Subgroups — C₂×C₄×#13093957 #26214323

Order 12032.

1358 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093957	C₂	1	2	yes	maximal
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#13093957	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688885	C₂	1	2	yes	maximal
6	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	—	2	1	no	—
11	C₂×#13093957	C₄	1	16	yes	direct product factor
12	C₂×#13093957	C₂²	1	8	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₄₇	—	2	2	no	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂×C₄₇⋊#95	—	2	1	no	—
18	#15688885	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	—	2	1	no	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
24	C₂×#95×C₄₇	—	2	1	no	—
25	C₂³×Dic₄₇	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	C₂×C₄×Dic₄₇	—	2	4	no	—
37	C₂×C₄×Dic₄₇	—	4	4	no	—
38	#13093957	C₂×C₄	1	32	yes	direct product factor
39	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
40	C₄₇⋊#95	—	2	4	no	—
41	C₂³×C₄×C₄₇	—	2	2	no	—
42	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	—	2	4	no	—
50	C₂×C₄²×C₄₇	—	4	4	no	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	D₄	1	2	yes	—
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	#95×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	—	4	4	no	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	D₈	1	4	yes	—
65	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	8	no	—
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
81	C₄²×C₄₇	—	4	8	no	—
82	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
85	C₄⋊Dic₄₇	—	2	8	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
88	C₄×Dic₄₇	—	4	8	no	—
89	C₂×C₄₇⋊C₈	—	2	8	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	C₄²	1	16	yes	complemented
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₄₇	C₄×D₄	1	4	yes	—
97	C₂³×C₄₇	C₂×D₈	1	2	yes	—
98	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
99	C₂³×C₄₇	#135	1	4	yes	—
100	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	2	8	no	—
104	C₂×C₄×C₄₇	—	4	4	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	4	16	no	—
107	C₂×C₄×C₄₇	#107	1	16	yes	complemented
108	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₄₇	#117	1	4	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
115	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	16	no	—
125	C₄₇⋊C₈	—	2	4	no	—
126	C₄₇⋊C₈	C₂×C₄²	1	8	yes	complemented
127	C₂×C₄×#107	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
129	C₂²×C₄₇	C₂×#107	1	4	yes	—
130	C₂²×C₄₇	#427	1	4	yes	—
131	C₂²×C₄₇	C₄×D₈	1	8	yes	—
132	C₂²×C₄₇	C₄×Q₁₆	1	8	yes	—
133	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
135	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
136	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#135	1	2	yes	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	4	8	no	—
140	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
141	C₄×C₄₇	C₂×#107	1	8	yes	complemented
142	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
144	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
145	Dic₄₇	—	4	8	no	—
146	C₂²×#107	—	47	2	no	—
147	C₂²×C₄×C₈	—	47	1	no	—
148	C₂×C₄×C₄⋊C₄	—	47	1	no	—
149	C₂×C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
150	C₄×#107	—	47	8	no	—
151	C₂×#340	—	47	2	no	—
152	C₂×C₄₇	C₄×#107	1	8	yes	complemented
153	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
156	C₂×C₄₇	C₂×C₄×D₈	1	2	yes	—
157	C₂×C₄₇	C₂×C₄×Q₁₆	1	2	yes	—
158	C₂³×C₈	—	47	1	no	—
159	C₂³×C₈	—	47	1	no	—
160	C₂²×C₄⋊C₄	—	47	2	no	—
161	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
162	C₂²×C₄²	—	2	1	no	—
163	C₂²×C₄²	—	94	1	no	—
164	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
165	C₂×#107	—	47	16	no	—
166	C₂×#107	—	47	8	no	—
167	C₂×C₄×C₈	—	47	8	no	—
168	C₂×C₄×C₈	—	47	4	no	—
169	C₄×C₄⋊C₄	—	2	2	no	—
170	C₄×C₄⋊C₄	—	47	4	no	—
171	C₄×C₄⋊C₄	—	94	2	no	—
172	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
173	C₂×#95	—	2	1	no	—
174	C₂×#95	—	94	1	no	—
175	#340	—	47	8	no	—
176	C₄₇	C₂×C₄×#107	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	—	94	2	no	—
179	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
182	C₂²×C₈	—	47	4	no	—
183	C₂²×C₈	—	47	8	no	—
184	C₂²×C₈	—	47	2	no	—
185	C₂²×C₈	—	47	4	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	94	4	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	47	8	no	—
191	C₂×C₄⋊C₄	—	47	4	no	—
192	C₂×C₄⋊C₄	—	94	4	no	—
193	C₂×C₄⋊C₄	—	94	2	no	—
194	C₂×C₄⋊C₄	C₄×D₄₇	1	8	yes	—
195	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
196	C₂×C₄²	—	2	4	no	—
197	C₂×C₄²	—	4	4	no	—
198	C₂×C₄²	—	94	4	no	—
199	C₂×C₄²	—	188	4	no	—
200	C₂×C₄²	Dic₉₄	1	4	yes	—
201	C₂×C₄²	C₄×D₄₇	1	4	yes	—
202	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
203	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
204	#107	—	47	32	no	—
205	C₄×C₈	—	47	16	no	—
206	#95	—	2	4	no	—
207	#95	—	94	4	no	—
208	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	94	8	no	—
215	C₂²×C₄	—	94	4	no	—
216	C₂²×C₄	—	188	4	no	—
217	C₂²×C₄	—	188	4	no	—
218	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
219	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
220	C₂²×C₄	#10625334	1	4	yes	—
221	C₂²×C₄	#10625337	1	4	yes	—
222	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
225	C₂²×C₄	#10625350	1	4	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
229	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
230	C₂×C₈	—	47	16	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	47	8	no	—
233	C₂×C₈	—	47	4	no	—
234	C₂×C₈	—	94	8	no	—
235	C₂×C₈	—	94	4	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	47	8	no	—
239	C₄⋊C₄	—	94	8	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₄⋊C₄	C₂×C₄×D₄₇	1	8	yes	—
242	C₄²	—	4	8	no	—
243	C₄²	—	188	8	no	—
244	C₄²	Dic₄₇⋊C₄	1	8	yes	—
245	C₄²	C₂×Dic₉₄	1	4	yes	—
246	C₄²	C₂×C₄×D₄₇	1	4	yes	—
247	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
248	C₂³	C₄×C₂²⋊D₄₇	1	4	yes	—
249	C₂³	C₂×#10625334	1	2	yes	—
250	C₂³	C₂×#10625337	1	2	yes	—
251	C₂³	#13094044	1	4	yes	—
252	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	94	8	no	—
260	C₂×C₄	—	188	4	no	—
261	C₂×C₄	—	188	4	no	—
262	C₂×C₄	—	188	16	no	—
263	C₂×C₄	#13093957	1	16	yes	direct product factor
264	C₂×C₄	C₄²×D₄₇	1	4	yes	—
265	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
266	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
267	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
268	C₂×C₄	#13093981	1	4	yes	—
269	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
270	C₂×C₄	C₄×Dic₉₄	1	8	yes	—
271	C₂×C₄	C₂×#10625334	1	4	yes	—
272	C₂×C₄	C₂×#10625337	1	4	yes	—
273	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
275	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
276	C₂×C₄	C₂×#10625350	1	2	yes	—
277	C₈	—	47	8	no	—
278	C₈	—	94	4	no	—
279	C₂²	C₄×Dic₄₇⋊C₄	1	4	yes	—
280	C₂²	C₂×#13093957	1	4	yes	—
281	C₂²	#15689159	1	4	yes	—
282	C₂²	C₄×#10625334	1	8	yes	—
283	C₂²	C₄×#10625337	1	8	yes	—
284	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
285	C₂²	C₂×C₄×C₂²⋊D₄₇	1	2	yes	—
286	C₂²	C₂²×#10625334	1	1	yes	characteristic
287	C₂²	C₂²×#10625337	1	1	yes	characteristic
288	C₂²	C₂×#13094044	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	4	8	no	—
291	C₄	—	188	8	no	—
292	C₄	C₄×Dic₄₇⋊C₄	1	4	yes	—
293	C₄	C₂×#13093957	1	8	yes	direct product factor
294	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
295	C₄	C₂×#13093981	1	1	yes	characteristic
296	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
297	C₂	C₄×#13093957	1	8	yes	direct product factor
298	C₂	C₂×C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
299	C₂	C₂²×#13093957	1	1	yes	characteristic
300	C₂	C₂×#15689159	1	1	yes	characteristic
301	C₂	C₂×C₄×#10625334	1	2	yes	—
302	C₂	C₂×C₄×#10625337	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄×#13093957	C₂	direct (N × H)
11. C₂×#13093957	C₄	direct (N × H)
13. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
38. #13093957	C₂×C₄	direct (N × H)
39. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
91. C₂×C₄₇⋊C₈	C₄²	semidirect
107. C₂×C₄×C₄₇	#107	semidirect
126. C₄₇⋊C₈	C₂×C₄²	semidirect
141. C₄×C₄₇	C₂×#107	semidirect
152. C₂×C₄₇	C₄×#107	semidirect
176. C₄₇	C₂×C₄×#107	semidirect
263. C₂×C₄	#13093957	direct (N × H)
293. C₄	C₂×#13093957	direct (N × H)
297. C₂	C₄×#13093957	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×#13093957	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×C₄×C₄⋊Dic₄₇	2
C₄×#13093957	2
C₂×#15688885	2
C₂×C₄×C₄⋊C₄×C₄₇	2
C₂×C₄×#107	47

Subgroups — C2×C4×#13093957 #26214323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13093957 #26214323