Subgroups — C₂×#20241269 #26106325

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093935	C₂	1	2	yes	maximal
2	#20241269	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	—	2	2	no	—
5	C₂×#13093935	—	2	2	no	—
6	C₂×#13093935	—	2	4	no	—
7	C₂×#13093935	C₄	1	8	yes	complemented
8	C₂×#13093935	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#382	—	2	2	no	—
12	C₄₇×#382	C₄	1	2	yes	—
13	C₄₇×#382	C₂²	1	2	yes	—
14	C₂×#95×C₄₇	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	16	no	—
16	C₂²×C₄₇⋊C₈	—	2	8	no	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	#13093935	—	2	8	no	—
19	#13093935	—	2	4	no	—
20	#13093935	C₂×C₄	1	8	yes	complemented
21	C₂³×C₄×C₄₇	—	2	2	no	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	D₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
26	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#95×C₄₇	—	2	4	no	—
34	#95×C₄₇	—	2	4	no	—
35	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	8	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	—	4	4	no	—
41	C₂²×C₄×C₄₇	—	4	8	no	—
42	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₄⋊C₄×C₄₇	—	4	4	no	—
51	C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₄²×C₄₇	C₄²	1	2	yes	—
53	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
54	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
55	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄₇⋊C₈	—	2	32	no	—
60	C₂×C₄₇⋊C₈	—	2	16	no	—
61	C₂×C₄₇⋊C₈	—	2	8	no	—
62	C₂³×C₄₇	#95	1	2	yes	—
63	C₂³×C₄₇	#100	1	2	yes	—
64	C₂³×C₄₇	#101	1	2	yes	—
65	C₂³×C₄₇	#104	1	8	yes	—
66	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	2	8	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	4	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	—	2	1	no	—
74	C₂×C₄×C₄₇	—	2	1	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	4	4	no	—
78	C₂×C₄×C₄₇	—	4	8	no	—
79	C₂×C₄×C₄₇	—	8	8	no	—
80	C₂×C₄×C₄₇	#95	1	4	yes	—
81	C₂×C₄×C₄₇	#95	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄₇⋊C₈	—	2	16	no	—
87	C₂×#1295	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	#329	1	8	yes	—
89	C₂²×C₄₇	#330	1	8	yes	—
90	C₂²×C₄₇	#339	1	8	yes	—
91	C₂²×C₄₇	#343	1	4	yes	—
92	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
94	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
95	C₂²×C₄₇	C₂×#104	1	4	yes	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	2	4	no	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	1	no	—
102	C₄×C₄₇	—	8	4	no	—
103	C₂²×#97	—	47	2	no	—
104	C₂×#382	D₄₇	1	1	yes	characteristic
105	#1295	—	47	4	no	—
106	C₂×C₄₇	#1295	1	8	yes	complemented
107	C₂×C₄₇	C₂×#329	1	2	yes	—
108	C₂×C₄₇	C₂×#330	1	2	yes	—
109	C₂×C₄₇	C₂×#339	1	2	yes	—
110	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
111	C₂³×C₈	—	94	2	no	—
112	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
113	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
114	C₂×#97	—	47	8	no	—
115	C₂×#97	—	47	4	no	—
116	C₂×#97	—	94	4	no	—
117	C₂×#97	—	94	2	no	—
118	#382	—	2	2	no	—
119	#382	Dic₄₇	1	2	yes	—
120	#382	C₂×D₄₇	1	2	yes	—
121	C₂×#95	—	2	2	no	—
122	C₄₇	C₂×#1295	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	—	2	2	no	—
124	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
125	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
126	C₂²×C₈	—	94	16	no	—
127	C₂²×C₈	—	94	8	no	—
128	C₂²×C₈	—	94	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	4	no	—
131	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
132	C₂×C₄²	Dic₉₄	1	2	yes	—
133	C₂×C₄²	C₄×D₄₇	1	4	yes	—
134	C₂×C₄²	D₁₈₈	1	2	yes	—
135	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
136	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
137	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
138	#97	—	47	8	no	—
139	#97	—	94	8	no	—
140	#97	—	94	4	no	—
141	#95	—	2	4	no	—
142	#95	—	2	4	no	—
143	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	—	4	8	no	—
150	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
151	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
152	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
153	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
154	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
155	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
158	C₂×C₈	—	94	16	no	—
159	C₂×C₈	—	94	32	no	—
160	C₂×C₈	—	94	8	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄²	C₄×Dic₄₇	1	2	yes	—
164	C₄²	Dic₄₇⋊C₄	1	4	yes	—
165	C₄²	C₄⋊Dic₄₇	1	2	yes	—
166	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
167	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
168	C₄²	C₂×C₄×D₄₇	1	2	yes	—
169	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
170	C₂³	C₄₇⋊#95	1	2	yes	—
171	C₂³	#13093941	1	2	yes	—
172	C₂³	#13093943	1	2	yes	—
173	C₂³	#13093951	1	8	yes	—
174	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	8	8	no	—
188	C₂×C₄	C₄₇⋊#95	1	4	yes	—
189	C₂×C₄	C₄₇⋊#95	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
191	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
192	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
193	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
194	C₈	—	94	16	no	—
195	C₂²	#15688863	1	8	yes	—
196	C₂²	#15688865	1	8	yes	—
197	C₂²	#15688882	1	8	yes	—
198	C₂²	#15688892	1	4	yes	—
199	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
200	C₂²	C₂×#13093941	1	1	yes	characteristic
201	C₂²	C₂×#13093943	1	1	yes	characteristic
202	C₂²	C₂×#13093951	1	4	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	8	4	no	—
210	C₂	#20241269	1	8	yes	direct product factor
211	C₂	C₂×#15688863	1	2	yes	—
212	C₂	C₂×#15688865	1	2	yes	—
213	C₂	C₂×#15688882	1	2	yes	—
214	C₂	C₂×#15688892	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#382	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	C₂×#1295
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
2. #20241269	C₂	direct (N × H)
7. C₂×#13093935	C₄	semidirect
20. #13093935	C₂×C₄	semidirect
106. C₂×C₄₇	#1295	semidirect
122. C₄₇	C₂×#1295	semidirect
210. C₂	#20241269	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×#13093935	2
#20241269	2
C₂×C₄₇×#382	2
C₂×#1295	47

Subgroups — C2×#20241269 #26106325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241269 #26106325