Subgroups — C₂²×Q₈×Dic₁₂₂ #30587080

Order 15616.

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×Q₈×Dic₁₂₂	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14519777	C₂	1	3	yes	maximal
3	C₂²×#14519776	C₂	1	6	yes	maximal
4	C₂²×Q₈×Dic₆₁	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₁₂₂	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₂	C₂²	1	6	yes	—
8	C₂³×Dic₁₂₂	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₂₂	C₂²	1	256	yes	direct product factor
10	C₂×#14519777	C₂²	1	72	yes	—
11	C₂×#14519776	C₂²	1	144	yes	—
12	C₂×Q₈×Dic₆₁	C₂²	1	48	yes	—
13	C₂×C₄×Dic₁₂₂	C₂²	1	72	yes	—
14	C₂²×C₄⋊Dic₆₁	C₂²	1	3	yes	—
15	C₂²×C₄⋊Dic₆₁	C₂²	1	6	yes	—
16	C₂²×#11758196	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₆₁	C₂²	1	6	yes	—
18	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₁	C₂²	1	24	yes	—
20	C₂²×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
21	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
22	C₂²×Dic₁₂₂	—	4	24	no	—
23	C₂²×Dic₁₂₂	Q₈	1	16	yes	direct product factor
24	C₂²×Dic₁₂₂	C₂³	1	72	yes	—
25	C₂²×Dic₁₂₂	C₂³	1	12	yes	—
26	C₂³×Dic₆₁	C₂³	1	6	yes	—
27	C₂³×Dic₆₁	C₂³	1	2	yes	—
28	#14519777	C₂³	1	192	yes	—
29	#14519776	C₂³	1	384	yes	—
30	Q₈×Dic₆₁	C₂³	1	128	yes	—
31	C₄×Dic₁₂₂	C₂³	1	192	yes	—
32	C₂×C₄⋊Dic₆₁	C₂³	1	36	yes	—
33	C₂×C₄⋊Dic₆₁	C₂³	1	72	yes	—
34	C₂×#11758196	C₂³	1	72	yes	—
35	C₂×C₄×Dic₆₁	C₂³	1	72	yes	—
36	C₂²×Q₈×C₆₁	Q₈	1	16	yes	complemented
37	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
38	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
39	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
40	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₆₁	C₂³	1	64	yes	—
42	C₂×C₄⋊C₄×C₆₁	C₂³	1	36	yes	—
43	C₂×C₄²×C₆₁	C₂³	1	36	yes	—
44	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₆₁	C₂×Q₈	1	96	yes	complemented
46	C₂×Q₈×C₆₁	C₂⁴	1	16	yes	—
47	C₂²×C₄×C₆₁	—	2	12	no	—
48	C₂²×C₄×C₆₁	C₂×Q₈	1	24	yes	—
49	C₂²×C₄×C₆₁	C₂×Q₈	1	8	yes	—
50	C₂²×C₄×C₆₁	C₂⁴	1	18	yes	—
51	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
52	C₂²×C₄×C₆₁	C₂⁴	1	18	yes	—
53	C₂²×Dic₆₁	—	2	24	no	—
54	C₂²×Dic₆₁	C₂×Q₈	1	16	yes	—
55	C₂²×Dic₆₁	C₂⁴	1	36	yes	—
56	C₂²×Dic₆₁	C₂⁴	1	12	yes	—
57	C₂×Dic₁₂₂	—	4	144	no	—
58	C₂×Dic₁₂₂	C₂×Q₈	1	96	yes	direct product factor
59	C₂×Dic₁₂₂	C₂⁴	1	96	yes	—
60	C₂×Dic₁₂₂	C₂⁴	1	16	yes	—
61	C₄⋊C₄×C₆₁	C₂⁴	1	48	yes	—
62	C₄²×C₆₁	C₂⁴	1	48	yes	—
63	C₄⋊Dic₆₁	C₂⁴	1	48	yes	—
64	C₄⋊Dic₆₁	C₂⁴	1	96	yes	—
65	#11758196	C₂⁴	1	96	yes	—
66	C₄×Dic₆₁	C₂⁴	1	96	yes	—
67	C₂³×C₆₁	C₂²×Q₈	1	4	yes	—
68	C₂³×C₆₁	C₂²×Q₈	1	4	yes	—
69	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
70	C₂³×C₆₁	C₂⁵	1	3	yes	—
71	Q₈×C₆₁	C₂²×Q₈	1	64	yes	complemented
72	C₂×C₄×C₆₁	—	2	36	no	—
73	C₂×C₄×C₆₁	C₂²×Q₈	1	24	yes	—
74	C₂×C₄×C₆₁	C₂²×Q₈	1	72	yes	—
75	C₂×C₄×C₆₁	C₂⁵	1	12	yes	—
76	C₂×C₄×C₆₁	C₂⁵	1	12	yes	—
77	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
78	C₂×Dic₆₁	—	2	72	no	—
79	C₂×Dic₆₁	C₂²×Q₈	1	48	yes	—
80	C₂×Dic₆₁	C₂⁵	1	24	yes	—
81	C₂×Dic₆₁	C₂⁵	1	8	yes	—
82	Dic₁₂₂	—	4	96	no	—
83	Dic₁₂₂	C₂²×Q₈	1	64	yes	direct product factor
84	C₂²×Q₈²	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	Q₈²	1	16	yes	complemented
86	C₂²×C₆₁	C₂³×Q₈	1	6	yes	—
87	C₂²×C₆₁	C₂³×Q₈	1	6	yes	—
88	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	6	yes	—
89	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
90	C₄×C₆₁	—	2	12	no	—
91	C₄×C₆₁	C₂³×Q₈	1	8	yes	—
92	C₄×C₆₁	C₂³×Q₈	1	24	yes	—
93	Dic₆₁	—	2	24	no	—
94	Dic₆₁	C₂³×Q₈	1	16	yes	—
95	C₂×Q₈²	—	61	48	no	—
96	C₂²×C₄⋊Q₈	—	61	3	no	—
97	C₂²×C₄⋊Q₈	—	61	6	no	—
98	C₂²×C₄×Q₈	—	61	3	no	—
99	C₂²×C₄×Q₈	—	61	2	no	—
100	C₂²×C₄×Q₈	D₆₁	1	1	yes	characteristic
101	C₂×C₆₁	C₂×Q₈²	1	12	yes	complemented
102	C₂×C₆₁	C₂⁴×Q₈	1	1	yes	characteristic
103	C₂×C₆₁	C₂⁴×Q₈	1	1	yes	characteristic
104	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
105	C₂³×Q₈	—	61	1	no	—
106	C₂³×Q₈	—	61	6	no	—
107	C₂³×Q₈	C₂×D₆₁	1	1	yes	characteristic
108	Q₈²	—	61	256	no	—
109	C₂×C₄⋊Q₈	—	61	144	no	—
110	C₂×C₄⋊Q₈	—	61	72	no	—
111	C₂×C₄×Q₈	—	61	48	no	—
112	C₂×C₄×Q₈	—	61	72	no	—
113	C₂×C₄×Q₈	C₂×D₆₁	1	24	yes	—
114	C₂²×C₄⋊C₄	—	61	6	no	—
115	C₂²×C₄⋊C₄	—	61	6	no	—
116	C₂²×C₄⋊C₄	—	61	3	no	—
117	C₂²×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
118	C₂²×C₄²	—	61	6	no	—
119	C₂²×C₄²	C₂×D₆₁	1	3	yes	—
120	C₆₁	C₂²×Q₈²	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	61	16	no	—
122	C₂²×Q₈	—	61	12	no	—
123	C₂²×Q₈	—	61	72	no	—
124	C₂²×Q₈	—	244	24	no	—
125	C₂²×Q₈	Dic₁₂₂	1	16	yes	direct product factor
126	C₂²×Q₈	C₂²×D₆₁	1	12	yes	—
127	C₂³×C₄	—	61	2	no	—
128	C₂³×C₄	—	61	6	no	—
129	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
130	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
131	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
132	C₄⋊Q₈	—	61	384	no	—
133	C₄⋊Q₈	—	61	192	no	—
134	C₄×Q₈	—	61	128	no	—
135	C₄×Q₈	—	61	192	no	—
136	C₄×Q₈	C₂²×D₆₁	1	64	yes	—
137	C₂×C₄⋊C₄	—	61	72	no	—
138	C₂×C₄⋊C₄	—	61	36	no	—
139	C₂×C₄⋊C₄	—	61	72	no	—
140	C₂×C₄⋊C₄	C₂²×D₆₁	1	36	yes	—
141	C₂×C₄²	—	61	72	no	—
142	C₂×C₄²	C₂²×D₆₁	1	36	yes	—
143	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
144	C₂×Q₈	—	61	96	no	—
145	C₂×Q₈	—	61	16	no	—
146	C₂×Q₈	—	61	96	no	—
147	C₂×Q₈	—	244	144	no	—
148	C₂×Q₈	C₂×Dic₁₂₂	1	96	yes	direct product factor
149	C₂×Q₈	C₂³×D₆₁	1	16	yes	—
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	—	61	16	no	—
152	C₂²×C₄	—	61	12	no	—
153	C₂²×C₄	—	61	36	no	—
154	C₂²×C₄	—	122	24	no	—
155	C₂²×C₄	C₂×Dic₁₂₂	1	24	yes	—
156	C₂²×C₄	Q₈×D₆₁	1	8	yes	—
157	C₂²×C₄	C₂³×D₆₁	1	18	yes	—
158	C₂²×C₄	C₂³×D₆₁	1	6	yes	—
159	C₂²×C₄	C₂³×D₆₁	1	18	yes	—
160	C₄⋊C₄	—	61	96	no	—
161	C₄⋊C₄	—	61	48	no	—
162	C₄⋊C₄	—	61	96	no	—
163	C₄⋊C₄	C₂³×D₆₁	1	48	yes	—
164	C₄²	—	61	96	no	—
165	C₄²	C₂³×D₆₁	1	48	yes	—
166	C₂³	C₂×Q₈×D₆₁	1	4	yes	—
167	C₂³	C₂²×Dic₁₂₂	1	4	yes	—
168	C₂³	#14519823	1	4	yes	—
169	C₂³	C₂⁴×D₆₁	1	3	yes	—
170	Q₈	—	61	64	no	—
171	Q₈	—	244	96	no	—
172	Q₈	C₂²×Dic₁₂₂	1	64	yes	direct product factor
173	C₂×C₄	—	2	36	no	—
174	C₂×C₄	—	61	48	no	—
175	C₂×C₄	—	61	8	no	—
176	C₂×C₄	—	61	24	no	—
177	C₂×C₄	—	122	72	no	—
178	C₂×C₄	C₂×Q₈×D₆₁	1	24	yes	—
179	C₂×C₄	C₂²×Dic₁₂₂	1	72	yes	—
180	C₂×C₄	C₂⁴×D₆₁	1	12	yes	—
181	C₂×C₄	C₂⁴×D₆₁	1	4	yes	—
182	C₂×C₄	C₂⁴×D₆₁	1	12	yes	—
183	C₂²	Q₈×Dic₁₂₂	1	16	yes	direct product factor
184	C₂²	C₂²×Q₈×D₆₁	1	6	yes	—
185	C₂²	C₂³×Dic₁₂₂	1	6	yes	—
186	C₂²	C₂×#14519823	1	6	yes	—
187	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
188	C₄	—	2	12	no	—
189	C₄	—	61	16	no	—
190	C₄	—	122	24	no	—
191	C₄	C₂²×Q₈×D₆₁	1	8	yes	—
192	C₄	C₂³×Dic₁₂₂	1	24	yes	—
193	C₂	C₂×Q₈×Dic₁₂₂	1	12	yes	direct product factor
194	C₂	C₂³×Q₈×D₆₁	1	1	yes	characteristic
195	C₂	C₂⁴×Dic₁₂₂	1	1	yes	characteristic
196	C₂	C₂²×#14519823	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator, cosocle	C₂²×C₆₁	C₂⁶
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂²×C₄×Q₈×C₆₁	C₂
socle, prime radical	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₆₁
hypocenter	C₆₁	C₂²×Q₈²
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₁₂₂	C₂	direct (N × H)
9. Q₈×Dic₁₂₂	C₂²	direct (N × H)
23. C₂²×Dic₁₂₂	Q₈	direct (N × H)
36. C₂²×Q₈×C₆₁	Q₈	semidirect
45. C₂×Q₈×C₆₁	C₂×Q₈	semidirect
58. C₂×Dic₁₂₂	C₂×Q₈	direct (N × H)
71. Q₈×C₆₁	C₂²×Q₈	semidirect
83. Dic₁₂₂	C₂²×Q₈	direct (N × H)
85. C₂²×C₆₁	Q₈²	semidirect
101. C₂×C₆₁	C₂×Q₈²	semidirect
120. C₆₁	C₂²×Q₈²	semidirect
125. C₂²×Q₈	Dic₁₂₂	direct (N × H)
148. C₂×Q₈	C₂×Dic₁₂₂	direct (N × H)
172. Q₈	C₂²×Dic₁₂₂	direct (N × H)
183. C₂²	Q₈×Dic₁₂₂	direct (N × H)
193. C₂	C₂×Q₈×Dic₁₂₂	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×Q₈×Dic₁₂₂	2
C₂²×#14519777	2
C₂²×#14519776	2
C₂²×Q₈×Dic₆₁	2
C₂²×C₄×Dic₁₂₂	2
C₂²×C₄×Q₈×C₆₁	2
C₂²×Q₈²	61

Subgroups — C22×Q8×Dic122 #30587080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×Dic₁₂₂ #30587080