Subgroups — C₄×#13093998 #20245901

Order 6016.

824 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13093998	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₂²⋊D₄₇	C₂	1	2	yes	maximal, complemented
4	#15689025	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689014	C₂	1	2	yes	maximal, complemented
7	#15689002	C₂	1	2	yes	maximal, complemented
8	#15688983	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×(C₂×D₄₇)⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
14	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
15	#13093998	C₄	1	8	yes	direct product factor
16	C₄×D₁₈₈	—	2	4	no	—
17	C₄×C₂²⋊D₄₇	—	2	8	no	—
18	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	—	2	4	no	—
35	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
40	C₂³×D₄₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	—	2	2	no	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	complemented
49	C₂²×C₄×C₄₇	D₄	1	4	yes	complemented
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₂²⋊D₄₇	—	2	4	no	—
57	C₂×C₂²⋊D₄₇	—	2	4	no	—
58	C₂×C₂²⋊D₄₇	C₂×C₄	1	8	yes	complemented
59	C₂²×Dic₄₇	C₂³	1	2	yes	—
60	C₂²×Dic₄₇	C₂³	1	2	yes	—
61	C₂²×Dic₄₇	C₂³	1	2	yes	—
62	C₂×D₁₈₈	—	2	2	no	—
63	C₂×D₁₈₈	—	2	2	no	—
64	C₂×D₁₈₈	C₂×C₄	1	4	yes	complemented
65	C₂×C₄×D₄₇	—	2	4	no	—
66	C₂×C₄×D₄₇	—	2	4	no	—
67	C₂×C₄×D₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	D₄	1	4	yes	complemented
72	C₂²⋊Dic₄₇	—	2	4	no	—
73	(C₂×D₄₇)⋊C₄	—	2	4	no	—
74	(C₂×D₄₇)⋊C₄	—	2	4	no	—
75	(C₂×D₄₇)⋊C₄	—	2	4	no	—
76	(C₂×D₄₇)⋊C₄	C₂×C₄	1	8	yes	complemented
77	C₄⋊Dic₄₇	—	2	2	no	—
78	C₄⋊Dic₄₇	—	2	2	no	—
79	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	complemented
80	Dic₄₇⋊C₄	—	2	4	no	—
81	C₄×Dic₄₇	—	2	4	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	—	2	1	no	—
86	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂³×C₄₇	C₂×D₄	1	2	yes	—
88	C₂³×C₄₇	#41	1	2	yes	—
89	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₂²×D₄₇	—	2	2	no	—
91	C₂²×D₄₇	—	2	2	no	—
92	C₂²×D₄₇	—	2	2	no	—
93	C₂²×D₄₇	—	2	2	no	—
94	C₂²×D₄₇	—	2	2	no	—
95	C₂²×D₄₇	—	2	2	no	—
96	C₂²×D₄₇	C₂²×C₄	1	4	yes	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	#41	1	2	yes	—
116	C₂×C₄×C₄₇	#41	1	2	yes	—
117	C₂×C₄×C₄₇	#41	1	2	yes	—
118	C₂²⋊D₄₇	—	2	16	no	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
128	D₁₈₈	—	2	8	no	—
129	C₄×D₄₇	—	4	8	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	C₄×D₄	1	4	yes	complemented
137	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×D₄₇	—	2	4	no	—
145	C₂×D₄₇	—	2	4	no	—
146	C₂×D₄₇	—	2	4	no	—
147	C₂×D₄₇	—	4	2	no	—
148	C₂×D₄₇	—	4	2	no	—
149	C₂×D₄₇	—	4	2	no	—
150	C₂×D₄₇	—	4	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	C₄×D₄	1	4	yes	complemented
156	C₄×C₄₇	#121	1	4	yes	complemented
157	Dic₄₇	—	2	4	no	—
158	Dic₄₇	—	4	2	no	—
159	Dic₄₇	—	4	2	no	—
160	C₄×#121	—	47	1	no	maximal, Sylow
161	C₂×C₄₇	—	2	4	no	—
162	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
166	C₂×C₄₇	#525	1	1	yes	characteristic
167	C₂×C₄₇	#529	1	1	yes	characteristic
168	C₂×C₄₇	#532	1	1	yes	characteristic
169	D₄₇	—	4	4	no	—
170	C₂×#121	—	47	1	no	—
171	C₂×C₄×D₄	—	47	2	no	—
172	C₂×C₄×D₄	—	47	1	no	—
173	C₂²×C₄²	D₄₇	1	1	yes	characteristic, complemented
174	#390	—	47	1	no	—
175	#389	—	47	1	no	—
176	#388	—	47	2	no	—
177	#386	—	47	2	no	—
178	#380	—	47	1	no	—
179	C₄×C₄⋊C₄	—	47	1	no	—
180	C₄×C₂²⋊C₄	—	47	2	no	—
181	C₄₇	C₄×#121	1	1	yes	characteristic, complemented, Sylow
182	C₂²×D₄	—	47	2	no	—
183	C₂²×D₄	—	47	1	no	—
184	C₂³×C₄	—	47	2	no	—
185	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
188	#121	—	47	8	no	—
189	C₄×D₄	—	94	8	no	—
190	C₄×D₄	—	94	4	no	—
191	C₂×C₄⋊C₄	—	47	1	no	—
192	C₂×C₄⋊C₄	—	47	2	no	—
193	C₂×C₄⋊C₄	—	47	1	no	—
194	C₂×C₄⋊C₄	—	47	1	no	—
195	C₂×C₂²⋊C₄	—	47	2	no	—
196	C₂×C₂²⋊C₄	—	47	2	no	—
197	C₂×C₂²⋊C₄	—	47	2	no	—
198	C₂×C₂²⋊C₄	—	47	2	no	—
199	C₂×C₂²⋊C₄	—	47	2	no	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
206	#95	—	47	2	no	—
207	#95	—	47	2	no	—
208	#95	—	47	2	no	—
209	C₂⁴	—	47	2	no	—
210	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
211	C₂×D₄	—	47	8	no	—
212	C₂×D₄	—	47	4	no	—
213	C₂×D₄	—	94	4	no	—
214	C₂×D₄	—	94	4	no	—
215	C₂×D₄	—	94	2	no	—
216	C₂×D₄	—	94	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	—	47	2	no	—
226	C₂²×C₄	—	47	2	no	—
227	C₂²×C₄	—	94	4	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	—	94	4	no	—
230	C₂²×C₄	C₄×D₄₇	1	4	yes	complemented
231	C₂²×C₄	D₁₈₈	1	4	yes	complemented
232	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
238	C₄⋊C₄	—	47	4	no	—
239	C₄⋊C₄	—	94	4	no	—
240	C₄⋊C₄	—	94	2	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₂²⋊C₄	—	47	8	no	—
243	C₂²⋊C₄	—	94	4	no	—
244	C₂²⋊C₄	—	94	4	no	—
245	C₂²⋊C₄	—	94	4	no	—
246	C₂²⋊C₄	—	94	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	94	4	no	—
251	C₄²	C₂²⋊D₄₇	1	4	yes	complemented
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	47	4	no	—
257	C₂³	—	94	2	no	—
258	C₂³	—	94	2	no	—
259	C₂³	—	94	2	no	—
260	C₂³	—	94	2	no	—
261	C₂³	—	94	2	no	—
262	C₂³	—	94	2	no	—
263	C₂³	C₂×C₄×D₄₇	1	2	yes	—
264	C₂³	C₂×D₁₈₈	1	2	yes	—
265	C₂³	#10625350	1	2	yes	—
266	C₂³	C₂³×D₄₇	1	1	yes	characteristic
267	D₄	—	94	16	no	—
268	D₄	—	94	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	47	4	no	—
281	C₂×C₄	—	94	2	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	94	2	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	188	8	no	—
290	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
291	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
292	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
293	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
294	C₂×C₄	#10625350	1	2	yes	—
295	C₂×C₄	#10625350	1	2	yes	—
296	C₂×C₄	#10625350	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
298	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
299	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	94	4	no	—
307	C₂²	—	94	4	no	—
308	C₂²	—	94	4	no	—
309	C₂²	—	188	2	no	—
310	C₂²	—	188	2	no	—
311	C₂²	—	188	2	no	—
312	C₂²	—	188	2	no	—
313	C₂²	C₄×D₁₈₈	1	4	yes	complemented
314	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
315	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
316	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
317	C₂²	C₂×#10625350	1	1	yes	characteristic
318	C₂²	C₂×#10625350	1	1	yes	characteristic
319	C₂²	C₂×#10625350	1	1	yes	characteristic
320	C₂²	C₂×#10625350	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	94	4	no	—
326	C₄	—	188	2	no	—
327	C₄	—	188	2	no	—
328	C₄	C₄×C₂²⋊D₄₇	1	4	yes	complemented
329	C₄	#13093998	1	4	yes	direct product factor
330	C₂	—	2	4	no	—
331	C₂	—	188	4	no	—
332	C₂	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₂	C₂×C₄×D₁₈₈	1	1	yes	characteristic
334	C₂	C₄×#10625350	1	1	yes	characteristic
335	C₂	C₂×#13093998	1	1	yes	characteristic
336	C₂	#15689634	1	1	yes	characteristic
337	C₂	#15689657	1	1	yes	characteristic
338	C₂	#15689673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Frattini	C₂³	C₂³×D₄₇
Fitting	C₂²×C₄²×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂²×C₄²	D₄₇
hypocenter	C₄₇	C₄×#121
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₁₈₈	C₂	semidirect
3. C₂×C₄×C₂²⋊D₄₇	C₂	semidirect
4. #15689025	C₂	semidirect
5. #15689021	C₂	semidirect
6. #15689014	C₂	semidirect
7. #15689002	C₂	semidirect
8. #15688983	C₂	semidirect
9. C₄×C₄⋊Dic₄₇	C₂	semidirect
10. C₄×(C₂×D₄₇)⋊C₄	C₂	semidirect
11. C₂²×C₄²×C₄₇	C₂	semidirect
15. #13093998	C₄	direct (N × H)
48. C₂²×C₄×C₄₇	C₂×C₄	semidirect
49. C₂²×C₄×C₄₇	D₄	semidirect
58. C₂×C₂²⋊D₄₇	C₂×C₄	semidirect
64. C₂×D₁₈₈	C₂×C₄	semidirect
71. C₄²×C₄₇	D₄	semidirect
76. (C₂×D₄₇)⋊C₄	C₂×C₄	semidirect
79. C₄⋊Dic₄₇	C₂×C₄	semidirect
136. C₂²×C₄₇	C₄×D₄	semidirect
155. C₄×C₄₇	C₄×D₄	semidirect
156. C₄×C₄₇	#121	semidirect
173. C₂²×C₄²	D₄₇	semidirect
181. C₄₇	C₄×#121	semidirect
230. C₂²×C₄	C₄×D₄₇	semidirect
231. C₂²×C₄	D₁₈₈	semidirect
251. C₄²	C₂²⋊D₄₇	semidirect
313. C₂²	C₄×D₁₈₈	semidirect
328. C₄	C₄×C₂²⋊D₄₇	semidirect
329. C₄	#13093998	direct (N × H)

Subgroups — C4×#13093998 #20245901

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13093998 #20245901