Subgroups — C₂²×#15689839 #27123298

Order 12032.

8710 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689839	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13094013	C₂	1	4	yes	maximal, complemented
3	C₂²×#13093999	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×D₁₈₈	C₂	1	2	yes	maximal, complemented
5	C₂²×#13093982	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₈₈	C₂²	1	2	yes	—
8	C₂³×D₁₈₈	C₂²	1	4	yes	—
9	C₂³×C₄×D₄₇	C₂²	1	4	yes	—
10	#15689839	C₂²	1	256	yes	direct product factor
11	C₂×#13094013	C₂²	1	96	yes	complemented
12	C₂×#13093999	C₂²	1	96	yes	complemented
13	C₂×C₄×D₁₈₈	C₂²	1	48	yes	complemented
14	C₂×#13093982	C₂²	1	96	yes	complemented
15	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
17	C₂²×(C₂×D₄₇)⋊C₄	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	complemented
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂⁴×D₄₇	C₂³	1	4	yes	—
24	C₂²×D₁₈₈	—	2	16	no	—
25	C₂²×D₁₈₈	—	4	16	no	—
26	C₂²×D₁₈₈	C₂³	1	24	yes	—
27	C₂²×D₁₈₈	C₂³	1	48	yes	—
28	C₂³×Dic₄₇	C₂³	1	4	yes	—
29	C₂²×C₄×D₄₇	—	2	32	no	—
30	C₂²×C₄×D₄₇	C₂³	1	48	yes	—
31	#13094013	C₂³	1	256	yes	complemented
32	#13093999	C₂³	1	256	yes	complemented
33	C₄×D₁₈₈	C₂³	1	128	yes	complemented
34	#13093982	C₂³	1	256	yes	complemented
35	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	96	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	48	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	48	yes	—
39	C₂²×Q₈×C₄₇	—	4	8	no	—
40	C₂²×Q₈×C₄₇	C₂³	1	24	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₄₇	C₂³	1	64	yes	complemented
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	48	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
47	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂³×D₄₇	—	2	16	no	—
49	C₂³×D₄₇	—	2	16	no	—
50	C₂³×D₄₇	—	2	16	no	—
51	C₂³×D₄₇	C₂⁴	1	24	yes	—
52	C₂×Q₈×C₄₇	—	4	48	no	—
53	C₂×Q₈×C₄₇	C₂⁴	1	32	yes	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	16	no	—
56	C₂²×C₄×C₄₇	#41	1	16	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	24	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
60	C₂²×Dic₄₇	—	2	16	no	—
61	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
62	C₂×D₁₈₈	—	2	96	no	—
63	C₂×D₁₈₈	—	4	96	no	—
64	C₂×D₁₈₈	C₂⁴	1	32	yes	—
65	C₂×D₁₈₈	C₂⁴	1	64	yes	—
66	C₂×C₄×D₄₇	—	2	192	no	—
67	C₂×C₄×D₄₇	C₂⁴	1	64	yes	—
68	C₄⋊C₄×C₄₇	C₂⁴	1	64	yes	—
69	C₄²×C₄₇	C₂⁴	1	16	yes	—
70	(C₂×D₄₇)⋊C₄	C₂⁴	1	128	yes	—
71	(C₂×D₄₇)⋊C₄	C₂⁴	1	64	yes	—
72	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
73	C₄×Dic₄₇	C₂⁴	1	64	yes	—
74	C₂³×C₄₇	C₂×#41	1	8	yes	—
75	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
76	C₂³×C₄₇	C₂⁵	1	3	yes	—
77	C₂²×D₄₇	—	2	48	no	—
78	C₂²×D₄₇	—	2	48	no	—
79	C₂²×D₄₇	—	2	48	no	—
80	C₂²×D₄₇	—	4	64	no	—
81	C₂²×D₄₇	C₂⁵	1	16	yes	—
82	Q₈×C₄₇	—	4	32	no	—
83	C₂×C₄×C₄₇	—	2	48	no	—
84	C₂×C₄×C₄₇	—	2	12	no	—
85	C₂×C₄×C₄₇	C₂×#41	1	48	yes	—
86	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
87	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
88	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
89	C₂×Dic₄₇	—	2	48	no	—
90	C₂×Dic₄₇	C₂⁵	1	16	yes	—
91	D₁₈₈	—	2	64	no	—
92	D₁₈₈	—	4	64	no	—
93	C₄×D₄₇	—	2	128	no	—
94	C₂²×#555	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	#555	1	16	yes	complemented
96	C₂²×C₄₇	C₂²×#41	1	12	yes	—
97	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	6	yes	—
98	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
99	C₂×D₄₇	—	2	16	no	—
100	C₂×D₄₇	—	2	16	no	—
101	C₂×D₄₇	—	2	16	no	—
102	C₂×D₄₇	—	4	96	no	—
103	C₄×C₄₇	—	2	16	no	—
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	C₂²×#41	1	16	yes	—
106	Dic₄₇	—	2	16	no	—
107	C₂×#555	—	47	48	no	—
108	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
109	C₂²×#124	—	47	4	no	—
110	C₂²×#121	—	47	4	no	—
111	C₂²×C₄×D₄	—	47	2	no	—
112	C₂²×#117	—	47	4	no	—
113	C₂×C₄₇	C₂×#555	1	12	yes	complemented
114	C₂×C₄₇	C₂³×#41	1	2	yes	—
115	C₂×C₄₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
116	D₄₇	—	4	16	no	—
117	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
118	C₂³×D₄	—	47	2	no	—
119	C₂³×D₄	—	47	4	no	—
120	C₂⁴×C₄	—	47	4	no	—
121	#555	—	47	256	no	—
122	C₂×C₄⋊Q₈	C₂×D₄₇	1	24	yes	complemented
123	C₂×#124	—	47	96	no	—
124	C₂×#121	—	47	96	no	—
125	C₂×C₄×D₄	—	47	48	no	—
126	C₂×#117	—	47	96	no	—
127	C₂²×C₄⋊C₄	—	47	2	no	—
128	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
129	C₂²×C₂²⋊C₄	—	47	8	no	—
130	C₂²×C₂²⋊C₄	—	47	4	no	—
131	C₂²×C₄²	—	47	4	no	—
132	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
133	C₄₇	C₂²×#555	1	1	yes	characteristic, complemented, Sylow
134	C₂⁵	—	47	4	no	—
135	C₂²×Q₈	—	4	8	no	—
136	C₂²×Q₈	C₂²×D₄₇	1	24	yes	—
137	C₂²×D₄	—	47	24	no	—
138	C₂²×D₄	—	47	48	no	—
139	C₂²×D₄	—	94	16	no	—
140	C₂²×D₄	—	188	16	no	—
141	C₂³×C₄	—	47	4	no	—
142	C₂³×C₄	—	47	48	no	—
143	C₂³×C₄	—	94	32	no	—
144	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
145	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
146	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
147	C₄⋊Q₈	C₂²×D₄₇	1	64	yes	complemented
148	#124	—	47	256	no	—
149	#121	—	47	256	no	—
150	C₄×D₄	—	47	128	no	—
151	#117	—	47	256	no	—
152	C₂×C₄⋊C₄	—	47	24	no	—
153	C₂×C₄⋊C₄	C₂²×D₄₇	1	48	yes	—
154	C₂×C₂²⋊C₄	—	47	96	no	—
155	C₂×C₂²⋊C₄	—	47	48	no	—
156	C₂×C₄²	—	47	48	no	—
157	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
158	C₂⁴	—	47	24	no	—
159	C₂⁴	—	94	16	no	—
160	C₂⁴	—	94	16	no	—
161	C₂⁴	—	94	16	no	—
162	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
163	C₂×Q₈	—	4	48	no	—
164	C₂×Q₈	C₂³×D₄₇	1	32	yes	—
165	C₂×D₄	—	47	32	no	—
166	C₂×D₄	—	47	64	no	—
167	C₂×D₄	—	94	96	no	—
168	C₂×D₄	—	188	96	no	—
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	47	24	no	—
172	C₂²×C₄	—	47	64	no	—
173	C₂²×C₄	—	94	16	no	—
174	C₂²×C₄	—	94	192	no	—
175	C₂²×C₄	#10625354	1	16	yes	—
176	C₂²×C₄	C₂³×D₄₇	1	24	yes	—
177	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
178	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
179	C₄⋊C₄	—	47	32	no	—
180	C₄⋊C₄	C₂³×D₄₇	1	64	yes	—
181	C₂²⋊C₄	—	47	128	no	—
182	C₂²⋊C₄	—	47	64	no	—
183	C₄²	—	47	64	no	—
184	C₄²	C₂³×D₄₇	1	16	yes	—
185	C₂³	—	47	16	no	—
186	C₂³	—	94	48	no	—
187	C₂³	—	94	48	no	—
188	C₂³	—	94	48	no	—
189	C₂³	—	188	64	no	—
190	C₂³	C₂×#10625354	1	8	yes	—
191	C₂³	#13094069	1	4	yes	—
192	C₂³	C₂⁴×D₄₇	1	3	yes	—
193	Q₈	—	4	32	no	—
194	D₄	—	94	64	no	—
195	D₄	—	188	64	no	—
196	C₂×C₄	—	2	48	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	47	16	no	—
199	C₂×C₄	—	94	48	no	—
200	C₂×C₄	—	94	128	no	—
201	C₂×C₄	C₂×#10625354	1	48	yes	—
202	C₂×C₄	C₂⁴×D₄₇	1	16	yes	—
203	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
204	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
205	C₂²	—	94	16	no	—
206	C₂²	—	94	16	no	—
207	C₂²	—	94	16	no	—
208	C₂²	—	188	96	no	—
209	C₂²	#15689839	1	16	yes	direct product factor
210	C₂²	C₂²×#10625354	1	12	yes	—
211	C₂²	C₂×#13094069	1	6	yes	—
212	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
213	C₄	—	2	16	no	—
214	C₄	—	2	4	no	—
215	C₄	—	94	16	no	—
216	C₄	C₂²×#10625354	1	16	yes	—
217	C₂	—	188	16	no	—
218	C₂	C₂×#15689839	1	12	yes	direct product factor
219	C₂	C₂³×#10625354	1	2	yes	—
220	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	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×#555
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×#15689839	C₂	direct (N × H)
2. C₂²×#13094013	C₂	semidirect
3. C₂²×#13093999	C₂	semidirect
4. C₂²×C₄×D₁₈₈	C₂	semidirect
5. C₂²×#13093982	C₂	semidirect
6. C₂²×C₄⋊Q₈×C₄₇	C₂	semidirect
10. #15689839	C₂²	direct (N × H)
11. C₂×#13094013	C₂²	semidirect
12. C₂×#13093999	C₂²	semidirect
13. C₂×C₄×D₁₈₈	C₂²	semidirect
14. C₂×#13093982	C₂²	semidirect
20. C₂×C₄⋊Q₈×C₄₇	C₂²	semidirect
31. #13094013	C₂³	semidirect
32. #13093999	C₂³	semidirect
33. C₄×D₁₈₈	C₂³	semidirect
34. #13093982	C₂³	semidirect
44. C₄⋊Q₈×C₄₇	C₂³	semidirect
95. C₂²×C₄₇	#555	semidirect
108. C₂²×C₄⋊Q₈	D₄₇	semidirect
113. C₂×C₄₇	C₂×#555	semidirect
122. C₂×C₄⋊Q₈	C₂×D₄₇	semidirect
133. C₄₇	C₂²×#555	semidirect
147. C₄⋊Q₈	C₂²×D₄₇	semidirect
209. C₂²	#15689839	direct (N × H)
218. C₂	C₂×#15689839	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×#15689839	2
C₂²×#13094013	2
C₂²×#13093999	2
C₂²×C₄×D₁₈₈	2
C₂²×#13093982	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂²×#555	47

Subgroups — C22×#15689839 #27123298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689839 #27123298