Subgroups — C₂×#20251784 #26422374

Order 12032.

3774 conjugacy classes of subgroups.

The subgroup lattice has 228 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	C₂²×#13094022	C₂	1	4	yes	maximal
3	#20251784	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15688992	C₂	1	8	yes	maximal
5	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	2	yes	—
8	C₂×#13094024	—	2	8	no	—
9	C₂×#13094024	C₂²	1	8	yes	—
10	C₂×#13094022	—	2	32	no	—
11	C₂×#13094022	C₂²	1	32	yes	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
13	C₂²×Dic₄₇⋊C₄	C₂²	1	16	yes	—
14	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
16	#15688992	C₂²	1	64	yes	—
17	C₄²×Dic₄₇	C₂²	1	8	yes	—
18	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Q₈×C₄₇	—	2	8	no	—
20	C₂×C₄⋊Q₈×C₄₇	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₉₄	—	2	16	no	—
24	C₂²×Dic₉₄	—	2	8	no	—
25	C₂²×Dic₉₄	C₂³	1	8	yes	—
26	C₂³×Dic₄₇	C₂³	1	4	yes	—
27	C₂³×Dic₄₇	C₂³	1	4	yes	—
28	#13094024	—	2	32	no	—
29	#13094022	—	2	128	no	—
30	C₂×C₄⋊Dic₄₇	—	2	16	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
32	C₂×Dic₄₇⋊C₄	—	2	64	no	—
33	C₂×Dic₄₇⋊C₄	C₂³	1	64	yes	—
34	C₂×C₄×Dic₄₇	D₄	1	16	yes	—
35	C₂×C₄×Dic₄₇	Q₈	1	32	yes	complemented
36	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
38	C₂²×Q₈×C₄₇	—	2	8	no	—
39	C₂²×Q₈×C₄₇	—	2	16	no	—
40	C₂²×Q₈×C₄₇	C₂³	1	8	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₄₇	—	2	32	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
47	C₂×C₄²×C₄₇	D₄	1	8	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
49	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₄₇	—	2	16	no	—
51	C₂×Q₈×C₄₇	—	2	32	no	—
52	C₂×Q₈×C₄₇	—	4	32	no	—
53	C₂²×C₄×C₄₇	—	2	8	no	—
54	C₂²×C₄×C₄₇	—	2	16	no	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
62	C₂²×Dic₄₇	—	2	16	no	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	C₂×D₄	1	16	yes	—
65	C₂²×Dic₄₇	C₂×Q₈	1	32	yes	—
66	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
67	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
68	C₂×Dic₉₄	—	2	32	no	—
69	C₂×Dic₉₄	—	2	16	no	—
70	C₂×Dic₉₄	—	4	32	no	—
71	C₄⋊C₄×C₄₇	—	2	32	no	—
72	C₄²×C₄₇	C₂×D₄	1	16	yes	—
73	C₄⋊Dic₄₇	—	2	32	no	—
74	Dic₄₇⋊C₄	—	2	128	no	—
75	C₄×Dic₄₇	C₂×D₄	1	32	yes	—
76	C₄×Dic₄₇	C₂×Q₈	1	64	yes	complemented
77	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
78	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
79	C₂³×C₄₇	C₂²×Q₈	1	8	yes	—
80	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
81	Q₈×C₄₇	—	4	32	no	—
82	C₂×C₄×C₄₇	—	2	16	no	—
83	C₂×C₄×C₄₇	—	2	8	no	—
84	C₂×C₄×C₄₇	—	4	16	no	—
85	C₂×C₄×C₄₇	#127	1	16	yes	—
86	C₂×C₄×C₄₇	C₄⋊Q₈	1	32	yes	complemented
87	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
88	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
89	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
90	C₂×C₄×C₄₇	C₂²×Q₈	1	16	yes	—
91	C₂×Dic₄₇	—	2	16	no	—
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	4	16	no	—
94	C₂×Dic₄₇	C₄⋊Q₈	1	64	yes	complemented
95	C₂×Dic₄₇	C₂²×D₄	1	16	yes	—
96	C₂×Dic₄₇	C₂²×Q₈	1	32	yes	—
97	Dic₉₄	—	4	32	no	—
98	C₂×#2868	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	C₂×#127	1	4	yes	—
100	C₂²×C₄₇	C₂×C₄⋊Q₈	1	16	yes	—
101	C₂²×C₄₇	C₂×C₄⋊Q₈	1	8	yes	—
102	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
103	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
104	C₂²×C₄₇	C₂³×Q₈	1	4	yes	—
105	C₄×C₄₇	—	4	8	no	—
106	C₄×C₄₇	C₂×#127	1	8	yes	—
107	C₄×C₄₇	C₂×C₄⋊Q₈	1	16	yes	complemented
108	Dic₄₇	—	4	8	no	—
109	Dic₄₇	C₂×C₄⋊Q₈	1	32	yes	complemented
110	C₂²×C₄⋊Q₈	—	47	1	no	—
111	C₂²×C₄⋊Q₈	—	47	4	no	—
112	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
113	#2868	—	47	16	no	—
114	C₂×#384	—	47	8	no	—
115	C₂×C₄³	—	47	1	no	—
116	C₂×C₄₇	#2868	1	8	yes	complemented
117	C₂×C₄₇	C₂²×#127	1	1	yes	characteristic
118	C₂×C₄₇	C₂²×C₄⋊Q₈	1	4	yes	—
119	C₂×C₄₇	C₂²×C₄⋊Q₈	1	2	yes	—
120	C₂³×Q₈	—	47	2	no	—
121	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
122	C₂×C₄⋊Q₈	—	2	8	no	—
123	C₂×C₄⋊Q₈	—	47	32	no	—
124	C₂×C₄⋊Q₈	—	47	8	no	—
125	C₂×C₄⋊Q₈	—	94	32	no	—
126	C₂×C₄⋊Q₈	—	94	8	no	—
127	C₂×C₄⋊Q₈	C₂×D₄₇	1	8	yes	—
128	C₂²×C₄⋊C₄	—	47	16	no	—
129	C₂²×C₄⋊C₄	—	47	4	no	—
130	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
131	C₂²×C₄²	—	47	2	no	—
132	C₂²×C₄²	—	47	4	no	—
133	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	#384	—	47	64	no	—
135	C₄³	—	47	8	no	—
136	C₄₇	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	2	16	no	—
138	C₂²×Q₈	—	2	8	no	—
139	C₂²×Q₈	—	47	8	no	—
140	C₂²×Q₈	—	94	16	no	—
141	C₂²×Q₈	—	94	8	no	—
142	C₂²×Q₈	C₂²×D₄₇	1	8	yes	—
143	C₂³×C₄	—	47	4	no	—
144	C₂³×C₄	—	47	4	no	—
145	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
146	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
147	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
148	C₄⋊Q₈	—	2	32	no	—
149	C₄⋊Q₈	—	94	128	no	—
150	C₄⋊Q₈	—	94	32	no	—
151	C₂×C₄⋊C₄	—	2	16	no	—
152	C₂×C₄⋊C₄	—	47	64	no	—
153	C₂×C₄⋊C₄	—	47	16	no	—
154	C₂×C₄⋊C₄	—	94	64	no	—
155	C₂×C₄⋊C₄	—	94	16	no	—
156	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
157	C₂×C₄²	—	47	16	no	—
158	C₂×C₄²	—	47	32	no	—
159	C₂×C₄²	—	47	8	no	—
160	C₂×C₄²	—	47	16	no	—
161	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
162	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
163	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
164	C₂×Q₈	—	2	32	no	—
165	C₂×Q₈	—	2	16	no	—
166	C₂×Q₈	—	4	32	no	—
167	C₂×Q₈	—	94	32	no	—
168	C₂×Q₈	—	94	16	no	—
169	C₂×Q₈	—	188	32	no	—
170	C₂²×C₄	—	2	16	no	—
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	47	16	no	—
173	C₂²×C₄	—	47	32	no	—
174	C₂²×C₄	—	47	8	no	—
175	C₂²×C₄	—	47	8	no	—
176	C₂²×C₄	—	94	16	no	—
177	C₂²×C₄	—	94	8	no	—
178	C₂²×C₄	D₄×D₄₇	1	8	yes	—
179	C₂²×C₄	Q₈×D₄₇	1	16	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
183	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
184	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
185	C₄⋊C₄	—	2	32	no	—
186	C₄⋊C₄	—	94	128	no	—
187	C₄⋊C₄	—	94	32	no	—
188	C₄²	—	47	32	no	—
189	C₄²	—	47	64	no	—
190	C₄²	C₂×C₂²⋊D₄₇	1	16	yes	—
191	C₂³	C₂×D₄×D₄₇	1	4	yes	—
192	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
193	C₂³	C₂×Q₈×D₄₇	1	8	yes	—
194	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
195	Q₈	—	4	32	no	—
196	Q₈	—	188	32	no	—
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	16	no	—
200	C₂×C₄	—	47	64	no	—
201	C₂×C₄	—	47	16	no	—
202	C₂×C₄	—	47	32	no	—
203	C₂×C₄	—	94	16	no	—
204	C₂×C₄	—	94	8	no	—
205	C₂×C₄	—	188	16	no	—
206	C₂×C₄	#13094020	1	16	yes	—
207	C₂×C₄	#13094022	1	32	yes	complemented
208	C₂×C₄	C₂×D₄×D₄₇	1	8	yes	—
209	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
210	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
211	C₂×C₄	C₂×Q₈×D₄₇	1	16	yes	—
212	C₂²	C₂×#13094020	1	4	yes	—
213	C₂²	C₄⋊Q₈×D₄₇	1	16	yes	—
214	C₂²	C₂×#13094022	1	8	yes	—
215	C₂²	C₂²×D₄×D₄₇	1	2	yes	—
216	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂²	C₂²×Q₈×D₄₇	1	4	yes	—
218	C₄	—	4	8	no	—
219	C₄	—	47	32	no	—
220	C₄	—	188	8	no	—
221	C₄	C₂×#13094020	1	8	yes	—
222	C₄	C₂×#13094022	1	16	yes	complemented
223	C₂	#20251784	1	8	yes	direct product factor
224	C₂	C₂²×#13094020	1	1	yes	characteristic
225	C₂	C₂×C₄⋊Q₈×D₄₇	1	4	yes	—
226	C₂	C₂²×#13094022	1	2	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×#2868
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
3. #20251784	C₂	direct (N × H)
35. C₂×C₄×Dic₄₇	Q₈	semidirect
76. C₄×Dic₄₇	C₂×Q₈	semidirect
86. C₂×C₄×C₄₇	C₄⋊Q₈	semidirect
94. C₂×Dic₄₇	C₄⋊Q₈	semidirect
107. C₄×C₄₇	C₂×C₄⋊Q₈	semidirect
109. Dic₄₇	C₂×C₄⋊Q₈	semidirect
116. C₂×C₄₇	#2868	semidirect
136. C₄₇	C₂×#2868	semidirect
207. C₂×C₄	#13094022	semidirect
222. C₄	C₂×#13094022	semidirect
223. C₂	#20251784	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
C₂²×#13094022	2
#20251784	2
C₂×#15688992	2
C₂×C₄²×Dic₄₇	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#2868	47

Subgroups — C2×#20251784 #26422374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251784 #26422374