Subgroups — C₂×#20251765 #26422290

Order 12032.

2814 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	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	#20251765	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689030	C₂	1	4	yes	maximal
4	C₂×#15689028	C₂	1	4	yes	maximal
5	C₂×#15688988	C₂	1	4	yes	maximal
6	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₉₄	C₂²	1	2	yes	—
9	C₂×#13094024	—	2	8	no	—
10	C₂×#13094024	C₂²	1	8	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
14	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
16	#15689030	C₂²	1	32	yes	—
17	#15689028	C₂²	1	32	yes	—
18	#15688988	C₂²	1	32	yes	—
19	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
20	C₄²×Dic₄₇	C₂²	1	8	yes	—
21	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈×C₄₇	—	2	8	no	—
23	C₂×C₄⋊Q₈×C₄₇	C₂²	1	8	yes	—
24	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×Dic₉₄	—	2	8	no	—
28	C₂²×Dic₉₄	—	2	16	no	—
29	C₂²×Dic₉₄	C₂³	1	8	yes	—
30	C₂³×Dic₄₇	C₂³	1	4	yes	—
31	C₂³×Dic₄₇	C₂³	1	4	yes	—
32	#13094024	—	2	32	no	—
33	C₂×C₄⋊Dic₄₇	—	2	8	no	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
37	C₂×C₄×Dic₄₇	—	2	8	no	—
38	C₂×C₄×Dic₄₇	—	2	8	no	—
39	C₂×C₄×Dic₄₇	Q₈	1	16	yes	complemented
40	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
43	C₄₇⋊#95	C₂³	1	64	yes	—
44	C₂²×Q₈×C₄₇	—	2	8	no	—
45	C₂²×Q₈×C₄₇	—	2	16	no	—
46	C₂²×Q₈×C₄₇	C₂³	1	8	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	—	2	32	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
54	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
56	C₂×C₄²×C₄₇	D₄	1	8	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
58	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂×Q₈×C₄₇	—	2	16	no	—
60	C₂×Q₈×C₄₇	—	2	32	no	—
61	C₂×Q₈×C₄₇	—	4	32	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	16	no	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
68	C₂²×C₄×C₄₇	#41	1	8	yes	—
69	C₂²×C₄×C₄₇	#41	1	8	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
74	C₂²×Dic₄₇	—	2	8	no	—
75	C₂²×Dic₄₇	—	2	8	no	—
76	C₂²×Dic₄₇	—	2	8	no	—
77	C₂²×Dic₄₇	—	2	16	no	—
78	C₂²×Dic₄₇	C₂×Q₈	1	16	yes	—
79	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
80	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
81	C₂×Dic₉₄	—	2	16	no	—
82	C₂×Dic₉₄	—	2	32	no	—
83	C₂×Dic₉₄	—	4	32	no	—
84	C₄⋊C₄×C₄₇	—	2	16	no	—
85	C₄⋊C₄×C₄₇	—	2	16	no	—
86	C₄²×C₄₇	C₂×D₄	1	16	yes	—
87	C₄⋊Dic₄₇	—	2	16	no	—
88	C₄⋊Dic₄₇	—	2	16	no	—
89	C₄×Dic₄₇	—	2	16	no	—
90	C₄×Dic₄₇	—	2	16	no	—
91	C₄×Dic₄₇	C₂×Q₈	1	32	yes	complemented
92	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
93	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
94	C₂³×C₄₇	C₂×#41	1	4	yes	—
95	C₂³×C₄₇	C₂×#41	1	4	yes	—
96	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
97	Q₈×C₄₇	—	4	32	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	16	no	—
100	C₂×C₄×C₄₇	—	4	16	no	—
101	C₂×C₄×C₄₇	#124	1	16	yes	—
102	C₂×C₄×C₄₇	#124	1	16	yes	—
103	C₂×C₄×C₄₇	C₄⋊Q₈	1	16	yes	—
104	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
105	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
108	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
109	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
110	C₂×Dic₄₇	—	2	32	no	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	2	16	no	—
115	C₂×Dic₄₇	—	4	16	no	—
116	C₂×Dic₄₇	C₂²×Q₈	1	16	yes	—
117	Dic₉₄	—	4	32	no	—
118	C₂×#2862	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	C₂×#124	1	4	yes	—
120	C₂²×C₄₇	C₂×#124	1	4	yes	—
121	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
122	C₂²×C₄₇	#533	1	16	yes	—
123	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
125	C₂²×C₄₇	C₂²×#41	1	2	yes	—
126	C₂²×C₄₇	C₂²×#41	1	2	yes	—
127	C₄×C₄₇	—	4	8	no	—
128	C₄×C₄₇	C₂×#124	1	8	yes	—
129	C₄×C₄₇	C₂×#124	1	8	yes	—
130	C₄×C₄₇	C₂×C₄⋊Q₈	1	8	yes	—
131	Dic₄₇	—	2	16	no	—
132	Dic₄₇	—	4	8	no	—
133	C₂²×C₄⋊Q₈	—	47	1	no	—
134	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
135	#2862	—	47	16	no	—
136	C₂×#391	—	47	4	no	—
137	C₂×#391	—	47	4	no	—
138	C₂×#382	—	47	4	no	—
139	C₂×C₄³	—	47	1	no	—
140	C₂×C₄₇	#2862	1	8	yes	complemented
141	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
142	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
143	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#533	1	4	yes	—
145	C₂³×Q₈	—	47	2	no	—
146	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
147	C₂×C₄⋊Q₈	—	2	8	no	—
148	C₂×C₄⋊Q₈	—	47	8	no	—
149	C₂×C₄⋊Q₈	—	94	8	no	—
150	C₂×C₄⋊Q₈	C₂×D₄₇	1	8	yes	—
151	C₂²×C₄⋊C₄	—	47	2	no	—
152	C₂²×C₄⋊C₄	—	47	2	no	—
153	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
154	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
155	C₂²×C₄²	—	47	2	no	—
156	C₂²×C₄²	—	47	2	no	—
157	C₂²×C₄²	—	47	2	no	—
158	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
159	#391	—	47	32	no	—
160	#391	—	47	32	no	—
161	#382	—	47	32	no	—
162	C₂×#95	—	47	16	no	—
163	C₄³	—	47	8	no	—
164	C₄₇	C₂×#2862	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	—	2	16	no	—
166	C₂²×Q₈	—	2	8	no	—
167	C₂²×Q₈	—	47	8	no	—
168	C₂²×Q₈	—	94	16	no	—
169	C₂²×Q₈	—	94	8	no	—
170	C₂²×Q₈	C₂²×D₄₇	1	8	yes	—
171	C₂³×C₄	—	47	4	no	—
172	C₂³×C₄	—	47	4	no	—
173	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₄⋊Q₈	—	2	32	no	—
178	C₄⋊Q₈	—	94	32	no	—
179	C₂×C₄⋊C₄	—	2	8	no	—
180	C₂×C₄⋊C₄	—	2	8	no	—
181	C₂×C₄⋊C₄	—	47	8	no	—
182	C₂×C₄⋊C₄	—	47	8	no	—
183	C₂×C₄⋊C₄	—	94	8	no	—
184	C₂×C₄⋊C₄	—	94	8	no	—
185	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
186	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
187	C₂×C₄²	—	47	16	no	—
188	C₂×C₄²	—	47	8	no	—
189	C₂×C₄²	—	47	8	no	—
190	C₂×C₄²	—	47	8	no	—
191	C₂×C₄²	—	94	8	no	—
192	C₂×C₄²	—	94	8	no	—
193	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
194	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
195	#95	—	47	64	no	—
196	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	2	32	no	—
199	C₂×Q₈	—	4	32	no	—
200	C₂×Q₈	—	94	16	no	—
201	C₂×Q₈	—	94	32	no	—
202	C₂×Q₈	—	188	32	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	47	16	no	—
206	C₂²×C₄	—	47	8	no	—
207	C₂²×C₄	—	47	8	no	—
208	C₂²×C₄	—	94	8	no	—
209	C₂²×C₄	—	94	8	no	—
210	C₂²×C₄	—	94	8	no	—
211	C₂²×C₄	—	94	16	no	—
212	C₂²×C₄	#10625352	1	8	yes	—
213	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
214	C₂²×C₄	#10625354	1	8	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
218	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
219	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
220	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
221	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
222	C₄⋊C₄	—	2	16	no	—
223	C₄⋊C₄	—	2	16	no	—
224	C₄⋊C₄	—	94	16	no	—
225	C₄⋊C₄	—	94	16	no	—
226	C₄²	—	47	32	no	—
227	C₄²	—	94	16	no	—
228	C₄²	—	94	16	no	—
229	C₄²	C₂×C₂²⋊D₄₇	1	16	yes	—
230	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
231	C₂³	C₂×Q₈×D₄₇	1	4	yes	—
232	C₂³	C₂×#10625352	1	4	yes	—
233	C₂³	C₂×#10625354	1	4	yes	—
234	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
235	Q₈	—	4	32	no	—
236	Q₈	—	188	32	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	—	4	16	no	—
240	C₂×C₄	—	47	16	no	—
241	C₂×C₄	—	94	32	no	—
242	C₂×C₄	—	94	8	no	—
243	C₂×C₄	—	94	8	no	—
244	C₂×C₄	—	94	8	no	—
245	C₂×C₄	—	94	16	no	—
246	C₂×C₄	—	188	16	no	—
247	C₂×C₄	#13094011	1	16	yes	—
248	C₂×C₄	#13094013	1	16	yes	—
249	C₂×C₄	#13094022	1	16	yes	—
250	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
252	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
253	C₂×C₄	C₂×Q₈×D₄₇	1	8	yes	—
254	C₂×C₄	C₂×#10625352	1	8	yes	—
255	C₂×C₄	C₂×#10625354	1	8	yes	—
256	C₂²	C₂×#13094011	1	4	yes	—
257	C₂²	C₂×#13094013	1	4	yes	—
258	C₂²	C₂×#13094022	1	4	yes	—
259	C₂²	#15689678	1	16	yes	—
260	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂²	C₂²×Q₈×D₄₇	1	2	yes	—
262	C₂²	C₂²×#10625352	1	2	yes	—
263	C₂²	C₂²×#10625354	1	2	yes	—
264	C₄	—	4	8	no	—
265	C₄	—	94	16	no	—
266	C₄	—	188	8	no	—
267	C₄	C₂×#13094011	1	8	yes	—
268	C₄	C₂×#13094013	1	8	yes	—
269	C₄	C₂×#13094022	1	8	yes	—
270	C₂	#20251765	1	8	yes	direct product factor
271	C₂	C₂²×#13094011	1	1	yes	characteristic
272	C₂	C₂²×#13094013	1	1	yes	characteristic
273	C₂	C₂²×#13094022	1	1	yes	characteristic
274	C₂	C₂×#15689678	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#2862
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20251765	C₂	direct (N × H)
39. C₂×C₄×Dic₄₇	Q₈	semidirect
91. C₄×Dic₄₇	C₂×Q₈	semidirect
140. C₂×C₄₇	#2862	semidirect
164. C₄₇	C₂×#2862	semidirect
270. C₂	#20251765	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×#13094024	2
#20251765	2
C₂×#15689030	2
C₂×#15689028	2
C₂×#15688988	2
C₂×C₄²×Dic₄₇	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#2862	47

Subgroups — C2×#20251765 #26422290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251765 #26422290