Subgroups — C₂×#20252325 #27080647

Order 12032.

1894 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689680	C₂	1	1	yes	maximal, characteristic
2	#20252325	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689252	C₂	1	8	yes	maximal, complemented
4	C₂×#15689151	C₂	1	2	yes	maximal
5	C₂×#15689081	C₂	1	2	yes	maximal
6	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15689680	C₄	1	8	yes	—
11	#15689680	C₂²	1	8	yes	—
12	C₂×#13094024	C₄	1	2	yes	—
13	C₂×#13094016	C₄	1	2	yes	—
14	C₂×#13094003	C₄	1	4	yes	—
15	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
16	C₂×#13093983	C₂²	1	2	yes	—
17	#15689252	C₂²	1	64	yes	complemented
18	#15689151	C₂²	1	16	yes	—
19	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
20	#15689081	C₂²	1	16	yes	—
21	C₂×#13093935	C₂²	1	4	yes	—
22	C₂×#13093935	C₂²	1	8	yes	—
23	C₂×#13093934	C₂²	1	4	yes	—
24	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
25	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#423	C₂²	1	8	yes	—
28	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
29	C₂²×Dic₉₄	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
36	#13094024	C₂×C₄	1	16	yes	—
37	#13094016	C₂×C₄	1	16	yes	—
38	#13094003	C₂×C₄	1	32	yes	—
39	C₄×Dic₉₄	C₂×C₄	1	16	yes	—
40	C₄×Dic₉₄	C₂³	1	16	yes	—
41	#13093983	C₂×C₄	1	8	yes	—
42	#13093983	C₂³	1	8	yes	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
45	C₂×Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
46	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
48	Dic₄₇⋊C₈	C₂³	1	32	yes	—
49	#13093935	C₂³	1	16	yes	—
50	#13093934	C₂³	1	16	yes	—
51	C₈×Dic₄₇	C₂³	1	16	yes	—
52	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₄₇	—	2	8	no	—
55	C₂×M₁₆×C₄₇	C₂³	1	8	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
62	C₄⋊C₈×C₄₇	C₂³	1	16	yes	—
63	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₄₇	#41	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
75	C₂×Dic₉₄	—	4	8	no	—
76	C₂×Dic₉₄	C₂²×C₄	1	16	yes	—
77	M₁₆×C₄₇	—	2	16	no	—
78	C₂×C₈×C₄₇	—	2	8	no	—
79	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
83	C₄²×C₄₇	C₂⁴	1	4	yes	—
84	C₂²⋊Dic₄₇	C₂²×C₄	1	16	yes	—
85	C₄⋊Dic₄₇	C₂²×C₄	1	16	yes	—
86	C₄⋊Dic₄₇	C₂²×C₄	1	16	yes	—
87	Dic₄₇⋊C₄	C₂²×C₄	1	32	yes	—
88	C₄×Dic₄₇	C₂²×C₄	1	8	yes	—
89	C₄×Dic₄₇	C₂⁴	1	8	yes	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	8	no	—
92	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
93	C₂×C₄₇⋊C₈	C₂×Q₈	1	16	yes	complemented
94	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
95	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
98	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
99	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	4	8	no	—
103	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
113	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
114	C₂×Dic₄₇	—	2	16	no	—
115	C₂×Dic₄₇	C₂³×C₄	1	16	yes	—
116	Dic₉₄	—	4	16	no	—
117	C₈×C₄₇	—	2	8	no	—
118	C₄₇⋊C₈	—	2	4	no	—
119	C₄₇⋊C₈	C₂²×Q₈	1	8	yes	—
120	C₂×#2961	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	8	no	—
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
124	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
125	C₂²×C₄₇	#568	1	4	yes	—
126	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
130	C₄×C₄₇	C₂×C₄×Q₈	1	4	yes	—
131	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
132	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
133	Dic₄₇	—	2	16	no	—
134	C₂×#533	—	47	1	no	—
135	#2961	—	47	16	no	—
136	C₂×#446	—	47	8	no	—
137	C₂×#424	—	47	2	no	—
138	C₂×#423	D₄₇	1	1	yes	characteristic
139	C₂×#405	—	47	2	no	—
140	C₂²×#97	—	47	1	no	—
141	C₂×C₄₇	—	2	4	no	—
142	C₂×C₄₇	#2961	1	4	yes	complemented
143	C₂×C₄₇	C₂²×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#568	1	2	yes	—
145	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
146	C₂³×C₈	—	47	1	no	—
147	C₂³×C₈	—	47	1	no	—
148	#533	—	47	8	no	—
149	#533	—	47	8	no	—
150	C₂×C₄⋊Q₈	—	47	2	no	—
151	C₂×#125	—	47	2	no	—
152	C₂×C₂²⋊Q₈	—	47	4	no	—
153	C₂×C₄×Q₈	—	47	4	no	—
154	C₂×#117	—	47	2	no	—
155	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
156	#446	—	47	64	no	—
157	#424	—	47	16	no	—
158	#423	C₂×D₄₇	1	8	yes	—
159	C₂×C₄⋊C₈	—	47	8	no	—
160	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
161	#405	—	47	16	no	—
162	C₂×#97	—	47	4	no	—
163	C₂×#97	—	47	4	no	—
164	C₂×#97	—	47	8	no	—
165	C₂×C₄×C₈	—	47	4	no	—
166	C₄₇	C₂×#2961	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	—	47	4	no	—
168	C₂³×C₄	C₄×D₄₇	1	2	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	C₂²×D₄₇	1	8	yes	—
172	C₂²×C₈	—	47	8	no	—
173	C₂²×C₈	—	47	4	no	—
174	C₂²×C₈	—	47	2	no	—
175	C₂²×C₈	—	47	4	no	—
176	C₂²×C₈	—	47	2	no	—
177	C₂²×C₈	—	94	4	no	—
178	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
179	C₄⋊Q₈	—	47	16	no	—
180	#125	—	47	16	no	—
181	C₂²⋊Q₈	—	47	32	no	—
182	C₄×Q₈	—	47	16	no	—
183	C₄×Q₈	—	47	16	no	—
184	#117	—	47	8	no	—
185	#117	—	47	8	no	—
186	C₂×C₄⋊C₄	—	47	4	no	—
187	C₂×C₄⋊C₄	—	47	8	no	—
188	C₂×C₄⋊C₄	—	47	4	no	—
189	C₂×C₂²⋊C₄	—	47	4	no	—
190	C₂×C₄²	—	47	4	no	—
191	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
192	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₄₇	1	4	yes	—
194	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
195	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
196	C₄⋊C₈	—	47	32	no	—
197	C₄⋊C₈	C₂²×D₄₇	1	16	yes	—
198	#97	—	47	16	no	—
199	#97	—	47	16	no	—
200	C₄×C₈	—	47	16	no	—
201	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₂×Q₈	—	47	16	no	—
203	C₂×Q₈	—	188	8	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	47	8	no	—
206	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
209	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
211	C₂²×C₄	#10625352	1	4	yes	—
212	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
213	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
214	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	M₁₆	—	2	16	no	—
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	47	8	no	—
218	C₂×C₈	—	47	16	no	—
219	C₂×C₈	—	47	4	no	—
220	C₂×C₈	—	47	4	no	—
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	94	8	no	—
223	C₂×C₈	C₂³×D₄₇	1	8	yes	—
224	C₄⋊C₄	—	47	32	no	—
225	C₄⋊C₄	—	47	16	no	—
226	C₄⋊C₄	—	47	16	no	—
227	C₂²⋊C₄	—	47	16	no	—
228	C₄²	—	47	8	no	—
229	C₄²	—	47	8	no	—
230	C₄²	C₂×C₄×D₄₇	1	4	yes	—
231	C₄²	C₂×C₄×D₄₇	1	4	yes	—
232	C₄²	C₂×C₄×D₄₇	1	4	yes	—
233	C₄²	C₂³×D₄₇	1	4	yes	—
234	C₂³	—	2	4	no	—
235	C₂³	#13093989	1	4	yes	—
236	C₂³	C₂²×C₄×D₄₇	1	2	yes	—
237	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
238	Q₈	—	188	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	47	16	no	—
243	C₂×C₄	—	94	16	no	—
244	C₂×C₄	#13093989	1	8	yes	—
245	C₂×C₄	#13093989	1	4	yes	—
246	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
247	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
248	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
250	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
251	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
252	C₂×C₄	C₂×#10625352	1	2	yes	—
253	C₂×C₄	C₂×#10625352	1	4	yes	—
254	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	47	8	no	—
257	C₈	—	94	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×#13093989	1	4	yes	—
261	C₂²	C₂×#13093989	1	2	yes	—
262	C₂²	#15689889	1	4	yes	—
263	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	94	16	no	—
267	C₄	C₂×#13093989	1	2	yes	—
268	C₄	C₂×#13093989	1	4	yes	—
269	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
270	C₄	C₂²×#10625352	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#20252325	1	4	yes	direct product factor
273	C₂	C₂²×#13093989	1	1	yes	characteristic
274	C₂	C₂×#15689889	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	C₂×#2961
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
2. #20252325	C₂	direct (N × H)
3. C₂×#15689252	C₂	semidirect
17. #15689252	C₂²	semidirect
31. C₂²×C₄₇⋊C₈	Q₈	semidirect
93. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
142. C₂×C₄₇	#2961	semidirect
166. C₄₇	C₂×#2961	semidirect
272. C₂	#20252325	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×#15689680	2
#20252325	2
C₂×#15689252	2
C₂×#15689151	2
C₂×#15689081	2
C₂²×#13093935	2
C₂×C₄₇×#423	2
C₂×#2961	47

Subgroups — C2×#20252325 #27080647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20252325 #27080647