Subgroups — C₂×C₄×#13093981 #26374464

Order 12032.

4462 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093981	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093981	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄×Dic₄₇⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×(C₂×D₄₇)⋊C₄	C₂	1	4	yes	maximal
6	C₂×#15688969	C₂	1	2	yes	maximal, complemented
7	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×D₄₇	C₂²	1	2	yes	—
10	C₂³×C₄×D₄₇	C₂²	1	1	yes	characteristic
11	C₂×#13093981	C₄	1	32	yes	direct product factor
12	C₂×#13093981	C₂²	1	16	yes	—
13	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
16	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
18	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₄²×D₄₇	C₄	1	16	yes	complemented
20	C₂×C₄²×D₄₇	C₂²	1	8	yes	—
21	C₄×Dic₄₇⋊C₄	C₂²	1	32	yes	complemented
22	C₄×(C₂×D₄₇)⋊C₄	C₂²	1	32	yes	—
23	#15688969	C₂²	1	16	yes	complemented
24	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
25	C₄²×Dic₄₇	C₂²	1	8	yes	complemented
26	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
29	C₄³×C₄₇	C₂²	1	8	yes	complemented
30	C₂⁴×D₄₇	C₂³	1	1	yes	characteristic
31	C₂³×Dic₄₇	C₂³	1	2	yes	—
32	C₂³×Dic₄₇	C₂³	1	4	yes	—
33	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×D₄₇	—	2	16	no	—
35	C₂²×C₄×D₄₇	C₂×C₄	1	16	yes	—
36	C₂²×C₄×D₄₇	C₂×C₄	1	8	yes	—
37	C₂²×C₄×D₄₇	C₂×C₄	1	16	yes	—
38	C₂²×C₄×D₄₇	C₂³	1	8	yes	—
39	C₂²×C₄×D₄₇	C₂³	1	4	yes	—
40	#13093981	C₂×C₄	1	128	yes	direct product factor
41	C₂×Dic₄₇⋊C₄	C₂×C₄	1	32	yes	complemented
42	C₂×Dic₄₇⋊C₄	C₂³	1	16	yes	—
43	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	32	yes	—
44	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	16	yes	—
45	C₂×C₄×Dic₄₇	—	2	16	no	—
46	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	16	yes	—
48	C₂×C₄×Dic₄₇	C₂×C₄	1	16	yes	complemented
49	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
52	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
53	C₄²×D₄₇	C₂×C₄	1	64	yes	complemented
54	C₄₇⋊#95	C₂³	1	32	yes	—
55	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₇	—	2	16	no	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
60	C₂×C₄²×C₄₇	C₂×C₄	1	16	yes	complemented
61	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
64	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂³×D₄₇	—	2	8	no	—
66	C₂³×D₄₇	C₂²×C₄	1	8	yes	—
67	C₂³×D₄₇	C₂²×C₄	1	4	yes	—
68	C₂³×D₄₇	C₂⁴	1	2	yes	—
69	C₂²×C₄×C₄₇	—	2	16	no	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₄₇	#41	1	16	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
78	C₂²×Dic₄₇	—	2	16	no	—
79	C₂²×Dic₄₇	—	2	8	no	—
80	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
81	C₂²×Dic₄₇	C₂²×C₄	1	16	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
84	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
85	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
86	C₂²×Dic₄₇	C₂⁴	1	2	yes	—
87	C₂×C₄×D₄₇	—	2	64	no	—
88	C₂×C₄×D₄₇	—	2	32	no	—
89	C₂×C₄×D₄₇	C₄²	1	64	yes	complemented
90	C₂×C₄×D₄₇	C₂²×C₄	1	32	yes	—
91	C₂×C₄×D₄₇	C₂²×C₄	1	16	yes	—
92	C₂×C₄×D₄₇	C₂²×C₄	1	32	yes	—
93	C₄²×C₄₇	—	2	32	no	—
94	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	32	yes	complemented
96	(C₂×D₄₇)⋊C₄	C₂²×C₄	1	64	yes	—
97	Dic₄₇⋊C₄	C₂²×C₄	1	64	yes	complemented
98	C₄×Dic₄₇	—	2	32	no	—
99	C₄×Dic₄₇	C₂²×C₄	1	16	yes	—
100	C₄×Dic₄₇	C₂²×C₄	1	32	yes	—
101	C₄×Dic₄₇	C₂²×C₄	1	32	yes	complemented
102	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
103	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
104	C₂³×C₄₇	C₂×#41	1	8	yes	—
105	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
106	C₂²×D₄₇	—	2	16	no	—
107	C₂²×D₄₇	—	2	32	no	—
108	C₂²×D₄₇	—	2	8	no	—
109	C₂²×D₄₇	C₂×C₄²	1	16	yes	—
110	C₂²×D₄₇	C₂³×C₄	1	8	yes	—
111	C₂²×D₄₇	C₂³×C₄	1	4	yes	—
112	C₂×C₄×C₄₇	—	2	32	no	—
113	C₂×C₄×C₄₇	—	2	16	no	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	16	yes	—
115	C₂×C₄×C₄₇	#117	1	32	yes	complemented
116	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
120	C₂×C₄×C₄₇	C₂×#41	1	16	yes	—
121	C₂×Dic₄₇	—	2	16	no	—
122	C₂×Dic₄₇	—	2	32	no	—
123	C₂×Dic₄₇	—	2	16	no	—
124	C₂×Dic₄₇	—	2	8	no	—
125	C₂×Dic₄₇	C₂×C₄²	1	16	yes	—
126	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
127	C₂×Dic₄₇	C₂³×C₄	1	16	yes	—
128	C₂×Dic₄₇	C₂³×C₄	1	4	yes	—
129	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
130	C₄×D₄₇	—	2	64	no	—
131	C₄×D₄₇	C₂×C₄²	1	64	yes	complemented
132	C₂×C₄×#117	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	C₂²×C₄²	1	4	yes	—
134	C₂²×C₄₇	C₂×#117	1	8	yes	—
135	C₂²×C₄₇	C₄×#41	1	16	yes	—
136	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₂⁴×C₄	1	2	yes	—
138	C₂²×C₄₇	C₂²×#41	1	4	yes	—
139	C₂×D₄₇	—	2	8	no	—
140	C₂×D₄₇	—	2	16	no	—
141	C₂×D₄₇	—	2	32	no	—
142	C₂×D₄₇	C₂²×C₄²	1	8	yes	—
143	C₄×C₄₇	—	2	16	no	—
144	C₄×C₄₇	C₂²×C₄²	1	8	yes	—
145	C₄×C₄₇	C₂×#117	1	16	yes	complemented
146	Dic₄₇	—	2	8	no	—
147	Dic₄₇	—	2	16	no	—
148	Dic₄₇	C₂²×C₄²	1	8	yes	—
149	C₂²×#117	—	47	2	no	—
150	C₂³×C₄²	—	47	1	no	—
151	C₄×#117	—	47	16	no	—
152	C₂×C₄×C₄⋊C₄	—	47	4	no	—
153	C₂×C₄×C₂²⋊C₄	—	47	4	no	—
154	C₂×#376	—	47	2	no	—
155	C₂×C₄³	—	47	1	no	—
156	C₂×C₄³	D₄₇	1	1	yes	characteristic, complemented
157	C₂×C₄₇	C₄×#117	1	8	yes	complemented
158	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
159	C₂×C₄₇	C₂²×#117	1	2	yes	—
160	C₂×C₄₇	C₂×C₄×#41	1	4	yes	—
161	D₄₇	—	2	8	no	—
162	C₂⁴×C₄	—	47	1	no	—
163	C₂⁴×C₄	—	47	2	no	—
164	C₂×#117	—	47	32	no	—
165	C₂×#117	—	47	16	no	—
166	C₂²×C₄⋊C₄	—	47	4	no	—
167	C₂²×C₂²⋊C₄	—	47	4	no	—
168	C₂²×C₄²	—	47	1	no	—
169	C₂²×C₄²	—	47	2	no	—
170	C₂²×C₄²	—	47	2	no	—
171	C₂²×C₄²	—	47	16	no	—
172	C₂²×C₄²	—	47	8	no	—
173	C₂²×C₄²	—	47	4	no	—
174	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
175	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
176	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
177	C₄×C₄⋊C₄	—	47	32	no	—
178	C₄×C₂²⋊C₄	—	47	32	no	—
179	#376	—	47	16	no	—
180	C₂×#95	—	47	8	no	—
181	C₄³	—	47	8	no	—
182	C₄³	C₂×D₄₇	1	8	yes	complemented
183	C₄₇	C₂×C₄×#117	1	1	yes	characteristic, complemented, Sylow
184	C₂⁵	—	47	1	no	—
185	C₂³×C₄	—	47	1	no	—
186	C₂³×C₄	—	47	16	no	—
187	C₂³×C₄	—	47	8	no	—
188	C₂³×C₄	—	47	4	no	—
189	C₂³×C₄	—	47	4	no	—
190	C₂³×C₄	—	47	2	no	—
191	C₂³×C₄	—	47	16	no	—
192	C₂³×C₄	—	47	8	no	—
193	C₂³×C₄	—	94	16	no	—
194	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
197	#117	—	47	128	no	—
198	C₂×C₄⋊C₄	—	47	32	no	—
199	C₂×C₄⋊C₄	—	47	16	no	—
200	C₂×C₂²⋊C₄	—	47	32	no	—
201	C₂×C₂²⋊C₄	—	47	16	no	—
202	C₂×C₄²	—	2	16	no	—
203	C₂×C₄²	—	47	16	no	—
204	C₂×C₄²	—	47	16	no	—
205	C₂×C₄²	—	47	8	no	—
206	C₂×C₄²	—	47	8	no	—
207	C₂×C₄²	—	47	4	no	—
208	C₂×C₄²	—	47	8	no	—
209	C₂×C₄²	—	47	64	no	—
210	C₂×C₄²	—	47	16	no	—
211	C₂×C₄²	—	94	16	no	—
212	C₂×C₄²	C₄×D₄₇	1	8	yes	—
213	C₂×C₄²	C₄×D₄₇	1	16	yes	complemented
214	C₂×C₄²	C₂²×D₄₇	1	16	yes	—
215	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
216	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
217	#95	—	47	32	no	—
218	C₂⁴	—	47	8	no	—
219	C₂⁴	—	47	4	no	—
220	C₂⁴	—	47	2	no	—
221	C₂⁴	—	94	8	no	—
222	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	—	2	16	no	—
224	C₂²×C₄	—	47	4	no	—
225	C₂²×C₄	—	47	8	no	—
226	C₂²×C₄	—	47	2	no	—
227	C₂²×C₄	—	47	64	no	—
228	C₂²×C₄	—	47	16	no	—
229	C₂²×C₄	—	47	32	no	—
230	C₂²×C₄	—	47	8	no	—
231	C₂²×C₄	—	47	16	no	—
232	C₂²×C₄	—	47	4	no	—
233	C₂²×C₄	—	47	8	no	—
234	C₂²×C₄	—	47	32	no	—
235	C₂²×C₄	—	94	16	no	—
236	C₂²×C₄	—	94	8	no	—
237	C₂²×C₄	—	94	64	no	—
238	C₂²×C₄	—	94	32	no	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	16	yes	—
240	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
241	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
242	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
243	C₂²×C₄	#10625350	1	16	yes	—
244	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
245	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
247	C₄⋊C₄	—	47	64	no	—
248	C₂²⋊C₄	—	47	64	no	—
249	C₄²	—	2	32	no	—
250	C₄²	—	47	32	no	—
251	C₄²	—	47	16	no	—
252	C₄²	—	47	32	no	—
253	C₄²	—	94	32	no	—
254	C₄²	C₂×C₄×D₄₇	1	16	yes	—
255	C₄²	C₂×C₄×D₄₇	1	32	yes	complemented
256	C₂³	—	47	16	no	—
257	C₂³	—	47	8	no	—
258	C₂³	—	47	4	no	—
259	C₂³	—	94	16	no	—
260	C₂³	—	94	32	no	—
261	C₂³	—	94	8	no	—
262	C₂³	C₂²×C₄×D₄₇	1	2	yes	—
263	C₂³	C₂²×C₄×D₄₇	1	4	yes	—
264	C₂³	C₂×#10625350	1	8	yes	—
265	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
266	C₂×C₄	—	2	32	no	—
267	C₂×C₄	—	2	16	no	—
268	C₂×C₄	—	47	16	no	—
269	C₂×C₄	—	47	4	no	—
270	C₂×C₄	—	47	8	no	—
271	C₂×C₄	—	47	64	no	—
272	C₂×C₄	—	47	8	no	—
273	C₂×C₄	—	47	16	no	—
274	C₂×C₄	—	94	16	no	—
275	C₂×C₄	—	94	32	no	—
276	C₂×C₄	—	94	16	no	—
277	C₂×C₄	—	94	8	no	—
278	C₂×C₄	—	94	64	no	—
279	C₂×C₄	C₄²×D₄₇	1	16	yes	—
280	C₂×C₄	#13093981	1	32	yes	direct product factor
281	C₂×C₄	C₂²×C₄×D₄₇	1	16	yes	—
282	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
283	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
284	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
285	C₂×C₄	C₂×#10625350	1	16	yes	—
286	C₂²	—	47	8	no	—
287	C₂²	—	94	8	no	—
288	C₂²	—	94	16	no	—
289	C₂²	—	94	32	no	—
290	C₂²	C₂×C₄²×D₄₇	1	4	yes	—
291	C₂²	C₂×#13093981	1	8	yes	—
292	C₂²	C₄×#10625350	1	16	yes	—
293	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
294	C₂²	C₂³×C₄×D₄₇	1	2	yes	—
295	C₂²	C₂²×#10625350	1	4	yes	—
296	C₄	—	2	16	no	—
297	C₄	—	47	8	no	—
298	C₄	—	94	8	no	—
299	C₄	—	94	16	no	—
300	C₄	C₂×C₄²×D₄₇	1	8	yes	—
301	C₄	C₂×#13093981	1	16	yes	direct product factor
302	C₂	—	94	8	no	—
303	C₂	C₄×#13093981	1	8	yes	direct product factor
304	C₂	C₂²×C₄²×D₄₇	1	1	yes	characteristic
305	C₂	C₂²×#13093981	1	2	yes	—
306	C₂	C₂×C₄×#10625350	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁴×D₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	C₂×C₄×#117
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. C₄×#13093981	C₂	direct (N × H)
4. C₂×C₄×Dic₄₇⋊C₄	C₂	semidirect
6. C₂×#15688969	C₂	semidirect
7. C₂×C₄²×Dic₄₇	C₂	semidirect
8. C₂×C₄³×C₄₇	C₂	semidirect
11. C₂×#13093981	C₄	direct (N × H)
19. C₂×C₄²×D₄₇	C₄	semidirect
21. C₄×Dic₄₇⋊C₄	C₂²	semidirect
23. #15688969	C₂²	semidirect
25. C₄²×Dic₄₇	C₂²	semidirect
29. C₄³×C₄₇	C₂²	semidirect
40. #13093981	C₂×C₄	direct (N × H)
41. C₂×Dic₄₇⋊C₄	C₂×C₄	semidirect
48. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
53. C₄²×D₄₇	C₂×C₄	semidirect
60. C₂×C₄²×C₄₇	C₂×C₄	semidirect
89. C₂×C₄×D₄₇	C₄²	semidirect
95. C₄²×C₄₇	C₂²×C₄	semidirect
97. Dic₄₇⋊C₄	C₂²×C₄	semidirect
101. C₄×Dic₄₇	C₂²×C₄	semidirect
115. C₂×C₄×C₄₇	#117	semidirect
131. C₄×D₄₇	C₂×C₄²	semidirect
145. C₄×C₄₇	C₂×#117	semidirect
156. C₂×C₄³	D₄₇	semidirect
157. C₂×C₄₇	C₄×#117	semidirect
182. C₄³	C₂×D₄₇	semidirect
183. C₄₇	C₂×C₄×#117	semidirect
213. C₂×C₄²	C₄×D₄₇	semidirect
255. C₄²	C₂×C₄×D₄₇	semidirect
280. C₂×C₄	#13093981	direct (N × H)
301. C₄	C₂×#13093981	direct (N × H)
303. C₂	C₄×#13093981	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×#13093981	2
C₂²×C₄²×D₄₇	2
C₄×#13093981	2
C₂×C₄×Dic₄₇⋊C₄	2
C₂×C₄×(C₂×D₄₇)⋊C₄	2
C₂×#15688969	2
C₂×C₄²×Dic₄₇	2
C₂×C₄³×C₄₇	2
C₂×C₄×#117	47

Subgroups — C2×C4×#13093981 #26374464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13093981 #26374464