Subgroups — C₂×#20241437 #26127518

Order 12032.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	#20241437	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
9	#15688876	—	2	1	no	—
10	#15688876	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₄₇	—	2	1	no	—
12	C₂²×Q₁₆×C₄₇	C₄	1	2	yes	—
13	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
14	C₄₇×#494	—	2	1	no	—
15	C₄₇×#494	—	2	1	no	—
16	C₄₇×#494	C₄	1	2	yes	—
17	C₄₇×#494	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄₇⋊C₈	—	4	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
24	C₂²×Q₈×C₄₇	—	2	2	no	—
25	C₂×Q₁₆×C₄₇	—	2	2	no	—
26	C₂×Q₁₆×C₄₇	—	2	2	no	—
27	C₂×Q₁₆×C₄₇	—	2	2	no	—
28	C₂×Q₁₆×C₄₇	—	4	4	no	—
29	C₂×Q₁₆×C₄₇	—	4	2	no	—
30	C₂×Q₁₆×C₄₇	—	4	2	no	—
31	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₄₇	—	2	2	no	—
35	C₄⋊Q₈×C₄₇	—	2	2	no	—
36	C₄⋊Q₈×C₄₇	—	4	1	no	—
37	C₄⋊Q₈×C₄₇	—	4	1	no	—
38	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
39	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₄₇	—	2	4	no	—
46	C₂×Q₈×C₄₇	—	4	2	no	—
47	C₂×Q₈×C₄₇	—	4	4	no	—
48	C₂×Q₈×C₄₇	—	4	4	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
53	Q₁₆×C₄₇	—	4	4	no	—
54	Q₁₆×C₄₇	—	4	4	no	—
55	Q₁₆×C₄₇	—	4	4	no	—
56	Q₁₆×C₄₇	—	4	2	no	—
57	Q₁₆×C₄₇	—	4	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	D₈	1	2	yes	—
62	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	—	4	2	no	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₄⋊C₄×C₄₇	—	4	2	no	—
69	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₄₇	D₈	1	1	yes	characteristic
71	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄₇⋊C₈	—	4	2	no	—
74	C₂×C₄₇⋊C₈	—	8	2	no	—
75	C₄₇⋊C₁₆	—	4	2	no	—
76	C₂³×C₄₇	#102	1	1	yes	characteristic
77	Q₈×C₄₇	—	4	4	no	—
78	Q₈×C₄₇	—	4	4	no	—
79	Q₈×C₄₇	—	8	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	8	4	no	—
83	C₂×C₄×C₄₇	#100	1	2	yes	—
84	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#102	1	2	yes	—
87	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
89	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₄₇	—	2	1	no	—
94	C₈×C₄₇	—	2	1	no	—
95	C₈×C₄₇	#102	1	2	yes	—
96	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
97	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₄₇⋊C₈	—	8	2	no	—
99	C₂×#1362	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	#331	1	2	yes	—
101	C₂²×C₄₇	#358	1	2	yes	—
102	C₂²×C₄₇	#362	1	2	yes	—
103	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	8	4	no	—
107	C₄×C₄₇	#331	1	2	yes	—
108	C₄×C₄₇	#358	1	2	yes	—
109	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
110	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
111	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
112	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
113	C₂×#494	D₄₇	1	1	yes	characteristic
114	C₂×C₄⋊C₁₆	—	47	1	no	—
115	C₂×#335	—	47	1	no	—
116	#1362	—	47	4	no	—
117	C₂×C₄₇	#1362	1	4	yes	complemented
118	C₂×C₄₇	C₂×#331	1	1	yes	characteristic
119	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#362	1	1	yes	characteristic
121	C₂²×Q₁₆	—	2	1	no	—
122	C₂²×Q₁₆	Dic₄₇	1	2	yes	—
123	C₂×C₄⋊Q₈	—	2	1	no	—
124	C₂²×C₁₆	—	94	1	no	—
125	#494	—	2	1	no	—
126	#494	—	2	1	no	—
127	#494	Dic₄₇	1	2	yes	—
128	#494	C₂×D₄₇	1	2	yes	—
129	C₂×C₄⋊C₈	—	94	1	no	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	C₄⋊C₁₆	—	47	2	no	—
132	C₄⋊C₁₆	—	94	1	no	—
133	#335	—	47	2	no	—
134	#335	—	94	1	no	—
135	C₄₇	C₂×#1362	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	4	4	no	—
141	C₂×Q₁₆	—	4	2	no	—
142	C₂×Q₁₆	—	4	2	no	—
143	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
144	C₂²×C₈	—	188	1	no	—
145	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	1	no	—
150	C₄⋊Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	4	1	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
154	C₂×C₁₆	—	94	2	no	—
155	C₂×C₁₆	—	188	2	no	—
156	C₄⋊C₈	—	94	2	no	—
157	C₄⋊C₈	—	188	1	no	—
158	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	#10625334	1	1	yes	characteristic
168	C₂²×C₄	#10625335	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
170	Q₁₆	—	4	4	no	—
171	Q₁₆	—	4	4	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	2	no	—
174	Q₁₆	—	4	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	376	2	no	—
177	C₂×C₈	—	188	2	no	—
178	C₂×C₈	#10625334	1	2	yes	—
179	C₂×C₈	#10625335	1	2	yes	—
180	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
182	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄²	#10625334	1	1	yes	characteristic
189	C₄²	#10625335	1	1	yes	characteristic
190	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
192	C₁₆	—	188	2	no	—
193	C₂³	#13093945	1	1	yes	characteristic
194	Q₈	—	4	4	no	—
195	Q₈	—	4	4	no	—
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	#13093941	1	2	yes	—
201	C₂×C₄	#13093945	1	1	yes	characteristic
202	C₂×C₄	#13093945	1	2	yes	—
203	C₂×C₄	#13093945	1	1	yes	characteristic
204	C₂×C₄	#13093945	1	1	yes	characteristic
205	C₂×C₄	#13093967	1	2	yes	—
206	C₂×C₄	#13093970	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
209	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	376	2	no	—
213	C₈	#13093945	1	2	yes	—
214	C₈	C₂×#10625334	1	1	yes	characteristic
215	C₈	C₂×#10625335	1	1	yes	characteristic
216	C₂²	#15688867	1	2	yes	—
217	C₂²	#15688927	1	2	yes	—
218	C₂²	#15688939	1	2	yes	—
219	C₂²	C₂×#13093945	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	4	no	—
223	C₄	#15688867	1	2	yes	—
224	C₄	#15688927	1	2	yes	—
225	C₄	C₂×#13093941	1	1	yes	characteristic
226	C₄	C₂×#13093945	1	1	yes	characteristic
227	C₄	C₂×#13093967	1	1	yes	characteristic
228	C₄	C₂×#13093970	1	1	yes	characteristic
229	C₂	#20241437	1	4	yes	direct product factor
230	C₂	C₂×#15688867	1	1	yes	characteristic
231	C₂	C₂×#15688927	1	1	yes	characteristic
232	C₂	C₂×#15688939	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle, prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	C₂×#1362
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #20241437	C₂	direct (N × H)
117. C₂×C₄₇	#1362	semidirect
135. C₄₇	C₂×#1362	semidirect
229. C₂	#20241437	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×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688876	2
#20241437	2
C₂×C₄₇×#494	2
C₂×#1362	47

Subgroups — C2×#20241437 #26127518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241437 #26127518