Subgroups — C₄×#15692220 #27173481

Order 12928.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692220	C₂	1	2	yes	maximal
3	C₄⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal
4	C₄×#13108053	C₂	1	2	yes	maximal
5	#20265028	C₂	1	4	yes	maximal
6	C₄²×Dic₁₀₁	C₂	1	2	yes	maximal
7	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₀₂	C₄	1	8	yes	complemented
11	#15692220	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×#13108053	C₂²	1	2	yes	—
14	C₂×#13108053	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
19	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
22	#15692162	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
26	#95×C₁₀₁	C₂²	1	2	yes	—
27	C₂×Dic₂₀₂	C₂×C₄	1	8	yes	—
28	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
30	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
31	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
32	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
33	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
34	#13108053	C₂×C₄	1	16	yes	—
35	#13108053	C₂×C₄	1	8	yes	—
36	C₄×Dic₁₀₁	—	2	2	no	—
37	C₄×Dic₁₀₁	—	2	4	no	—
38	C₄×Dic₁₀₁	—	2	8	no	—
39	C₄×Dic₁₀₁	C₂×C₄	1	8	yes	—
40	C₄×Dic₁₀₁	C₂×C₄	1	16	yes	—
41	C₄×Dic₁₀₁	C₂×C₄	1	8	yes	—
42	C₄×Dic₁₀₁	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	—
48	C₄²×C₁₀₁	—	2	4	no	—
49	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
50	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₁₀₁	—	2	4	no	—
52	C₂×C₄×C₁₀₁	—	2	2	no	—
53	C₂×C₄×C₁₀₁	—	2	4	no	—
54	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
56	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₀₁	#41	1	2	yes	—
61	C₂×Dic₁₀₁	—	2	2	no	—
62	C₂×Dic₁₀₁	—	2	4	no	—
63	C₂×Dic₁₀₁	—	2	2	no	—
64	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
65	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
67	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
69	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
70	C₂×Dic₁₀₁	#41	1	4	yes	—
71	Dic₂₀₂	C₄²	1	16	yes	complemented
72	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
73	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
76	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
77	C₄×C₁₀₁	—	2	8	no	—
78	C₄×C₁₀₁	—	2	2	no	—
79	C₄×C₁₀₁	C₂×C₄²	1	4	yes	—
80	C₄×C₁₀₁	C₄×Q₈	1	4	yes	complemented
81	Dic₁₀₁	—	2	4	no	—
82	Dic₁₀₁	—	2	4	no	—
83	Dic₁₀₁	C₂×C₄²	1	8	yes	—
84	Dic₁₀₁	C₄×Q₈	1	8	yes	complemented
85	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
86	C₂×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
87	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₀₁	C₄×#41	1	2	yes	—
89	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
90	C₄²×Q₈	—	101	1	no	maximal, Sylow
91	C₁₀₁	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄×Q₈	—	101	2	no	—
93	C₂×C₄×Q₈	—	101	1	no	—
94	C₄×C₄⋊C₄	—	101	2	no	—
95	C₄×C₄⋊C₄	—	101	2	no	—
96	C₄×C₄⋊C₄	—	101	4	no	—
97	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
98	C₄³	—	101	2	no	—
99	C₄³	—	101	1	no	—
100	C₂²×Q₈	—	101	1	no	—
101	C₄×Q₈	—	101	16	no	—
102	C₄×Q₈	—	101	8	no	—
103	C₂×C₄⋊C₄	—	101	4	no	—
104	C₂×C₄⋊C₄	—	101	2	no	—
105	C₂×C₄⋊C₄	—	101	1	no	—
106	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
107	C₂×C₄²	—	101	4	no	—
108	C₂×C₄²	—	101	2	no	—
109	C₂×C₄²	—	101	2	no	—
110	C₂×C₄²	—	101	1	no	—
111	C₂×C₄²	—	101	4	no	—
112	C₂×C₄²	—	101	2	no	—
113	C₂×C₄²	—	101	1	no	—
114	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
115	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
116	#95	—	101	4	no	—
117	#95	C₂×D₁₀₁	1	2	yes	—
118	C₂×Q₈	—	101	8	no	—
119	C₂×Q₈	—	101	4	no	—
120	C₂²×C₄	—	101	2	no	—
121	C₂²×C₄	—	101	2	no	—
122	C₂²×C₄	—	101	2	no	—
123	C₂²×C₄	—	101	2	no	—
124	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
128	C₄⋊C₄	—	101	16	no	—
129	C₄⋊C₄	—	101	8	no	—
130	C₄⋊C₄	—	101	4	no	—
131	C₄⋊C₄	C₄×D₁₀₁	1	8	yes	—
132	C₄²	—	2	4	no	—
133	C₄²	—	101	16	no	—
134	C₄²	—	101	8	no	—
135	C₄²	—	101	8	no	—
136	C₄²	—	101	4	no	—
137	C₄²	—	202	8	no	—
138	C₄²	—	202	4	no	—
139	C₄²	—	202	2	no	—
140	C₄²	C₄×D₁₀₁	1	4	yes	—
141	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
142	Q₈	—	101	16	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	101	4	no	—
147	C₂×C₄	—	101	8	no	—
148	C₂×C₄	—	101	4	no	—
149	C₂×C₄	—	101	4	no	—
150	C₂×C₄	—	101	4	no	—
151	C₂×C₄	—	101	4	no	—
152	C₂×C₄	—	101	4	no	—
153	C₂×C₄	—	202	2	no	—
154	C₂×C₄	—	202	4	no	—
155	C₂×C₄	—	202	2	no	—
156	C₂×C₄	C₂×C₄×D₁₀₁	1	8	yes	—
157	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
159	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
161	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
162	C₂×C₄	#13108073	1	2	yes	—
163	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
164	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
165	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
166	C₂²	#41×D₁₀₁	1	2	yes	—
167	C₂²	C₂×#13108073	1	1	yes	characteristic
168	C₄	—	2	8	no	—
169	C₄	—	2	2	no	—
170	C₄	—	101	8	no	—
171	C₄	—	101	8	no	—
172	C₄	—	202	4	no	—
173	C₄	—	202	4	no	—
174	C₄	C₄²×D₁₀₁	1	4	yes	—
175	C₄	#15692220	1	4	yes	direct product factor
176	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
177	C₂	C₄×Q₈×D₁₀₁	1	2	yes	—
178	C₂	C₂×#15692220	1	1	yes	characteristic
179	C₂	C₄×#13108073	1	2	yes	—
180	C₂	#20265645	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂×C₁₀₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₄×C₄⋊C₄×C₁₀₁	C₂
socle, prime radical, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₄²×Q₈
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₀₂	C₄	semidirect
11. #15692220	C₄	direct (N × H)
42. C₄×Dic₁₀₁	Q₈	semidirect
71. Dic₂₀₂	C₄²	semidirect
80. C₄×C₁₀₁	C₄×Q₈	semidirect
84. Dic₁₀₁	C₄×Q₈	semidirect
91. C₁₀₁	C₄²×Q₈	semidirect
175. C₄	#15692220	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×C₄×Dic₂₀₂	2
C₂×#15692220	2
C₄⋊C₄×Dic₁₀₁	2
C₄×#13108053	2
#20265028	2
C₄²×Dic₁₀₁	2
C₄×C₄⋊C₄×C₁₀₁	2
C₄²×Q₈	101

Subgroups — C4×#15692220 #27173481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692220 #27173481