Subgroups — C₄⋊C₄×Dic₁₂₂ #21448734

Order 7808.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₂₂	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
3	#16839660	C₂	1	2	yes	maximal
4	#16839659	C₂	1	4	yes	maximal
5	#16839632	C₂	1	1	yes	maximal, characteristic
6	#16839631	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₆₁	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₂₂	—	2	8	no	—
11	C₄×Dic₁₂₂	C₄	1	8	yes	complemented
12	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×#11758196	C₂²	1	4	yes	—
17	C₂×#11758196	C₂²	1	4	yes	—
18	C₂×#11758196	C₂²	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂²	1	4	yes	—
20	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
21	#14519684	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
26	#95×C₆₁	C₂²	1	2	yes	—
27	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×Dic₆₁	C₂³	1	4	yes	—
33	C₂²×Dic₆₁	C₂³	1	2	yes	—
34	C₂²×Dic₆₁	C₂³	1	2	yes	—
35	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
36	C₂×Dic₁₂₂	D₄	1	4	yes	—
37	C₂×Dic₁₂₂	Q₈	1	4	yes	—
38	C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₄⋊C₄×C₆₁	Q₈	1	4	yes	complemented
40	C₄²×C₆₁	—	2	4	no	—
41	C₄²×C₆₁	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₆₁	—	2	4	no	—
43	C₄⋊Dic₆₁	—	2	4	no	—
44	C₄⋊Dic₆₁	—	4	2	no	—
45	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
46	#11758196	—	2	8	no	—
47	#11758196	—	4	4	no	—
48	#11758196	C₂×C₄	1	8	yes	—
49	C₄×Dic₆₁	—	2	8	no	—
50	C₄×Dic₆₁	C₂×C₄	1	8	yes	—
51	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₆₁	#41	1	2	yes	—
67	C₂×Dic₆₁	—	2	2	no	—
68	C₂×Dic₆₁	—	2	2	no	—
69	C₂×Dic₆₁	—	2	4	no	—
70	C₂×Dic₆₁	—	2	4	no	—
71	C₂×Dic₆₁	—	2	4	no	—
72	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
73	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
74	C₂×Dic₆₁	C₂×D₄	1	4	yes	—
75	C₂×Dic₆₁	C₂×Q₈	1	4	yes	—
76	Dic₁₂₂	C₄⋊C₄	1	16	yes	direct product factor
77	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
78	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
82	C₂²×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	2	4	no	—
86	C₄×C₆₁	—	4	2	no	—
87	C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₄×C₆₁	C₄×Q₈	1	4	yes	complemented
89	Dic₆₁	—	2	4	no	—
90	Dic₆₁	—	4	4	no	—
91	Dic₆₁	C₂×C₄⋊C₄	1	8	yes	—
92	Q₈×C₄⋊C₄	—	61	1	no	maximal, Sylow
93	C₂×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₂×C₆₁	#520	1	1	yes	characteristic
96	C₂×C₆₁	D₄×Q₈	1	1	yes	characteristic
97	C₂×C₆₁	#550	1	1	yes	characteristic
98	C₂×C₆₁	#557	1	1	yes	characteristic
99	C₂×C₆₁	Q₈²	1	1	yes	characteristic
100	C₂×C₄×Q₈	—	61	1	no	—
101	C₂×C₄×Q₈	—	61	2	no	—
102	#389	—	61	2	no	—
103	#389	—	61	4	no	—
104	#384	—	61	2	no	—
105	#384	—	61	1	no	—
106	C₄×C₄⋊C₄	—	61	2	no	—
107	C₄×C₄⋊C₄	D₆₁	1	1	yes	characteristic
108	C₆₁	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
109	C₂²×Q₈	—	61	1	no	—
110	C₄×Q₈	—	61	8	no	—
111	C₄×Q₈	—	122	8	no	—
112	C₂×C₄⋊C₄	—	61	2	no	—
113	C₂×C₄⋊C₄	—	61	1	no	—
114	C₂×C₄⋊C₄	—	61	4	no	—
115	C₂×C₄⋊C₄	—	61	2	no	—
116	C₂×C₄⋊C₄	—	61	2	no	—
117	C₂×C₄⋊C₄	—	61	4	no	—
118	C₂×C₄⋊C₄	—	61	2	no	—
119	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
121	C₂×C₄²	—	61	2	no	—
122	C₂×C₄²	—	61	4	no	—
123	C₂×C₄²	C₂×D₆₁	1	2	yes	—
124	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
125	#95	—	61	4	no	—
126	#95	C₂×D₆₁	1	2	yes	—
127	C₂×Q₈	—	61	4	no	—
128	C₂×Q₈	—	61	4	no	—
129	C₂×Q₈	—	61	4	no	—
130	C₂²×C₄	—	61	2	no	—
131	C₂²×C₄	—	61	2	no	—
132	C₂²×C₄	—	61	4	no	—
133	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
134	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
135	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	61	8	no	—
140	C₄⋊C₄	—	61	4	no	—
141	C₄⋊C₄	—	122	8	no	—
142	C₄⋊C₄	—	122	4	no	—
143	C₄⋊C₄	—	122	4	no	—
144	C₄⋊C₄	—	244	4	no	—
145	C₄⋊C₄	—	244	2	no	—
146	C₄⋊C₄	Dic₁₂₂	1	4	yes	direct product factor
147	C₄²	—	2	4	no	—
148	C₄²	—	61	8	no	—
149	C₄²	—	122	8	no	—
150	C₄²	C₄×D₆₁	1	4	yes	—
151	C₂³	C₂³×D₆₁	1	1	yes	characteristic
152	Q₈	—	61	16	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	61	4	no	—
161	C₂×C₄	—	61	4	no	—
162	C₂×C₄	—	61	4	no	—
163	C₂×C₄	—	61	4	no	—
164	C₂×C₄	—	122	2	no	—
165	C₂×C₄	—	122	2	no	—
166	C₂×C₄	—	122	4	no	—
167	C₂×C₄	—	122	4	no	—
168	C₂×C₄	—	122	4	no	—
169	C₂×C₄	C₂×Dic₁₂₂	1	4	yes	—
170	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
173	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
174	C₂×C₄	#11758222	1	2	yes	—
175	C₂×C₄	D₄×D₆₁	1	2	yes	—
176	C₂×C₄	Q₈×D₆₁	1	2	yes	—
177	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
179	C₂²	C₂×Q₈×D₆₁	1	1	yes	characteristic
180	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
181	C₂²	C₂×#11758222	1	1	yes	characteristic
182	C₂²	#14519823	1	1	yes	characteristic
183	C₂²	#14519824	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	—	61	8	no	—
188	C₄	—	122	4	no	—
189	C₄	—	244	4	no	—
190	C₄	C₄⋊C₄×D₆₁	1	4	yes	—
191	C₄	C₄×Dic₁₂₂	1	4	yes	complemented
192	C₂	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
193	C₂	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
194	C₂	#16840242	1	1	yes	characteristic
195	C₂	D₄×Dic₁₂₂	1	1	yes	characteristic
196	C₂	#16840433	1	1	yes	characteristic
197	C₂	#16840490	1	1	yes	characteristic
198	C₂	Q₈×Dic₁₂₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₆₁	C₂
socle	C₂³×C₆₁	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₆₁
hypocenter	C₆₁	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₁₂₂	C₄	semidirect
39. C₄⋊C₄×C₆₁	Q₈	semidirect
76. Dic₁₂₂	C₄⋊C₄	direct (N × H)
88. C₄×C₆₁	C₄×Q₈	semidirect
108. C₆₁	Q₈×C₄⋊C₄	semidirect
146. C₄⋊C₄	Dic₁₂₂	direct (N × H)
191. C₄	C₄×Dic₁₂₂	semidirect

Subgroups — C4⋊C4×Dic122 #21448734

Conjugacy classes of subgroups

Complements

Subgroups — C₄⋊C₄×Dic₁₂₂ #21448734