	47	4	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
204	C₄×D₄	—	47	16	no	—
205	C₄×D₄	—	47	8	no	—
206	C₄×D₄	—	94	8	no	—
207	C₄×D₄	—	94	4	no	—
208	C₂×C₄⋊C₄	—	47	2	no	—
209	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
210	C₂×C₂²⋊C₄	—	94	4	no	—
211	C₂×C₂²⋊C₄	—	94	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	47	1	no	—
214	C₂×C₄²	—	47	2	no	—
215	C₂×C₄²	—	47	2	no	—
216	C₂×C₄²	—	47	1	no	—
217	C₂×C₄²	—	94	4	no	—
218	C₂×C₄²	—	188	8	no	—
219	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
220	#107	—	47	4	no	—
221	#107	C₄×D₄₇	1	8	yes	complemented
222	#102	—	47	8	no	—
223	#102	—	47	16	no	—
224	C₄×C₈	—	47	8	no	—
225	C₄×C₈	—	94	4	no	—
226	C₄×C₈	—	94	2	no	—
227	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
228	#95	—	2	2	no	—
229	#95	—	94	4	no	—
230	C₂⁴	—	94	2	no	—
231	C₂×D₄	—	47	4	no	—
232	C₂×D₄	—	47	8	no	—
233	C₂×D₄	—	94	4	no	—
234	C₂×D₄	—	94	2	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	47	2	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	94	16	no	—
239	C₂²×C₄	—	94	2	no	—
240	C₂²×C₄	—	94	8	no	—
241	C₂²×C₄	—	188	8	no	—
242	C₂²×C₄	—	188	8	no	—
243	C₂²×C₄	—	188	4	no	—
244	C₂²×C₄	—	188	4	no	—
245	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	D₈	—	47	16	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	94	2	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
254	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
255	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	47	4	no	—
258	C₄⋊C₄	—	94	2	no	—
259	C₄⋊C₄	C₂×C₄×D₄₇	1	8	yes	—
260	C₂²⋊C₄	—	94	16	no	—
261	C₂²⋊C₄	—	94	8	no	—
262	C₄²	—	4	4	no	—
263	C₄²	—	47	4	no	—
264	C₄²	—	47	4	no	—
265	C₄²	—	47	4	no	—
266	C₄²	—	94	4	no	—
267	C₄²	—	94	2	no	—
268	C₄²	—	188	8	no	—
269	C₄²	C₂×C₄×D₄₇	1	2	yes	—
270	C₄²	#10625354	1	2	yes	—
271	C₂³	—	94	8	no	—
272	C₂³	—	94	4	no	—
273	C₂³	—	188	2	no	—
274	C₂³	—	188	4	no	—
275	C₂³	—	188	2	no	—
276	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
277	D₄	—	47	8	no	—
278	D₄	—	94	4	no	—
279	D₄	—	94	8	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	47	4	no	—
285	C₂×C₄	—	47	4	no	—
286	C₂×C₄	—	47	4	no	—
287	C₂×C₄	—	47	4	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	—	188	32	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	16	no	—
295	C₂×C₄	#13093986	1	2	yes	—
296	C₂×C₄	#13093986	1	2	yes	—
297	C₂×C₄	D₈×D₄₇	1	2	yes	—
298	C₂×C₄	#13094047	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
301	C₂×C₄	#41×D₄₇	1	2	yes	—
302	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	94	4	no	—
305	C₈	C₄²×D₄₇	1	4	yes	complemented
306	C₂²	—	94	8	no	—
307	C₂²	—	188	8	no	—
308	C₂²	—	188	8	no	—
309	C₂²	—	188	4	no	—
310	C₂²	—	188	4	no	—
311	C₂²	C₄×D₄×D₄₇	1	2	yes	—
312	C₂²	C₂×#13093986	1	1	yes	characteristic
313	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
314	C₂²	#135×D₄₇	1	2	yes	—
315	C₂²	C₂×#13094047	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	8	no	—
318	C₄	—	47	8	no	—
319	C₄	—	94	4	no	—
320	C₄	—	188	4	no	—
321	C₄	#15689205	1	4	yes	direct product factor
322	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
323	C₄	#15689591	1	1	yes	characteristic
324	C₄	C₄×#10625354	1	2	yes	—
325	C₂	—	188	8	no	—
326	C₂	C₄×#13093986	1	1	yes	characteristic
327	C₂	C₄×D₈×D₄₇	1	2	yes	—
328	C₂	C₂×#15689205	1	1	yes	characteristic
329	C₂	#20252159	1	1	yes	characteristic
330	C₂	C₄×#13094047	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#107×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#107	D₄₇
hypocenter	C₄₇	C₄²×D₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
3. C₄×#13093986	C₂	semidirect
4. #107×Dic₄₇	C₂	semidirect
5. C₄×#13093944	C₂	semidirect
6. #20243202	C₂	semidirect
7. C₄×C₈×Dic₄₇	C₂	semidirect
8. C₄×#107×C₄₇	C₂	semidirect
13. C₄×D₃₇₆	C₄	semidirect
14. #15689205	C₄	direct (N × H)
37. C₄×D₁₈₈	C₂×C₄	semidirect
39. #13093986	C₂×C₄	semidirect
49. #13093958	C₂×C₄	semidirect
50. #13093946	C₂×C₄	semidirect
51. #13093944	C₂×C₄	semidirect
54. C₈×Dic₄₇	C₂×C₄	semidirect
61. #107×C₄₇	C₂×C₄	semidirect
62. C₄×C₈×C₄₇	C₂×C₄	semidirect
98. C₄×Dic₄₇	D₈	semidirect
104. D₃₇₆	C₄²	semidirect
134. D₁₈₈	C₂×C₄²	semidirect
138. C₈×C₄₇	C₂×C₄²	semidirect
153. C₄×C₄₇	C₄×D₈	semidirect
159. Dic₄₇	C₄×D₈	semidirect
164. C₄×#107	D₄₇	semidirect
195. C₄₇	C₄²×D₈	semidirect
221. #107	C₄×D₄₇	semidirect
227. C₄×C₈	C₄×D₄₇	semidirect
305. C₈	C₄²×D₄₇	semidirect
321. C₄	#15689205	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×C₄×D₃₇₆	2
C₂×#15689205	2
C₄×#13093986	2
#107×Dic₄₇	2
C₄×#13093944	2
#20243202	2
C₄×C₈×Dic₄₇	2
C₄×#107×C₄₇	2
C₄²×D₈	47

Subgroups — C4×#15689205 #26214741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689205 #26214741