Subgroups — C₄×#17984807 #31808753

Order 17792.

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×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984807	C₂	1	2	yes	maximal
3	C₄⋊C₄×Dic₁₃₉	C₂	1	2	yes	maximal
4	C₄×Dic₁₃₉⋊C₄	C₂	1	2	yes	maximal
5	#22580804	C₂	1	4	yes	maximal
6	C₄²×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₁₃₉	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₇₈	C₄	1	8	yes	complemented
11	#17984807	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
14	C₂×Dic₁₃₉⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
19	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
22	C₁₃₉⋊#95	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
26	#95×C₁₃₉	C₂²	1	2	yes	—
27	C₂×Dic₂₇₈	C₂×C₄	1	8	yes	—
28	C₂×Dic₂₇₈	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
30	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
31	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
32	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
33	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
34	Dic₁₃₉⋊C₄	C₂×C₄	1	16	yes	—
35	Dic₁₃₉⋊C₄	C₂×C₄	1	8	yes	—
36	C₄×Dic₁₃₉	—	2	4	no	—
37	C₄×Dic₁₃₉	—	2	8	no	—
38	C₄×Dic₁₃₉	—	2	2	no	—
39	C₄×Dic₁₃₉	C₂×C₄	1	16	yes	—
40	C₄×Dic₁₃₉	C₂×C₄	1	8	yes	—
41	C₄×Dic₁₃₉	C₂×C₄	1	8	yes	—
42	C₄×Dic₁₃₉	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	—
48	C₄²×C₁₃₉	—	2	4	no	—
49	C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
50	Dic₂₇₈	C₄²	1	16	yes	complemented
51	C₂×Dic₁₃₉	—	2	2	no	—
52	C₂×Dic₁₃₉	—	2	4	no	—
53	C₂×Dic₁₃₉	—	2	2	no	—
54	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
55	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
56	C₂×Dic₁₃₉	C₂²×C₄	1	8	yes	—
57	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
59	C₂×Dic₁₃₉	C₂×Q₈	1	4	yes	—
60	C₂×Dic₁₃₉	#41	1	4	yes	—
61	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₁₃₉	—	2	4	no	—
63	C₂×C₄×C₁₃₉	—	2	2	no	—
64	C₂×C₄×C₁₃₉	—	2	4	no	—
65	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
67	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₁₃₉	#41	1	2	yes	—
72	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
73	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₁₃₉	C₂×#41	1	2	yes	—
76	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
77	C₄×C₁₃₉	—	2	8	no	—
78	C₄×C₁₃₉	—	2	2	no	—
79	C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
80	C₄×C₁₃₉	C₄×Q₈	1	4	yes	complemented
81	Dic₁₃₉	—	2	4	no	—
82	Dic₁₃₉	—	2	4	no	—
83	Dic₁₃₉	C₂×C₄²	1	8	yes	—
84	Dic₁₃₉	C₄×Q₈	1	8	yes	complemented
85	C₂×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
86	C₂×C₁₃₉	C₂×C₄×Q₈	1	2	yes	—
87	C₂×C₁₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₃₉	C₄×#41	1	2	yes	—
89	C₂×C₁₃₉	C₄×#41	1	1	yes	characteristic
90	C₁₃₉	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
91	C₄²×Q₈	—	139	1	no	maximal, Sylow
92	C₂×C₄×Q₈	—	139	2	no	—
93	C₂×C₄×Q₈	—	139	1	no	—
94	C₄×C₄⋊C₄	—	139	2	no	—
95	C₄×C₄⋊C₄	—	139	2	no	—
96	C₄×C₄⋊C₄	—	139	4	no	—
97	C₄×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic
98	C₄³	—	139	1	no	—
99	C₄³	—	139	2	no	—
100	C₂²×Q₈	—	139	1	no	—
101	C₄×Q₈	—	139	16	no	—
102	C₄×Q₈	—	139	8	no	—
103	C₂×C₄⋊C₄	—	139	4	no	—
104	C₂×C₄⋊C₄	—	139	2	no	—
105	C₂×C₄⋊C₄	—	139	1	no	—
106	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
107	C₂×C₄²	—	139	4	no	—
108	C₂×C₄²	—	139	2	no	—
109	C₂×C₄²	—	139	1	no	—
110	C₂×C₄²	—	139	2	no	—
111	C₂×C₄²	—	139	4	no	—
112	C₂×C₄²	—	139	2	no	—
113	C₂×C₄²	—	139	1	no	—
114	C₂×C₄²	C₂×D₁₃₉	1	2	yes	—
115	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
116	#95	—	139	4	no	—
117	#95	C₂×D₁₃₉	1	2	yes	—
118	C₂×Q₈	—	139	8	no	—
119	C₂×Q₈	—	139	4	no	—
120	C₂²×C₄	—	139	2	no	—
121	C₂²×C₄	—	139	2	no	—
122	C₂²×C₄	—	139	2	no	—
123	C₂²×C₄	—	139	2	no	—
124	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
128	C₄⋊C₄	—	139	16	no	—
129	C₄⋊C₄	—	139	8	no	—
130	C₄⋊C₄	—	139	4	no	—
131	C₄⋊C₄	C₄×D₁₃₉	1	8	yes	—
132	C₄²	—	2	4	no	—
133	C₄²	—	278	8	no	—
134	C₄²	—	278	4	no	—
135	C₄²	—	278	2	no	—
136	C₄²	—	139	8	no	—
137	C₄²	—	139	8	no	—
138	C₄²	—	139	16	no	—
139	C₄²	—	139	4	no	—
140	C₄²	C₄×D₁₃₉	1	4	yes	—
141	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
142	Q₈	—	139	16	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	278	2	no	—
147	C₂×C₄	—	278	4	no	—
148	C₂×C₄	—	278	2	no	—
149	C₂×C₄	—	139	4	no	—
150	C₂×C₄	—	139	4	no	—
151	C₂×C₄	—	139	4	no	—
152	C₂×C₄	—	139	8	no	—
153	C₂×C₄	—	139	4	no	—
154	C₂×C₄	—	139	4	no	—
155	C₂×C₄	—	139	4	no	—
156	C₂×C₄	C₂×C₄×D₁₃₉	1	8	yes	—
157	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
159	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
161	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
162	C₂×C₄	#14555784	1	2	yes	—
163	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
164	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
165	C₂²	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
166	C₂²	#41×D₁₃₉	1	2	yes	—
167	C₂²	C₂×#14555784	1	1	yes	characteristic
168	C₄	—	2	8	no	—
169	C₄	—	2	2	no	—
170	C₄	—	278	4	no	—
171	C₄	—	278	4	no	—
172	C₄	—	139	8	no	—
173	C₄	—	139	8	no	—
174	C₄	C₄²×D₁₃₉	1	4	yes	—
175	C₄	#17984807	1	4	yes	direct product factor
176	C₂	C₂×C₄²×D₁₃₉	1	1	yes	characteristic
177	C₂	C₄×Q₈×D₁₃₉	1	2	yes	—
178	C₂	C₂×#17984807	1	1	yes	characteristic
179	C₂	C₄×#14555784	1	2	yes	—
180	C₂	#22581421	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂×C₁₃₉	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₃₉
Fitting	C₄×C₄⋊C₄×C₁₃₉	C₂
socle, prime radical, cosocle	C₂³×C₁₃₉	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	C₄²×Q₈
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
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×Dic₂₇₈	C₄	semidirect
11. #17984807	C₄	direct (N × H)
42. C₄×Dic₁₃₉	Q₈	semidirect
50. Dic₂₇₈	C₄²	semidirect
80. C₄×C₁₃₉	C₄×Q₈	semidirect
84. Dic₁₃₉	C₄×Q₈	semidirect
90. C₁₃₉	C₄²×Q₈	semidirect
175. C₄	#17984807	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×Dic₂₇₈	2
C₂×#17984807	2
C₄⋊C₄×Dic₁₃₉	2
C₄×Dic₁₃₉⋊C₄	2
#22580804	2
C₄²×Dic₁₃₉	2
C₄×C₄⋊C₄×C₁₃₉	2
C₄²×Q₈	139

Subgroups — C4×#17984807 #31808753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984807 #31808753