Subgroups — C₂×#20251978 #27079939

Order 12032.

2878 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093983	C₂	1	2	yes	maximal
2	#20251978	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689165	C₂	1	8	yes	maximal, complemented
4	C₂×#15689161	C₂	1	4	yes	maximal, complemented
5	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	—	2	2	no	—
7	C₂×#13093983	—	2	2	no	—
8	C₂×#13093983	—	2	8	no	—
9	C₂×#13093983	C₂²	1	4	yes	—
10	C₂×#13093983	C₂²	1	4	yes	—
11	C₂×#13093983	C₂²	1	16	yes	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
13	C₂²×C₂²⋊Dic₄₇	—	2	2	no	—
14	C₂²×C₄×Dic₄₇	—	2	2	no	—
15	#15689165	C₂²	1	64	yes	complemented
16	#15689161	C₂²	1	32	yes	complemented
17	C₂×#13093958	C₂²	1	8	yes	complemented
18	C₂×#13093956	C₂²	1	8	yes	complemented
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₄₇	C₄	1	16	yes	complemented
21	C₂²×M₁₆×C₄₇	C₂²	1	4	yes	—
22	C₂²×M₁₆×C₄₇	C₂²	1	8	yes	—
23	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
24	C₂³×Dic₄₇	—	2	4	no	—
25	#13093983	—	2	16	no	—
26	#13093983	—	2	8	no	—
27	#13093983	—	2	8	no	—
28	#13093983	—	2	16	no	—
29	#13093983	C₂³	1	16	yes	—
30	#13093983	C₂³	1	16	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
36	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
37	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
38	C₂×C₄×Dic₄₇	—	2	4	no	—
39	C₂×C₄×Dic₄₇	—	2	16	no	—
40	C₂×C₄×Dic₄₇	—	2	4	no	—
41	#13093958	C₂³	1	32	yes	complemented
42	#13093956	C₂³	1	32	yes	complemented
43	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
45	C₂³×C₄×C₄₇	D₄	1	2	yes	—
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	D₄	1	4	yes	—
48	C₂³×C₄×C₄₇	Q₈	1	8	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×M₁₆×C₄₇	C₂×C₄	1	32	yes	complemented
53	C₂×M₁₆×C₄₇	C₂×C₄	1	64	yes	complemented
54	C₂×M₁₆×C₄₇	C₂³	1	16	yes	—
55	C₂²×C₈×C₄₇	C₂×C₄	1	16	yes	complemented
56	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	8	yes	—
58	C₂⁴×C₄₇	—	4	4	no	—
59	C₂⁴×C₄₇	C₄⋊C₄	1	8	yes	—
60	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
62	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	16	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	32	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
81	C₂²×Dic₄₇	—	2	16	no	—
82	C₂²×Dic₄₇	—	2	8	no	—
83	C₂²×Dic₄₇	—	4	16	no	—
84	M₁₆×C₄₇	C₂²×C₄	1	64	yes	complemented
85	C₂×C₈×C₄₇	C₂²×C₄	1	16	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	32	yes	complemented
87	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
88	C₂²⋊Dic₄₇	—	2	16	no	—
89	C₂²⋊Dic₄₇	—	2	16	no	—
90	C₄⋊Dic₄₇	—	2	16	no	—
91	C₄⋊Dic₄₇	—	2	8	no	—
92	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
93	C₄×Dic₄₇	—	2	16	no	—
94	C₄×Dic₄₇	—	2	16	no	—
95	C₂³×C₄₇	—	2	8	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	4	16	no	—
98	C₂³×C₄₇	—	4	8	no	—
99	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
101	C₂³×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
102	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
103	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
104	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	32	yes	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
119	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	16	no	—
122	C₂×Dic₄₇	—	4	32	no	—
123	C₂×Dic₄₇	—	4	16	no	—
124	C₈×C₄₇	C₂³×C₄	1	16	yes	—
125	C₂×#2911	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	16	no	—
127	C₂²×C₄₇	—	2	8	no	—
128	C₂²×C₄₇	—	2	8	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	4	16	no	—
131	C₂²×C₄₇	C₂²×C₄⋊C₄	1	4	yes	—
132	C₂²×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
133	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
134	C₂²×C₄₇	#575	1	4	yes	—
135	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
136	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
137	C₄×C₄₇	C₂²×C₄⋊C₄	1	4	yes	—
138	C₄×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
139	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
140	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
141	Dic₄₇	—	4	16	no	—
142	C₂³×M₁₆	D₄₇	1	1	yes	characteristic
143	C₂²×#117	—	47	2	no	—
144	#2911	—	47	16	no	—
145	C₂×#428	—	47	8	no	—
146	C₂×#427	—	47	4	no	—
147	C₂×C₄₇	—	2	8	no	—
148	C₂×C₄₇	—	2	4	no	—
149	C₂×C₄₇	#2911	1	4	yes	complemented
150	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#575	1	2	yes	—
152	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
153	C₂²×M₁₆	Dic₄₇	1	16	yes	complemented
154	C₂²×M₁₆	C₂×D₄₇	1	4	yes	—
155	C₂²×M₁₆	C₂×D₄₇	1	8	yes	—
156	C₂³×C₈	C₂×D₄₇	1	2	yes	—
157	C₂×#117	—	47	4	no	—
158	C₂×#117	—	47	4	no	—
159	C₂×#117	—	47	16	no	—
160	C₂×#117	—	94	2	no	—
161	C₂×#117	—	94	8	no	—
162	C₂×#117	—	94	2	no	—
163	C₂²×C₄⋊C₄	—	47	4	no	—
164	C₂²×C₂²⋊C₄	—	94	2	no	—
165	C₂²×C₄²	—	94	2	no	—
166	#428	—	47	64	no	—
167	#427	—	47	32	no	—
168	C₂×#107	—	47	8	no	—
169	C₂×#106	—	47	8	no	—
170	C₄₇	C₂×#2911	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	D₁₈₈	1	1	yes	characteristic
172	C₂³×C₄	—	94	4	no	—
173	C₂³×C₄	Dic₉₄	1	8	yes	—
174	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
175	C₂³×C₄	D₁₈₈	1	4	yes	—
176	C₂³×C₄	D₁₈₈	1	2	yes	—
177	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
178	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
179	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
180	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
181	C₂×M₁₆	C₂×Dic₄₇	1	32	yes	complemented
182	C₂×M₁₆	C₂×Dic₄₇	1	64	yes	complemented
183	C₂×M₁₆	C₂²×D₄₇	1	16	yes	—
184	C₂²×C₈	C₂×Dic₄₇	1	16	yes	complemented
185	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
186	C₂²×C₈	C₂²×D₄₇	1	8	yes	—
187	#117	—	47	16	no	—
188	#117	—	47	16	no	—
189	#117	—	94	16	no	—
190	#117	—	94	8	no	—
191	#117	—	94	8	no	—
192	#117	—	94	16	no	—
193	C₂×C₄⋊C₄	—	47	8	no	—
194	C₂×C₄⋊C₄	—	47	16	no	—
195	C₂×C₄⋊C₄	—	94	4	no	—
196	C₂×C₄⋊C₄	—	94	8	no	—
197	C₂×C₂²⋊C₄	—	94	4	no	—
198	C₂×C₂²⋊C₄	—	94	4	no	—
199	C₂×C₂²⋊C₄	—	94	16	no	—
200	C₂×C₄²	—	94	4	no	—
201	C₂×C₄²	—	94	16	no	—
202	C₂×C₄²	—	94	4	no	—
203	#107	—	47	32	no	—
204	#106	—	47	32	no	—
205	C₂⁴	—	4	4	no	—
206	C₂⁴	C₄⋊Dic₄₇	1	8	yes	—
207	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
208	C₂⁴	C₂×D₁₈₈	1	2	yes	—
209	C₂⁴	C₂×D₁₈₈	1	4	yes	—
210	C₂²×C₄	—	94	16	no	—
211	C₂²×C₄	—	94	8	no	—
212	C₂²×C₄	—	188	16	no	—
213	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
214	C₂²×C₄	C₄⋊Dic₄₇	1	32	yes	—
215	C₂²×C₄	C₄⋊Dic₄₇	1	16	yes	—
216	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
217	C₂²×C₄	C₂×Dic₉₄	1	16	yes	—
218	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
219	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
220	C₂²×C₄	C₂×D₁₈₈	1	8	yes	—
221	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
222	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
223	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
224	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
230	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
231	M₁₆	C₂²×Dic₄₇	1	64	yes	complemented
232	C₂×C₈	C₂²×Dic₄₇	1	16	yes	—
233	C₂×C₈	C₂²×Dic₄₇	1	32	yes	complemented
234	C₂×C₈	C₂³×D₄₇	1	8	yes	—
235	C₄⋊C₄	—	47	16	no	—
236	C₄⋊C₄	—	94	16	no	—
237	C₄⋊C₄	—	94	8	no	—
238	C₂²⋊C₄	—	94	16	no	—
239	C₂²⋊C₄	—	94	16	no	—
240	C₄²	—	94	16	no	—
241	C₄²	—	94	16	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	4	16	no	—
245	C₂³	—	4	8	no	—
246	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
247	C₂³	C₂×C₄⋊Dic₄₇	1	8	yes	—
248	C₂³	C₂×C₄⋊Dic₄₇	1	16	yes	—
249	C₂³	C₂²×D₁₈₈	1	2	yes	—
250	C₂³	C₂²×D₁₈₈	1	4	yes	—
251	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
252	C₂×C₄	—	94	16	no	—
253	C₂×C₄	—	188	32	no	—
254	C₂×C₄	—	188	16	no	—
255	C₂×C₄	C₂×C₄⋊Dic₄₇	1	32	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₄₇	1	16	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₄₇	1	16	yes	—
259	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
261	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
262	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
263	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
264	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
265	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
266	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
268	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
269	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
270	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
271	C₈	C₂³×Dic₄₇	1	16	yes	—
272	C₂²	—	2	16	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	8	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	16	no	—
277	C₂²	C₂²×C₄⋊Dic₄₇	1	4	yes	—
278	C₂²	C₂²×C₄⋊Dic₄₇	1	8	yes	—
279	C₂²	C₂²×C₄⋊Dic₄₇	1	2	yes	—
280	C₂²	#15689923	1	4	yes	—
281	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
282	C₄	—	188	16	no	—
283	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
284	C₄	C₂²×C₄⋊Dic₄₇	1	8	yes	—
285	C₄	C₂²×C₄⋊Dic₄₇	1	4	yes	—
286	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
287	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
288	C₂	—	2	8	no	—
289	C₂	—	2	4	no	—
290	C₂	#20251978	1	4	yes	direct product factor
291	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
292	C₂	C₂×#15689923	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	C₂×#2911
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20251978	C₂	direct (N × H)
3. C₂×#15689165	C₂	semidirect
4. C₂×#15689161	C₂	semidirect
15. #15689165	C₂²	semidirect
16. #15689161	C₂²	semidirect
17. C₂×#13093958	C₂²	semidirect
18. C₂×#13093956	C₂²	semidirect
20. C₂²×M₁₆×C₄₇	C₄	semidirect
41. #13093958	C₂³	semidirect
42. #13093956	C₂³	semidirect
52. C₂×M₁₆×C₄₇	C₂×C₄	semidirect
53. C₂×M₁₆×C₄₇	C₂×C₄	semidirect
55. C₂²×C₈×C₄₇	C₂×C₄	semidirect
84. M₁₆×C₄₇	C₂²×C₄	semidirect
86. C₂×C₈×C₄₇	C₂²×C₄	semidirect
149. C₂×C₄₇	#2911	semidirect
153. C₂²×M₁₆	Dic₄₇	semidirect
170. C₄₇	C₂×#2911	semidirect
181. C₂×M₁₆	C₂×Dic₄₇	semidirect
182. C₂×M₁₆	C₂×Dic₄₇	semidirect
184. C₂²×C₈	C₂×Dic₄₇	semidirect
231. M₁₆	C₂²×Dic₄₇	semidirect
233. C₂×C₈	C₂²×Dic₄₇	semidirect
290. C₂	#20251978	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×#13093983	2
#20251978	2
C₂×#15689165	2
C₂×#15689161	2
C₂³×M₁₆×C₄₇	2
C₂×#2911	47

Subgroups — C2×#20251978 #27079939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251978 #27079939