Subgroups — C₂×#20245232 #26369536

Order 12032.

806 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	C₂²×#13093965	C₂	1	1	yes	maximal, characteristic
2	C₂×D₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689205	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245232	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688925	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×D₄×Dic₄₇	—	2	1	no	—
11	C₂×#13093986	—	2	1	no	—
12	C₂×#13093965	C₄	1	8	yes	complemented
13	C₂×#13093965	C₂²	1	4	yes	—
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	D₈×Dic₄₇	—	2	2	no	—
16	D₈×Dic₄₇	C₂²	1	4	yes	complemented
17	#15689205	—	2	2	no	—
18	#15689205	C₂²	1	4	yes	complemented
19	C₂×#13093958	C₂²	1	1	yes	characteristic
20	C₂×#13093945	—	2	1	no	—
21	C₂×#13093944	—	2	1	no	—
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15688948	C₂²	1	4	yes	complemented
24	#15688925	C₂²	1	4	yes	complemented
25	C₁₆×Dic₄₇	C₂²	1	4	yes	complemented
26	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#357	C₂²	1	4	yes	complemented
30	C₂²×D₁₈₈	—	2	1	no	—
31	C₂³×Dic₄₇	—	4	1	no	—
32	C₂²×C₄×D₄₇	—	4	1	no	—
33	C₂×D₃₇₆	—	2	1	no	—
34	C₂×D₃₇₆	—	2	2	no	—
35	C₂×D₃₇₆	C₂×C₄	1	4	yes	complemented
36	C₂×D₃₇₆	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
38	D₄×Dic₄₇	—	2	4	no	—
39	D₄×Dic₄₇	—	4	2	no	—
40	#13093986	—	2	4	no	—
41	#13093986	—	4	2	no	—
42	C₂×C₄⋊Dic₄₇	—	2	1	no	—
43	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
44	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
45	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
46	#13093965	C₂×C₄	1	16	yes	complemented
47	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
48	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
49	#13093958	—	2	1	no	—
50	#13093958	C₂³	1	2	yes	—
51	#13093945	—	2	4	no	—
52	#13093944	—	2	4	no	—
53	C₈×Dic₄₇	D₄	1	2	yes	—
54	C₈×Dic₄₇	C₂³	1	2	yes	—
55	C₂²×D₄×C₄₇	—	2	1	no	—
56	C₂×D₈×C₄₇	—	2	1	no	—
57	C₂×D₈×C₄₇	—	2	2	no	—
58	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	complemented
59	C₂×D₈×C₄₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
62	C₂×C₁₆×C₄₇	—	2	2	no	—
63	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
64	#107×C₄₇	—	2	1	no	—
65	#107×C₄₇	C₂³	1	2	yes	—
66	C₂⁴×C₄₇	—	4	1	no	—
67	C₂³×D₄₇	—	4	1	no	—
68	C₂×D₄×C₄₇	—	2	4	no	—
69	C₂×D₄×C₄₇	—	2	2	no	—
70	C₂×D₄×C₄₇	—	4	1	no	—
71	C₂×D₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	—	4	1	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	4	1	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	8	4	no	—
77	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
78	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
79	C₂×D₁₈₈	—	2	4	no	—
80	C₂×D₁₈₈	—	2	2	no	—
81	C₂×D₁₈₈	—	4	1	no	—
82	C₂×D₁₈₈	—	4	2	no	—
83	C₂×C₄×D₄₇	—	4	4	no	—
84	C₂×C₄×D₄₇	—	8	4	no	—
85	D₈×C₄₇	—	2	4	no	—
86	D₈×C₄₇	—	2	2	no	—
87	D₈×C₄₇	C₂²×C₄	1	4	yes	complemented
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	#41	1	2	yes	—
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	4	1	no	—
94	C₂²⋊Dic₄₇	—	4	4	no	—
95	(C₂×D₄₇)⋊C₄	—	4	4	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	4	1	no	—
98	C₄×Dic₄₇	D₈	1	2	yes	—
99	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
102	D₃₇₆	—	2	4	no	—
103	D₃₇₆	—	2	2	no	—
104	D₃₇₆	C₂²×C₄	1	4	yes	complemented
105	C₁₆×C₄₇	—	2	2	no	—
106	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	complemented
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	—	4	4	no	—
109	C₂³×C₄₇	—	8	2	no	—
110	C₂³×C₄₇	—	8	2	no	—
111	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₄₇	—	4	2	no	—
113	C₂²×D₄₇	—	4	4	no	—
114	C₂²×D₄₇	—	8	2	no	—
115	C₂²×D₄₇	—	8	2	no	—
116	D₄×C₄₇	—	2	4	no	—
117	D₄×C₄₇	—	4	2	no	—
118	D₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	8	2	no	—
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#135	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	8	2	no	—
128	C₂×Dic₄₇	—	8	8	no	—
129	C₂×Dic₄₇	D₁₆	1	4	yes	complemented
130	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
131	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
132	D₁₈₈	—	2	4	no	—
133	D₁₈₈	—	4	2	no	—
134	D₁₈₈	—	4	4	no	—
135	C₄×D₄₇	—	8	8	no	—
136	C₈×C₄₇	C₄×D₄	1	2	yes	—
137	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₄₇⋊C₈	—	2	2	no	—
140	C₂×C₄×D₁₆	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	4	4	no	—
142	C₂²×C₄₇	—	8	2	no	—
143	C₂²×C₄₇	—	8	2	no	—
144	C₂²×C₄₇	—	8	8	no	—
145	C₂²×C₄₇	C₄×D₈	1	2	yes	—
146	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
147	C₂²×C₄₇	#508	1	2	yes	—
148	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₄₇	—	4	4	no	—
150	C₂×D₄₇	—	8	2	no	—
151	C₂×D₄₇	—	8	2	no	—
152	C₂×D₄₇	—	8	8	no	—
153	C₄×C₄₇	—	8	2	no	—
154	C₄×C₄₇	C₄×D₈	1	2	yes	—
155	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	—	8	2	no	—
159	Dic₄₇	C₂×D₁₆	1	4	yes	complemented
160	C₂²×D₁₆	—	47	1	no	—
161	C₂×C₄×D₈	—	47	1	no	—
162	C₂×C₄×D₈	—	47	1	no	—
163	C₄×D₁₆	—	47	8	no	—
164	C₂×#366	—	47	1	no	—
165	C₂×#357	—	47	1	no	—
166	C₂×#357	D₄₇	1	1	yes	characteristic, complemented
167	C₂×C₄×C₁₆	—	47	1	no	—
168	C₂×C₄₇	—	8	4	no	—
169	C₂×C₄₇	C₄×D₁₆	1	4	yes	complemented
170	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
171	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
173	D₄₇	—	8	4	no	—
174	C₂²×D₈	—	47	1	no	—
175	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂×C₄×D₄	—	94	1	no	—
177	C₂×C₄×D₄	—	94	1	no	—
178	C₂×D₁₆	—	47	8	no	—
179	C₂×D₁₆	—	47	4	no	—
180	C₂²×C₁₆	—	47	1	no	—
181	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
182	C₄×D₈	—	47	4	no	—
183	C₄×D₈	—	47	4	no	—
184	C₄×D₈	—	94	2	no	—
185	C₄×D₈	—	94	2	no	—
186	C₂×#107	—	47	1	no	—
187	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
188	C₂×#102	—	94	1	no	—
189	C₂×#102	—	94	1	no	—
190	C₂×C₄×C₈	—	47	1	no	—
191	#366	—	47	4	no	—
192	#357	—	47	4	no	—
193	#357	C₂×D₄₇	1	4	yes	complemented
194	C₄×C₁₆	—	47	4	no	—
195	C₄₇	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	94	1	no	—
198	C₂³×C₄	—	188	1	no	—
199	C₂³×C₄	—	188	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	—	47	4	no	—
203	C₂×D₈	—	47	2	no	—
204	C₂×D₈	—	94	1	no	—
205	C₂×D₈	—	94	2	no	—
206	C₂×D₈	C₄×D₄₇	1	4	yes	complemented
207	C₂×D₈	C₂²×D₄₇	1	2	yes	—
208	C₂²×C₈	—	47	1	no	—
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	C₄×D₄	—	94	4	no	—
211	C₄×D₄	—	94	4	no	—
212	C₄×D₄	—	188	2	no	—
213	C₄×D₄	—	188	2	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	94	1	no	—
216	C₂×C₂²⋊C₄	—	188	1	no	—
217	C₂×C₂²⋊C₄	—	188	1	no	—
218	C₂×C₄²	—	47	1	no	—
219	D₁₆	—	47	16	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	47	4	no	—
222	C₂×C₁₆	—	47	2	no	—
223	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
224	#107	—	2	1	no	—
225	#107	—	47	2	no	—
226	#107	—	94	1	no	—
227	#107	C₂²×D₄₇	1	2	yes	—
228	#102	—	94	4	no	—
229	#102	—	94	4	no	—
230	C₄×C₈	—	47	2	no	—
231	C₄×C₈	—	47	2	no	—
232	C₂⁴	—	4	1	no	—
233	C₂⁴	—	188	1	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	94	4	no	—
239	C₂×D₄	—	94	2	no	—
240	C₂×D₄	—	188	1	no	—
241	C₂×D₄	—	188	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	47	1	no	—
244	C₂²×C₄	—	47	1	no	—
245	C₂²×C₄	—	376	4	no	—
246	C₂²×C₄	—	376	4	no	—
247	C₂²×C₄	—	188	4	no	—
248	C₂²×C₄	—	188	4	no	—
249	C₂²×C₄	—	188	1	no	—
250	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
251	D₈	—	2	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	47	4	no	—
254	D₈	—	94	4	no	—
255	D₈	—	94	2	no	—
256	D₈	C₂×C₄×D₄₇	1	4	yes	complemented
257	C₂×C₈	—	47	2	no	—
258	C₂×C₈	—	94	2	no	—
259	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
260	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
261	C₂×C₈	#10625352	1	2	yes	—
262	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	94	2	no	—
266	C₄⋊C₄	—	188	1	no	—
267	C₂²⋊C₄	—	188	4	no	—
268	C₂²⋊C₄	—	188	4	no	—
269	C₄²	—	47	2	no	—
270	C₄²	—	47	2	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	47	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	4	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	—	376	2	no	—
278	C₂³	—	376	2	no	—
279	C₂³	—	188	2	no	—
280	C₂³	—	188	4	no	—
281	C₂³	D₈×D₄₇	1	1	yes	characteristic
282	D₄	—	2	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	4	no	—
285	D₄	—	94	4	no	—
286	D₄	—	188	2	no	—
287	D₄	—	188	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	47	4	no	—
291	C₂×C₄	—	47	2	no	—
292	C₂×C₄	—	47	2	no	—
293	C₂×C₄	—	94	2	no	—
294	C₂×C₄	—	376	8	no	—
295	C₂×C₄	—	376	8	no	—
296	C₂×C₄	—	376	2	no	—
297	C₂×C₄	—	188	2	no	—
298	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
299	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
300	C₂×C₄	#13094045	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
302	C₈	—	94	2	no	—
303	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
304	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
305	C₈	C₂×#10625352	1	1	yes	characteristic
306	C₂²	—	4	4	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	8	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	376	2	no	—
311	C₂²	—	376	8	no	—
312	C₂²	—	376	2	no	—
313	C₂²	—	188	4	no	—
314	C₂²	#15689203	1	2	yes	—
315	C₂²	D₁₆×D₄₇	1	2	yes	—
316	C₂²	#15689554	1	2	yes	—
317	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
318	C₄	—	8	2	no	—
319	C₄	—	47	4	no	—
320	C₄	—	94	2	no	—
321	C₄	—	376	2	no	—
322	C₄	#15689203	1	2	yes	—
323	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
324	C₄	C₂×#13094045	1	1	yes	characteristic
325	C₂	—	8	4	no	—
326	C₂	—	376	4	no	—
327	C₂	#20245232	1	4	yes	direct product factor
328	C₂	C₂×#15689203	1	1	yes	characteristic
329	C₂	C₂×D₁₆×D₄₇	1	1	yes	characteristic
330	C₂	C₂×#15689554	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#357	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×#13093965	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#357	D₄₇
hypocenter	C₄₇	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
2. C₂×D₈×Dic₄₇	C₂	semidirect
3. C₂×#15689205	C₂	semidirect
4. #20245232	C₂	direct (N × H)
5. C₂×#15688948	C₂	semidirect
6. C₂×#15688925	C₂	semidirect
7. C₂×C₁₆×Dic₄₇	C₂	semidirect
8. C₂×C₄₇×#357	C₂	semidirect
12. C₂×#13093965	C₄	semidirect
16. D₈×Dic₄₇	C₂²	semidirect
18. #15689205	C₂²	semidirect
23. #15688948	C₂²	semidirect
24. #15688925	C₂²	semidirect
25. C₁₆×Dic₄₇	C₂²	semidirect
29. C₄₇×#357	C₂²	semidirect
35. C₂×D₃₇₆	C₂×C₄	semidirect
46. #13093965	C₂×C₄	semidirect
47. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
58. C₂×D₈×C₄₇	C₂×C₄	semidirect
87. D₈×C₄₇	C₂²×C₄	semidirect
104. D₃₇₆	C₂²×C₄	semidirect
106. C₄₇⋊C₁₆	C₂²×C₄	semidirect
129. C₂×Dic₄₇	D₁₆	semidirect
159. Dic₄₇	C₂×D₁₆	semidirect
166. C₂×#357	D₄₇	semidirect
169. C₂×C₄₇	C₄×D₁₆	semidirect
193. #357	C₂×D₄₇	semidirect
195. C₄₇	C₂×C₄×D₁₆	semidirect
206. C₂×D₈	C₄×D₄₇	semidirect
256. D₈	C₂×C₄×D₄₇	semidirect
327. C₂	#20245232	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×#13093965	2
C₂×D₈×Dic₄₇	2
C₂×#15689205	2
#20245232	2
C₂×#15688948	2
C₂×#15688925	2
C₂×C₁₆×Dic₄₇	2
C₂×C₄₇×#357	2
C₂×C₄×D₁₆	47

Subgroups — C2×#20245232 #26369536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245232 #26369536