Subgroups — C₄×C₂₉×#135 #16816653

Order 3712.

500 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₄×#41×C₂₉	C₂	1	2	yes	maximal, complemented
3	C₄×Q₁₆×C₂₉	C₂	1	2	yes	maximal, complemented
4	C₄×SD₁₆×C₂₉	C₂	1	4	yes	maximal, complemented
5	C₄×D₈×C₂₉	C₂	1	2	yes	maximal, complemented
6	C₂₉×#427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#416	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₉	C₂²	1	2	yes	—
10	C₂₉×#135	C₄	1	8	yes	direct product factor
11	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₂₉	C₂²	1	2	yes	—
13	C₂×D₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₂₉	C₂²	1	4	yes	—
17	C₄×D₄×C₂₉	—	2	4	no	—
18	C₄×D₄×C₂₉	C₂²	1	4	yes	—
19	C₂₉×#117	—	2	2	no	—
20	C₂₉×#117	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₉	—	2	2	no	—
22	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
23	C₂₉×#107	C₂²	1	2	yes	—
24	C₂₉×#106	C₂²	1	2	yes	—
25	C₂₉×#103	C₂²	1	4	yes	—
26	C₂₉×#102	C₂²	1	4	yes	—
27	C₄×C₈×C₂₉	C₂²	1	2	yes	—
28	C₄×C₈×C₂₉	C₂²	1	2	yes	—
29	#41×C₂₉	—	2	4	no	—
30	#41×C₂₉	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
32	C₂×D₄×C₂₉	—	2	2	no	—
33	C₂×D₄×C₂₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₉	—	2	2	no	—
35	C₂²×C₄×C₂₉	—	2	4	no	—
36	C₂²×C₄×C₂₉	D₄	1	2	yes	—
37	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
38	Q₁₆×C₂₉	C₂×C₄	1	4	yes	complemented
39	SD₁₆×C₂₉	C₂×C₄	1	8	yes	complemented
40	D₈×C₂₉	C₂×C₄	1	4	yes	complemented
41	C₂×C₈×C₂₉	—	2	2	no	—
42	C₂×C₈×C₂₉	C₂×C₄	1	4	yes	complemented
43	C₂×C₈×C₂₉	C₂³	1	2	yes	—
44	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₉	—	2	4	no	—
47	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₂₉	—	2	2	no	—
49	C₂²⋊C₄×C₂₉	—	2	4	no	—
50	C₄²×C₂₉	—	2	2	no	—
51	C₄²×C₂₉	—	2	4	no	—
52	C₄²×C₂₉	D₄	1	1	yes	characteristic
53	C₄²×C₂₉	D₄	1	1	yes	characteristic
54	C₄²×C₂₉	C₂³	1	2	yes	—
55	C₂³×C₂₉	—	2	2	no	—
56	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₂₉	—	2	2	no	—
58	Q₈×C₂₉	C₂²×C₄	1	4	yes	—
59	D₄×C₂₉	—	2	8	no	—
60	D₄×C₂₉	—	2	2	no	—
61	D₄×C₂₉	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₂₉	—	2	2	no	—
63	C₂×C₄×C₂₉	—	2	8	no	—
64	C₂×C₄×C₂₉	—	2	4	no	—
65	C₂×C₄×C₂₉	—	2	4	no	—
66	C₂×C₄×C₂₉	—	4	8	no	—
67	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₂₉	#41	1	2	yes	—
72	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
73	C₈×C₂₉	—	2	2	no	—
74	C₈×C₂₉	C₂²×C₄	1	2	yes	—
75	C₈×C₂₉	C₂²×C₄	1	2	yes	—
76	C₄×#135	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₂₉	—	2	2	no	—
78	C₂²×C₂₉	—	2	4	no	—
79	C₂²×C₂₉	—	4	4	no	—
80	C₂²×C₂₉	C₄×D₄	1	2	yes	—
81	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₂₉	—	2	4	no	—
83	C₄×C₂₉	—	2	2	no	—
84	C₄×C₂₉	—	4	4	no	—
85	C₄×C₂₉	C₄×D₄	1	2	yes	—
86	C₄×C₂₉	#135	1	4	yes	direct product factor
87	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
89	C₂×#135	C₂×C₂₉	1	1	yes	characteristic
90	C₄×#41	C₂×C₂₉	1	2	yes	complemented
91	C₄×Q₁₆	C₂×C₂₉	1	2	yes	complemented
92	C₄×SD₁₆	C₂×C₂₉	1	4	yes	complemented
93	C₄×D₈	C₂×C₂₉	1	2	yes	complemented
94	#427	C₂×C₂₉	1	1	yes	characteristic, complemented
95	#416	C₂×C₂₉	1	2	yes	complemented
96	C₂×C₄×C₈	C₂×C₂₉	1	1	yes	characteristic, complemented
97	C₂×C₂₉	—	2	2	no	—
98	C₂×C₂₉	—	4	4	no	—
99	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₂₉	C₂×#135	1	2	yes	—
101	C₂×#41	C₂²×C₂₉	1	2	yes	—
102	#135	C₄×C₂₉	1	8	yes	direct product factor
103	C₂×Q₁₆	C₂²×C₂₉	1	1	yes	characteristic
104	C₂×SD₁₆	C₂²×C₂₉	1	2	yes	—
105	C₂×D₈	C₂²×C₂₉	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₂₉	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₂₉	1	1	yes	characteristic
108	C₄×Q₈	C₂²×C₂₉	1	4	yes	—
109	C₄×D₄	—	2	4	no	—
110	C₄×D₄	C₂²×C₂₉	1	4	yes	—
111	#117	—	2	2	no	—
112	#117	C₂²×C₂₉	1	2	yes	—
113	C₂×C₄²	—	2	2	no	—
114	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
115	#107	C₂²×C₂₉	1	2	yes	—
116	#106	C₂²×C₂₉	1	2	yes	—
117	#103	C₂²×C₂₉	1	4	yes	—
118	#102	C₂²×C₂₉	1	4	yes	—
119	C₄×C₈	C₂²×C₂₉	1	2	yes	—
120	C₄×C₈	C₂²×C₂₉	1	2	yes	—
121	C₂₉	C₄×#135	1	1	yes	characteristic, direct product factor, Sylow
122	#41	—	2	4	no	—
123	#41	C₂×C₄×C₂₉	1	8	yes	complemented
124	C₂×Q₈	C₂³×C₂₉	1	2	yes	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	C₂³×C₂₉	1	2	yes	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	D₄×C₂₉	1	2	yes	—
130	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
131	Q₁₆	C₂×C₄×C₂₉	1	4	yes	complemented
132	SD₁₆	C₂×C₄×C₂₉	1	8	yes	complemented
133	D₈	C₂×C₄×C₂₉	1	4	yes	complemented
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	C₂×C₄×C₂₉	1	4	yes	complemented
136	C₂×C₈	C₂³×C₂₉	1	2	yes	—
137	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
138	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	4	no	—
145	C₄²	D₄×C₂₉	1	1	yes	characteristic
146	C₄²	D₄×C₂₉	1	1	yes	characteristic
147	C₄²	C₂³×C₂₉	1	2	yes	—
148	C₂³	—	2	2	no	—
149	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
150	Q₈	—	2	2	no	—
151	Q₈	C₂²×C₄×C₂₉	1	4	yes	—
152	D₄	—	2	8	no	—
153	D₄	—	2	2	no	—
154	D₄	C₂²×C₄×C₂₉	1	4	yes	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
161	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
162	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
163	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
164	C₂×C₄	#41×C₂₉	1	2	yes	—
165	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	C₂²×C₄×C₂₉	1	2	yes	—
168	C₈	C₂²×C₄×C₂₉	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	4	4	no	—
172	C₂²	C₄×D₄×C₂₉	1	2	yes	—
173	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
174	C₄	—	2	4	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	4	no	—
177	C₄	C₄×D₄×C₂₉	1	2	yes	—
178	C₄	C₂₉×#135	1	4	yes	direct product factor
179	C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
180	C₄	C₂×#41×C₂₉	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	—	4	4	no	—
183	C₂	C₂×C₄×D₄×C₂₉	1	1	yes	characteristic
184	C₂	C₂×C₂₉×#135	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₉	D₄
commutator	C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×C₂₉×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#41×C₂₉	C₂	semidirect
3. C₄×Q₁₆×C₂₉	C₂	semidirect
4. C₄×SD₁₆×C₂₉	C₂	semidirect
5. C₄×D₈×C₂₉	C₂	semidirect
6. C₂₉×#427	C₂	semidirect
7. C₂₉×#416	C₂	semidirect
8. C₂×C₄×C₈×C₂₉	C₂	semidirect
10. C₂₉×#135	C₄	direct (N × H)
30. #41×C₂₉	C₂×C₄	semidirect
38. Q₁₆×C₂₉	C₂×C₄	semidirect
39. SD₁₆×C₂₉	C₂×C₄	semidirect
40. D₈×C₂₉	C₂×C₄	semidirect
42. C₂×C₈×C₂₉	C₂×C₄	semidirect
76. C₄×#135	C₂₉	direct (N × H)
86. C₄×C₂₉	#135	direct (N × H)
90. C₄×#41	C₂×C₂₉	semidirect
91. C₄×Q₁₆	C₂×C₂₉	semidirect
92. C₄×SD₁₆	C₂×C₂₉	semidirect
93. C₄×D₈	C₂×C₂₉	semidirect
94. #427	C₂×C₂₉	semidirect
95. #416	C₂×C₂₉	semidirect
96. C₂×C₄×C₈	C₂×C₂₉	semidirect
102. #135	C₄×C₂₉	direct (N × H)
121. C₂₉	C₄×#135	direct (N × H)
123. #41	C₂×C₄×C₂₉	semidirect
131. Q₁₆	C₂×C₄×C₂₉	semidirect
132. SD₁₆	C₂×C₄×C₂₉	semidirect
133. D₈	C₂×C₄×C₂₉	semidirect
135. C₂×C₈	C₂×C₄×C₂₉	semidirect
178. C₄	C₂₉×#135	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×C₂₉×#135	2
C₄×#41×C₂₉	2
C₄×Q₁₆×C₂₉	2
C₄×SD₁₆×C₂₉	2
C₄×D₈×C₂₉	2
C₂₉×#427	2
C₂₉×#416	2
C₂×C₄×C₈×C₂₉	2
C₄×#135	29

Subgroups — C4×C29×#135 #16816653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₉×#135 #16816653