Subgroups — C₂×#20241396 #26126496

Order 12032.

558 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×#15689359	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241396	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₇₆	C₄	1	1	yes	characteristic, complemented
6	C₂×#13093998	—	2	1	no	—
7	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
8	#15689359	—	2	1	no	—
9	#15689359	—	2	1	no	—
10	#15689359	C₄	1	2	yes	complemented
11	#15689359	C₂²	1	2	yes	—
12	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
13	C₂×#13093946	—	2	1	no	—
14	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
15	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₄₇×#340	—	2	1	no	—
19	C₄₇×#340	C₂²	1	2	yes	complemented
20	C₂²×D₁₈₈	—	2	1	no	—
21	C₂²×C₂²⋊D₄₇	—	4	1	no	—
22	C₂×D₃₇₆	—	2	1	no	—
23	C₂×D₃₇₆	—	4	2	no	—
24	C₂×D₃₇₆	C₂×C₄	1	2	yes	complemented
25	#13093998	—	2	2	no	—
26	#13093998	—	2	2	no	—
27	#13093998	—	4	1	no	—
28	#13093998	—	4	1	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
31	C₂×C₄₇⋊C₁₆	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	—	4	2	no	—
33	#13093958	—	2	1	no	—
34	#13093958	C₂×C₄	1	2	yes	complemented
35	#13093946	—	2	2	no	—
36	#13093946	—	2	2	no	—
37	C₂³×C₄×C₄₇	—	4	1	no	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	4	2	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
50	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
51	C₂³×D₄₇	—	4	1	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	8	2	no	—
57	C₂²×C₄×C₄₇	—	8	2	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₇	—	4	2	no	—
65	C₂×C₂²⋊D₄₇	—	4	2	no	—
66	C₂×C₂²⋊D₄₇	—	8	2	no	—
67	C₂×C₂²⋊D₄₇	—	8	2	no	—
68	C₂²×Dic₄₇	—	4	1	no	—
69	C₂×D₁₈₈	—	2	2	no	—
70	C₂×D₁₈₈	—	4	2	no	—
71	C₂×D₁₈₈	—	4	2	no	—
72	C₂×D₁₈₈	—	4	1	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	4	1	no	—
76	C₂×C₈×C₄₇	—	4	1	no	—
77	C₂×C₈×C₄₇	—	4	1	no	—
78	C₂×C₈×C₄₇	—	4	1	no	—
79	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	4	4	no	—
85	C₄⋊C₄×C₄₇	—	4	2	no	—
86	(C₂×D₄₇)⋊C₄	—	4	2	no	—
87	(C₂×D₄₇)⋊C₄	—	4	2	no	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	4	1	no	—
90	D₃₇₆	—	4	2	no	—
91	D₃₇₆	—	4	2	no	—
92	C₄₇⋊C₁₆	—	4	2	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	#102	1	1	yes	characteristic
96	C₂³×C₄₇	#102	1	1	yes	characteristic
97	C₂³×C₄₇	D₁₆	1	2	yes	—
98	C₂³×C₄₇	SD₃₂	1	2	yes	—
99	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₄₇	—	4	2	no	—
101	C₂²×D₄₇	—	8	2	no	—
102	C₂²×D₄₇	—	8	2	no	—
103	C₂²×D₄₇	—	8	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	8	8	no	—
113	C₂×C₄×C₄₇	—	8	2	no	—
114	C₂×C₄×C₄₇	—	8	2	no	—
115	C₂×C₄×C₄₇	#99	1	2	yes	—
116	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#104	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₂²⋊D₄₇	—	8	4	no	—
122	C₂²⋊D₄₇	—	8	2	no	—
123	C₂²⋊D₄₇	—	8	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	8	2	no	—
126	D₁₈₈	—	4	2	no	—
127	D₁₈₈	—	4	2	no	—
128	D₁₈₈	—	8	2	no	—
129	C₈×C₄₇	—	2	1	no	—
130	C₈×C₄₇	—	2	1	no	—
131	C₈×C₄₇	—	2	1	no	—
132	C₈×C₄₇	—	2	1	no	—
133	C₂×#1347	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	2	4	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	#327	1	2	yes	—
141	C₂²×C₄₇	#357	1	2	yes	—
142	C₂²×C₄₇	#357	1	2	yes	—
143	C₂²×C₄₇	#360	1	2	yes	—
144	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
147	C₂×D₄₇	—	8	2	no	—
148	C₂×D₄₇	—	8	2	no	—
149	C₂×D₄₇	—	8	2	no	—
150	C₂×D₄₇	—	16	4	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	8	4	no	—
154	C₄×C₄₇	#327	1	2	yes	—
155	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
157	Dic₄₇	—	8	2	no	—
158	C₂×#466	—	47	1	no	—
159	C₂×C₂²⋊C₁₆	—	47	1	no	—
160	C₂×#340	D₄₇	1	1	yes	characteristic, complemented
161	#1347	—	47	4	no	—
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	—	2	1	no	—
165	C₂×C₄₇	#1347	1	4	yes	complemented
166	C₂×C₄₇	C₂×#327	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#357	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#360	1	1	yes	characteristic
169	D₄₇	—	16	2	no	—
170	C₂²×D₈	—	47	1	no	—
171	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
172	C₂×#121	—	94	1	no	—
173	C₂²×C₄⋊C₄	—	2	1	no	—
174	C₂²×C₁₆	—	94	1	no	—
175	#466	—	47	2	no	—
176	#466	—	47	2	no	—
177	#466	—	94	1	no	—
178	#466	—	94	1	no	—
179	C₂×#107	—	47	1	no	—
180	C₂×#102	—	94	1	no	—
181	C₂²⋊C₁₆	—	47	2	no	—
182	C₂²⋊C₁₆	—	94	1	no	—
183	#340	—	2	1	no	—
184	#340	C₂×D₄₇	1	2	yes	complemented
185	C₄₇	C₂×#1347	1	1	yes	characteristic, complemented, Sylow
186	C₂²×D₄	—	94	1	no	—
187	C₂²×D₄	—	188	1	no	—
188	C₂³×C₄	—	4	1	no	—
189	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×D₈	—	47	2	no	—
191	C₂×D₈	—	94	1	no	—
192	C₂×D₈	—	188	2	no	—
193	C₂²×C₈	—	4	2	no	—
194	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
197	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	#121	—	94	2	no	—
201	#121	—	94	2	no	—
202	#121	—	188	1	no	—
203	#121	—	188	1	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	—	4	1	no	—
208	C₂×C₄⋊C₄	—	94	1	no	—
209	C₂×C₂²⋊C₄	—	188	1	no	—
210	C₂×C₁₆	—	94	2	no	—
211	C₂×C₁₆	—	188	2	no	—
212	#107	—	47	2	no	—
213	#107	—	94	1	no	—
214	#102	—	94	2	no	—
215	#102	—	94	2	no	—
216	C₂⁴	—	188	1	no	—
217	C₂⁴	#10625334	1	1	yes	characteristic
218	C₂×D₄	—	94	2	no	—
219	C₂×D₄	—	376	2	no	—
220	C₂×D₄	—	376	2	no	—
221	C₂×D₄	—	188	2	no	—
222	C₂×D₄	—	188	2	no	—
223	C₂×D₄	—	188	2	no	—
224	C₂×D₄	—	188	1	no	—
225	C₂×D₄	—	188	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	8	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	—	188	1	no	—
233	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	#10625334	1	1	yes	characteristic
236	C₂²×C₄	#10625336	1	1	yes	characteristic
237	C₂²×C₄	#10625336	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	D₈	—	188	2	no	—
240	D₈	—	188	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
248	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
250	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	188	1	no	—
256	C₂²⋊C₄	—	188	2	no	—
257	C₂²⋊C₄	—	188	2	no	—
258	C₁₆	—	188	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	376	2	no	—
262	C₂³	—	376	2	no	—
263	C₂³	—	376	2	no	—
264	C₂³	—	188	2	no	—
265	C₂³	#13093944	1	1	yes	characteristic
266	C₂³	#13093944	1	1	yes	characteristic
267	C₂³	#13093965	1	2	yes	—
268	C₂³	#13093967	1	2	yes	—
269	C₂³	C₂×#10625334	1	1	yes	characteristic
270	D₄	—	376	2	no	—
271	D₄	—	376	2	no	—
272	D₄	—	376	4	no	—
273	D₄	—	376	2	no	—
274	D₄	—	188	2	no	—
275	D₄	—	188	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	376	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	#13093938	1	2	yes	—
290	C₂×C₄	#13093944	1	1	yes	characteristic
291	C₂×C₄	#13093944	1	1	yes	characteristic
292	C₂×C₄	#13093952	1	2	yes	—
293	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	376	2	no	—
306	C₂²	—	376	2	no	—
307	C₂²	—	376	2	no	—
308	C₂²	—	752	4	no	—
309	C₂²	#15688856	1	2	yes	—
310	C₂²	#15688923	1	2	yes	—
311	C₂²	#15688923	1	2	yes	—
312	C₂²	#15688932	1	2	yes	—
313	C₂²	C₂×#13093944	1	1	yes	characteristic
314	C₂²	C₂×#13093965	1	1	yes	characteristic
315	C₂²	C₂×#13093967	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	4	no	—
319	C₄	—	376	2	no	—
320	C₄	#15688856	1	2	yes	—
321	C₄	C₂×#13093938	1	1	yes	characteristic
322	C₄	C₂×#13093952	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	752	2	no	—
327	C₂	#20241396	1	4	yes	direct product factor
328	C₂	C₂×#15688856	1	1	yes	characteristic
329	C₂	C₂×#15688923	1	1	yes	characteristic
330	C₂	C₂×#15688932	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093944
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle	C₂³×C₄₇	#102
prime radical	C₂×#15689359	C₂
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	C₂×#1347
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
3. #20241396	C₂	direct (N × H)
4. C₂×C₄₇×#340	C₂	semidirect
5. C₂²×D₃₇₆	C₄	semidirect
10. #15689359	C₄	semidirect
12. C₂×#13093958	C₄	semidirect
15. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
19. C₄₇×#340	C₂²	semidirect
24. C₂×D₃₇₆	C₂×C₄	semidirect
34. #13093958	C₂×C₄	semidirect
160. C₂×#340	D₄₇	semidirect
165. C₂×C₄₇	#1347	semidirect
184. #340	C₂×D₄₇	semidirect
185. C₄₇	C₂×#1347	semidirect
327. C₂	#20241396	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×#15689359	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241396	2
C₂×C₄₇×#340	2
C₂×#1347	47

Subgroups — C2×#20241396 #26126496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241396 #26126496