Subgroups — C₂×#20243528 #26216507

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20243528	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688887	C₂	1	4	yes	maximal
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15689147	—	2	2	no	—
9	#15689147	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
11	C₂×#13093935	C₄	1	4	yes	—
12	C₂×#13093935	C₄	1	4	yes	—
13	C₂×#13093935	C₂²	1	4	yes	—
14	#15688887	C₂²	1	16	yes	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#423	—	2	2	no	—
18	C₄₇×#423	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	8	no	—
22	C₂×#10625329	C₂³	1	4	yes	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
30	#13093935	C₂×C₄	1	16	yes	—
31	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₄₇	—	2	2	no	—
34	C₂×M₁₆×C₄₇	—	2	8	no	—
35	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	D₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
43	C₄⋊C₈×C₄₇	—	2	8	no	—
44	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	4	4	no	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	#41	1	4	yes	—
55	C₂²×C₄×C₄₇	#41	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₄₇	—	2	8	no	—
58	M₁₆×C₄₇	—	4	4	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	4	4	no	—
61	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
63	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
64	C₄²×C₄₇	C₂×D₄	1	4	yes	—
65	#10625329	—	2	8	no	—
66	#10625329	—	4	4	no	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂×C₄₇⋊C₈	—	2	8	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	4	4	no	—
73	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
74	C₂³×C₄₇	—	2	4	no	—
75	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₄₇	#125	1	4	yes	—
78	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#117	1	4	yes	—
86	C₂×C₄×C₄₇	#117	1	4	yes	—
87	C₂×C₄×C₄₇	#124	1	2	yes	—
88	C₂×C₄×C₄₇	#124	1	2	yes	—
89	C₂×C₄×C₄₇	#124	1	2	yes	—
90	C₂×C₄×C₄₇	#124	1	2	yes	—
91	C₂×C₄×C₄₇	#125	1	4	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
96	C₈×C₄₇	—	4	4	no	—
97	C₄₇⋊C₈	—	2	8	no	—
98	C₄₇⋊C₈	—	4	4	no	—
99	C₂×#1838	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	8	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	#382	1	2	yes	—
104	C₂²×C₄₇	#382	1	2	yes	—
105	C₂²×C₄₇	#430	1	4	yes	—
106	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×#125	1	2	yes	—
108	C₄×C₄₇	—	2	8	no	—
109	C₄×C₄₇	#382	1	2	yes	—
110	C₄×C₄₇	#382	1	2	yes	—
111	C₄×C₄₇	C₂×#117	1	2	yes	—
112	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
113	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
114	C₂×#423	—	47	1	no	—
115	C₂×#423	D₄₇	1	1	yes	characteristic
116	C₂²×#97	—	47	1	no	—
117	#1838	—	47	8	no	—
118	C₂×#341	—	47	4	no	—
119	C₂×C₄₇	—	2	2	no	—
120	C₂×C₄₇	—	2	2	no	—
121	C₂×C₄₇	#1838	1	4	yes	complemented
122	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
123	C₂×C₄₇	C₂×#430	1	2	yes	—
124	C₂²×M₁₆	—	47	2	no	—
125	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
126	C₂³×C₈	—	47	2	no	—
127	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
128	#423	—	2	2	no	—
129	#423	—	47	4	no	—
130	#423	—	94	2	no	—
131	#423	C₂×D₄₇	1	4	yes	—
132	C₂×C₄⋊C₈	—	2	2	no	—
133	C₂×C₄⋊C₈	—	94	2	no	—
134	C₂×#97	—	47	4	no	—
135	C₂×#97	—	47	4	no	—
136	C₂×#97	—	47	4	no	—
137	#341	—	47	16	no	—
138	C₄₇	C₂×#1838	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	Dic₉₄	1	2	yes	—
140	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	8	no	—
143	C₂×M₁₆	—	47	4	no	—
144	C₂×M₁₆	—	94	2	no	—
145	C₂×M₁₆	—	94	8	no	—
146	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	47	4	no	—
149	C₂²×C₈	—	47	4	no	—
150	C₂²×C₈	—	47	4	no	—
151	C₂²×C₈	—	94	4	no	—
152	C₂²×C₈	—	94	4	no	—
153	C₂²×C₈	—	94	2	no	—
154	C₂×C₄²	Dic₉₄	1	2	yes	—
155	C₂×C₄²	C₄×D₄₇	1	2	yes	—
156	C₂×C₄²	C₄×D₄₇	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
159	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
160	C₄⋊C₈	—	2	8	no	—
161	C₄⋊C₈	—	94	8	no	—
162	#97	—	47	16	no	—
163	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
166	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
167	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
168	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₄	#10625352	1	4	yes	—
172	C₂²×C₄	#10625354	1	4	yes	—
173	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
176	M₁₆	—	2	8	no	—
177	M₁₆	—	4	4	no	—
178	M₁₆	—	94	8	no	—
179	M₁₆	—	188	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	47	8	no	—
183	C₂×C₈	—	94	8	no	—
184	C₂×C₈	—	94	8	no	—
185	C₂×C₈	—	94	4	no	—
186	C₂×C₈	—	94	4	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	—	188	4	no	—
189	C₄²	Dic₄₇⋊C₄	1	4	yes	—
190	C₄²	Dic₄₇⋊C₄	1	4	yes	—
191	C₄²	C₂×C₄×D₄₇	1	4	yes	—
192	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
195	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
196	C₂³	#13094015	1	4	yes	—
197	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
202	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
204	C₂×C₄	#13093982	1	4	yes	—
205	C₂×C₄	#13093982	1	4	yes	—
206	C₂×C₄	#13094011	1	2	yes	—
207	C₂×C₄	#13094011	1	2	yes	—
208	C₂×C₄	#13094013	1	2	yes	—
209	C₂×C₄	#13094013	1	2	yes	—
210	C₂×C₄	#13094015	1	4	yes	—
211	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₄	C₂×#10625352	1	2	yes	—
214	C₂×C₄	C₂×#10625354	1	2	yes	—
215	C₈	—	4	4	no	—
216	C₈	—	94	8	no	—
217	C₈	—	188	4	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	#15688988	1	2	yes	—
222	C₂²	#15688988	1	2	yes	—
223	C₂²	#15689171	1	4	yes	—
224	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
225	C₂²	C₂×#13094015	1	2	yes	—
226	C₄	—	2	8	no	—
227	C₄	#15688988	1	2	yes	—
228	C₄	#15688988	1	2	yes	—
229	C₄	C₂×#13093982	1	2	yes	—
230	C₄	C₂×#13094011	1	1	yes	characteristic
231	C₄	C₂×#13094013	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	2	2	no	—
234	C₂	#20243528	1	4	yes	direct product factor
235	C₂	C₂×#15688988	1	1	yes	characteristic
236	C₂	C₂×#15689171	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	C₂×#1838
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20243528	C₂	direct (N × H)
121. C₂×C₄₇	#1838	semidirect
138. C₄₇	C₂×#1838	semidirect
234. C₂	#20243528	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×#15689147	2
C₂²×#13093935	2
#20243528	2
C₂×#15688887	2
C₂×C₄₇×#423	2
C₂×#1838	47

Subgroups — C2×#20243528 #26216507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243528 #26216507