Subgroups — C₂²×#15689853 #27123344

Order 12032.

7846 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689853	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13094023	C₂	1	4	yes	maximal
3	C₂²×#13094022	C₂	1	4	yes	maximal
4	C₂²×C₄×Dic₉₄	C₂	1	2	yes	maximal
5	C₂²×#13093989	C₂	1	4	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	4	yes	—
8	C₂³×Dic₉₄	C₂²	1	2	yes	—
9	#15689853	C₂²	1	256	yes	direct product factor
10	C₂×#13094023	C₂²	1	96	yes	—
11	C₂×#13094022	C₂²	1	96	yes	—
12	C₂×C₄×Dic₉₄	C₂²	1	48	yes	—
13	C₂×#13093989	C₂²	1	96	yes	—
14	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂²×Dic₄₇⋊C₄	C₂²	1	8	yes	—
16	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
17	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
18	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
19	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₄₇	C₂²	1	24	yes	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₉₄	—	4	16	no	—
24	C₂²×Dic₉₄	Q₈	1	32	yes	complemented
25	C₂²×Dic₉₄	C₂³	1	48	yes	—
26	C₂²×Dic₉₄	C₂³	1	24	yes	—
27	C₂³×Dic₄₇	C₂³	1	4	yes	—
28	C₂³×Dic₄₇	C₂³	1	4	yes	—
29	#13094023	C₂³	1	256	yes	—
30	#13094022	C₂³	1	256	yes	—
31	C₄×Dic₉₄	C₂³	1	128	yes	—
32	#13093989	C₂³	1	256	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
34	C₂×Dic₄₇⋊C₄	C₂³	1	96	yes	—
35	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	48	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	48	yes	—
38	C₂²×Q₈×C₄₇	—	4	8	no	—
39	C₂²×Q₈×C₄₇	C₂³	1	24	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
43	C₄⋊Q₈×C₄₇	C₂³	1	64	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	48	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
46	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	—	4	48	no	—
48	C₂×Q₈×C₄₇	C₂⁴	1	32	yes	—
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	2	16	no	—
51	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
52	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	24	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
55	C₂²×Dic₄₇	—	2	16	no	—
56	C₂²×Dic₄₇	C₂×Q₈	1	32	yes	—
57	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
58	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
59	C₂×Dic₉₄	—	4	96	no	—
60	C₂×Dic₉₄	C₂×Q₈	1	192	yes	complemented
61	C₂×Dic₉₄	C₂⁴	1	64	yes	—
62	C₂×Dic₉₄	C₂⁴	1	32	yes	—
63	C₄⋊C₄×C₄₇	C₂⁴	1	64	yes	—
64	C₄²×C₄₇	C₂⁴	1	16	yes	—
65	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
66	Dic₄₇⋊C₄	C₂⁴	1	128	yes	—
67	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
68	C₄×Dic₄₇	C₂⁴	1	64	yes	—
69	C₄×Dic₄₇	C₂⁴	1	64	yes	—
70	C₂³×C₄₇	C₂²×Q₈	1	8	yes	—
71	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
72	C₂³×C₄₇	C₂⁵	1	3	yes	—
73	Q₈×C₄₇	—	4	32	no	—
74	C₂×C₄×C₄₇	—	2	48	no	—
75	C₂×C₄×C₄₇	—	2	12	no	—
76	C₂×C₄×C₄₇	C₂²×Q₈	1	48	yes	—
77	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
78	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
79	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
80	C₂×Dic₄₇	—	2	48	no	—
81	C₂×Dic₄₇	C₂²×Q₈	1	96	yes	—
82	C₂×Dic₄₇	C₂⁵	1	16	yes	—
83	C₂×Dic₄₇	C₂⁵	1	16	yes	—
84	Dic₉₄	—	4	64	no	—
85	Dic₉₄	C₂²×Q₈	1	128	yes	complemented
86	C₂²×Q₈²	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	Q₈²	1	16	yes	complemented
88	C₂²×C₄₇	C₂³×Q₈	1	12	yes	—
89	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	6	yes	—
90	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
91	C₄×C₄₇	—	2	16	no	—
92	C₄×C₄₇	—	2	4	no	—
93	C₄×C₄₇	C₂³×Q₈	1	16	yes	—
94	Dic₄₇	—	2	16	no	—
95	Dic₄₇	C₂³×Q₈	1	32	yes	—
96	C₂×Q₈²	—	47	48	no	—
97	C₂²×C₄⋊Q₈	—	47	4	no	—
98	C₂²×C₄⋊Q₈	—	47	4	no	—
99	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
100	C₂²×C₄×Q₈	—	47	2	no	—
101	C₂²×C₄×Q₈	—	47	4	no	—
102	C₂×C₄₇	C₂×Q₈²	1	12	yes	complemented
103	C₂×C₄₇	C₂⁴×Q₈	1	2	yes	—
104	C₂×C₄₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
105	C₂³×Q₈	—	47	2	no	—
106	C₂³×Q₈	—	47	4	no	—
107	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
108	Q₈²	—	47	256	no	—
109	C₂×C₄⋊Q₈	—	47	96	no	—
110	C₂×C₄⋊Q₈	—	47	96	no	—
111	C₂×C₄⋊Q₈	C₂×D₄₇	1	24	yes	—
112	C₂×C₄×Q₈	—	47	96	no	—
113	C₂×C₄×Q₈	—	47	48	no	—
114	C₂²×C₄⋊C₄	—	47	4	no	—
115	C₂²×C₄⋊C₄	—	47	8	no	—
116	C₂²×C₄⋊C₄	—	47	2	no	—
117	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
118	C₂²×C₄²	—	47	4	no	—
119	C₂²×C₄²	—	47	4	no	—
120	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
121	C₄₇	C₂²×Q₈²	1	1	yes	characteristic, complemented, Sylow
122	C₂²×Q₈	—	4	8	no	—
123	C₂²×Q₈	—	47	32	no	—
124	C₂²×Q₈	—	47	24	no	—
125	C₂²×Q₈	—	47	48	no	—
126	C₂²×Q₈	—	188	16	no	—
127	C₂²×Q₈	C₂²×D₄₇	1	24	yes	—
128	C₂³×C₄	—	47	4	no	—
129	C₂³×C₄	—	47	4	no	—
130	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
132	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
133	C₄⋊Q₈	—	47	256	no	—
134	C₄⋊Q₈	—	47	256	no	—
135	C₄⋊Q₈	C₂²×D₄₇	1	64	yes	—
136	C₄×Q₈	—	47	256	no	—
137	C₄×Q₈	—	47	128	no	—
138	C₂×C₄⋊C₄	—	47	96	no	—
139	C₂×C₄⋊C₄	—	47	48	no	—
140	C₂×C₄⋊C₄	—	47	24	no	—
141	C₂×C₄⋊C₄	C₂²×D₄₇	1	48	yes	—
142	C₂×C₄²	—	47	48	no	—
143	C₂×C₄²	—	47	48	no	—
144	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
145	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
146	C₂×Q₈	—	4	48	no	—
147	C₂×Q₈	—	47	192	no	—
148	C₂×Q₈	—	47	32	no	—
149	C₂×Q₈	—	47	64	no	—
150	C₂×Q₈	—	188	96	no	—
151	C₂×Q₈	C₂³×D₄₇	1	32	yes	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	2	16	no	—
154	C₂²×C₄	—	47	32	no	—
155	C₂²×C₄	—	47	24	no	—
156	C₂²×C₄	—	47	24	no	—
157	C₂²×C₄	—	94	16	no	—
158	C₂²×C₄	Q₈×D₄₇	1	16	yes	—
159	C₂²×C₄	C₂³×D₄₇	1	24	yes	—
160	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
161	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
162	C₄⋊C₄	—	47	128	no	—
163	C₄⋊C₄	—	47	64	no	—
164	C₄⋊C₄	—	47	32	no	—
165	C₄⋊C₄	C₂³×D₄₇	1	64	yes	—
166	C₄²	—	47	64	no	—
167	C₄²	—	47	64	no	—
168	C₄²	C₂³×D₄₇	1	16	yes	—
169	C₂³	C₂×Q₈×D₄₇	1	8	yes	—
170	C₂³	#13094069	1	4	yes	—
171	C₂³	C₂⁴×D₄₇	1	3	yes	—
172	Q₈	—	4	32	no	—
173	Q₈	—	47	128	no	—
174	Q₈	—	188	64	no	—
175	C₂×C₄	—	2	12	no	—
176	C₂×C₄	—	2	48	no	—
177	C₂×C₄	—	47	96	no	—
178	C₂×C₄	—	47	16	no	—
179	C₂×C₄	—	47	16	no	—
180	C₂×C₄	—	94	48	no	—
181	C₂×C₄	C₂×Q₈×D₄₇	1	48	yes	—
182	C₂×C₄	C₂⁴×D₄₇	1	16	yes	—
183	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
184	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
185	C₂²	#15689853	1	16	yes	direct product factor
186	C₂²	C₂²×Q₈×D₄₇	1	12	yes	—
187	C₂²	C₂×#13094069	1	6	yes	—
188	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	16	no	—
191	C₄	—	47	32	no	—
192	C₄	—	94	16	no	—
193	C₄	C₂²×Q₈×D₄₇	1	16	yes	—
194	C₂	C₂×#15689853	1	12	yes	direct product factor
195	C₂	C₂³×Q₈×D₄₇	1	2	yes	—
196	C₂	C₂²×#13094069	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×Q₈²
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15689853	C₂	direct (N × H)
9. #15689853	C₂²	direct (N × H)
24. C₂²×Dic₉₄	Q₈	semidirect
60. C₂×Dic₉₄	C₂×Q₈	semidirect
85. Dic₉₄	C₂²×Q₈	semidirect
87. C₂²×C₄₇	Q₈²	semidirect
102. C₂×C₄₇	C₂×Q₈²	semidirect
121. C₄₇	C₂²×Q₈²	semidirect
185. C₂²	#15689853	direct (N × H)
194. C₂	C₂×#15689853	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×#15689853	2
C₂²×#13094023	2
C₂²×#13094022	2
C₂²×C₄×Dic₉₄	2
C₂²×#13093989	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂²×Q₈²	47

Subgroups — C22×#15689853 #27123344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689853 #27123344