Subgroups — C₂×C₂₉×#548 #16817179

Order 3712.

1620 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#548	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₉×#124	C₂	1	3	yes	maximal, complemented
5	C₂×C₂₉×C₂²⋊Q₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₉×#121	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₉	C₂	1	3	yes	maximal, complemented
9	C₂×#41×C₂₉	—	2	8	no	—
10	C₂×#41×C₂₉	C₂²	1	8	yes	complemented
11	C₂²×Q₈×C₂₉	—	2	8	no	—
12	C₂²×Q₈×C₂₉	C₂²	1	3	yes	—
13	C₂²×Q₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₉	C₂²	1	8	yes	complemented
15	C₂²×D₄×C₂₉	C₂²	1	3	yes	—
16	C₂²×D₄×C₂₉	C₂²	1	3	yes	—
17	C₂³×C₄×C₂₉	C₂²	1	3	yes	—
18	C₂³×C₄×C₂₉	C₂²	1	3	yes	—
19	C₂₉×#124	C₂²	1	24	yes	complemented
20	C₂₉×C₂²⋊Q₈	C₂²	1	24	yes	complemented
21	C₂₉×#121	C₂²	1	24	yes	complemented
22	C₄×Q₈×C₂₉	C₂²	1	8	yes	complemented
23	C₄×D₄×C₂₉	C₂²	1	24	yes	complemented
24	C₂×C₄⋊C₄×C₂₉	C₂²	1	3	yes	—
25	C₂×C₄⋊C₄×C₂₉	C₂²	1	3	yes	—
26	C₂×C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₂₉	C₂²	1	3	yes	—
28	C₂×C₂²⋊C₄×C₂₉	C₂²	1	6	yes	—
29	C₂×C₄²×C₂₉	C₂²	1	3	yes	—
30	C₂⁴×C₂₉	C₂³	1	3	yes	—
31	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
32	#41×C₂₉	—	2	32	no	—
33	C₂×Q₈×C₂₉	—	2	32	no	—
34	C₂×Q₈×C₂₉	—	2	12	no	—
35	C₂×Q₈×C₂₉	D₄	1	8	yes	complemented
36	C₂×Q₈×C₂₉	C₂³	1	12	yes	—
37	C₂×Q₈×C₂₉	C₂³	1	4	yes	—
38	C₂×D₄×C₂₉	—	2	12	no	—
39	C₂×D₄×C₂₉	—	2	12	no	—
40	C₂×D₄×C₂₉	C₂³	1	12	yes	—
41	C₂×D₄×C₂₉	C₂³	1	12	yes	—
42	C₂²×C₄×C₂₉	—	2	12	no	—
43	C₂²×C₄×C₂₉	—	2	12	no	—
44	C₂²×C₄×C₂₉	C₂³	1	12	yes	—
45	C₂²×C₄×C₂₉	C₂³	1	12	yes	—
46	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
48	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
49	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
51	C₄⋊C₄×C₂₉	C₂³	1	12	yes	—
52	C₄⋊C₄×C₂₉	C₂³	1	12	yes	—
53	C₂²⋊C₄×C₂₉	C₂³	1	4	yes	—
54	C₂²⋊C₄×C₂₉	C₂³	1	12	yes	—
55	C₂²⋊C₄×C₂₉	C₂³	1	24	yes	—
56	C₄²×C₂₉	C₂³	1	12	yes	—
57	C₂³×C₂₉	—	2	2	no	—
58	C₂³×C₂₉	—	2	2	no	—
59	C₂³×C₂₉	—	2	6	no	—
60	C₂³×C₂₉	—	2	6	no	—
61	C₂³×C₂₉	—	2	6	no	—
62	C₂³×C₂₉	#41	1	4	yes	complemented
63	C₂³×C₂₉	C₂⁴	1	6	yes	—
64	C₂³×C₂₉	C₂⁴	1	2	yes	—
65	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
66	Q₈×C₂₉	—	2	24	no	—
67	Q₈×C₂₉	C₂×D₄	1	16	yes	complemented
68	D₄×C₂₉	—	2	24	no	—
69	D₄×C₂₉	—	2	24	no	—
70	C₂×C₄×C₂₉	—	2	2	no	—
71	C₂×C₄×C₂₉	—	2	6	no	—
72	C₂×C₄×C₂₉	—	2	6	no	—
73	C₂×C₄×C₂₉	—	2	2	no	—
74	C₂×C₄×C₂₉	—	2	24	no	—
75	C₂×C₄×C₂₉	—	2	24	no	—
76	C₂×C₄×C₂₉	C₂×D₄	1	12	yes	—
77	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
78	C₂×C₄×C₂₉	C₂⁴	1	6	yes	—
79	C₂×C₄×C₂₉	C₂⁴	1	6	yes	—
80	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
81	C₂×C₄×C₂₉	C₂⁴	1	6	yes	—
82	C₂×#548	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₂₉	—	2	8	no	—
84	C₂²×C₂₉	—	2	2	no	—
85	C₂²×C₂₉	—	2	2	no	—
86	C₂²×C₂₉	—	2	6	no	—
87	C₂²×C₂₉	—	2	6	no	—
88	C₂²×C₂₉	—	2	6	no	—
89	C₂²×C₂₉	—	4	12	no	—
90	C₂²×C₂₉	C₂²×D₄	1	2	yes	—
91	C₂²×C₂₉	C₂×#41	1	2	yes	—
92	C₂²×C₂₉	C₂×#41	1	4	yes	complemented
93	C₂²×C₂₉	2₋¹⁺⁴	1	2	yes	—
94	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
95	C₄×C₂₉	—	2	2	no	—
96	C₄×C₂₉	—	2	6	no	—
97	C₄×C₂₉	—	2	6	no	—
98	C₄×C₂₉	—	2	2	no	—
99	C₄×C₂₉	C₂²×D₄	1	12	yes	—
100	C₂²×#41	C₂×C₂₉	1	1	yes	characteristic, complemented
101	C₂³×Q₈	C₂×C₂₉	1	1	yes	characteristic, complemented
102	#548	C₂×C₂₉	1	16	yes	direct product factor
103	C₂×#124	C₂×C₂₉	1	3	yes	complemented
104	C₂×C₂²⋊Q₈	C₂×C₂₉	1	3	yes	complemented
105	C₂×#121	C₂×C₂₉	1	3	yes	complemented
106	C₂×C₄×Q₈	C₂×C₂₉	1	1	yes	characteristic, complemented
107	C₂×C₄×D₄	C₂×C₂₉	1	3	yes	complemented
108	C₂×C₂₉	—	2	4	no	—
109	C₂×C₂₉	—	4	6	no	—
110	C₂×C₂₉	#548	1	4	yes	direct product factor
111	C₂×C₂₉	C₂³×D₄	1	1	yes	characteristic
112	C₂×C₂₉	C₂²×#41	1	1	yes	characteristic
113	C₂×C₂₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×#41	—	2	8	no	—
115	C₂×#41	C₂²×C₂₉	1	8	yes	complemented
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	C₂²×C₂₉	1	3	yes	—
118	C₂²×Q₈	C₂²×C₂₉	1	1	yes	characteristic
119	C₂²×Q₈	C₂²×C₂₉	1	8	yes	complemented
120	C₂²×D₄	C₂²×C₂₉	1	3	yes	—
121	C₂²×D₄	C₂²×C₂₉	1	3	yes	—
122	C₂³×C₄	C₂²×C₂₉	1	3	yes	—
123	C₂³×C₄	C₂²×C₂₉	1	3	yes	—
124	#124	C₂²×C₂₉	1	24	yes	complemented
125	C₂²⋊Q₈	C₂²×C₂₉	1	24	yes	complemented
126	#121	C₂²×C₂₉	1	24	yes	complemented
127	C₄×Q₈	C₂²×C₂₉	1	8	yes	complemented
128	C₄×D₄	C₂²×C₂₉	1	24	yes	complemented
129	C₂×C₄⋊C₄	C₂²×C₂₉	1	3	yes	—
130	C₂×C₄⋊C₄	C₂²×C₂₉	1	3	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	C₂²×C₂₉	1	3	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₂₉	1	6	yes	—
134	C₂×C₄²	C₂²×C₂₉	1	3	yes	—
135	C₂₉	C₂×#548	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	C₂³×C₂₉	1	3	yes	—
137	C₂⁴	C₂³×C₂₉	1	1	yes	characteristic
138	#41	—	2	32	no	—
139	C₂×Q₈	—	2	32	no	—
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	D₄×C₂₉	1	8	yes	complemented
142	C₂×Q₈	C₂³×C₂₉	1	12	yes	—
143	C₂×Q₈	C₂³×C₂₉	1	4	yes	—
144	C₂×D₄	—	2	12	no	—
145	C₂×D₄	—	2	12	no	—
146	C₂×D₄	C₂³×C₂₉	1	12	yes	—
147	C₂×D₄	C₂³×C₂₉	1	12	yes	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	2	12	no	—
150	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₂₉	1	3	yes	—
152	C₂²×C₄	C₂³×C₂₉	1	3	yes	—
153	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₂₉	1	12	yes	—
155	C₂²×C₄	C₂³×C₂₉	1	12	yes	—
156	C₂²×C₄	C₂³×C₂₉	1	3	yes	—
157	C₄⋊C₄	C₂³×C₂₉	1	12	yes	—
158	C₄⋊C₄	C₂³×C₂₉	1	12	yes	—
159	C₂²⋊C₄	C₂³×C₂₉	1	4	yes	—
160	C₂²⋊C₄	C₂³×C₂₉	1	12	yes	—
161	C₂²⋊C₄	C₂³×C₂₉	1	24	yes	—
162	C₄²	C₂³×C₂₉	1	12	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	6	no	—
166	C₂³	—	2	6	no	—
167	C₂³	—	2	6	no	—
168	C₂³	#41×C₂₉	1	4	yes	complemented
169	C₂³	C₂⁴×C₂₉	1	6	yes	—
170	C₂³	C₂⁴×C₂₉	1	2	yes	—
171	C₂³	C₂⁴×C₂₉	1	1	yes	characteristic
172	Q₈	—	2	24	no	—
173	Q₈	C₂×D₄×C₂₉	1	16	yes	complemented
174	D₄	—	2	24	no	—
175	D₄	—	2	24	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	24	no	—
181	C₂×C₄	—	2	24	no	—
182	C₂×C₄	C₂×D₄×C₂₉	1	12	yes	—
183	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
184	C₂×C₄	C₂⁴×C₂₉	1	6	yes	—
185	C₂×C₄	C₂⁴×C₂₉	1	6	yes	—
186	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
187	C₂×C₄	C₂⁴×C₂₉	1	6	yes	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	6	no	—
192	C₂²	—	2	6	no	—
193	C₂²	—	2	6	no	—
194	C₂²	—	4	12	no	—
195	C₂²	C₂²×D₄×C₂₉	1	2	yes	—
196	C₂²	C₂×#41×C₂₉	1	2	yes	—
197	C₂²	C₂×#41×C₂₉	1	4	yes	complemented
198	C₂²	C₂₉×2₋¹⁺⁴	1	2	yes	—
199	C₂²	C₂⁵×C₂₉	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	6	no	—
202	C₄	—	2	6	no	—
203	C₄	—	2	2	no	—
204	C₄	C₂²×D₄×C₂₉	1	12	yes	—
205	C₂	—	2	4	no	—
206	C₂	—	4	6	no	—
207	C₂	C₂₉×#548	1	4	yes	direct product factor
208	C₂	C₂³×D₄×C₂₉	1	1	yes	characteristic
209	C₂	C₂²×#41×C₂₉	1	1	yes	characteristic
210	C₂	C₂×C₂₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₂₉	C₂	semidirect
2. C₂³×Q₈×C₂₉	C₂	semidirect
3. C₂₉×#548	C₂	direct (N × H)
4. C₂×C₂₉×#124	C₂	semidirect
5. C₂×C₂₉×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₂₉×#121	C₂	semidirect
7. C₂×C₄×Q₈×C₂₉	C₂	semidirect
8. C₂×C₄×D₄×C₂₉	C₂	semidirect
10. C₂×#41×C₂₉	C₂²	semidirect
14. C₂²×Q₈×C₂₉	C₂²	semidirect
19. C₂₉×#124	C₂²	semidirect
20. C₂₉×C₂²⋊Q₈	C₂²	semidirect
21. C₂₉×#121	C₂²	semidirect
22. C₄×Q₈×C₂₉	C₂²	semidirect
23. C₄×D₄×C₂₉	C₂²	semidirect
35. C₂×Q₈×C₂₉	D₄	semidirect
62. C₂³×C₂₉	#41	semidirect
67. Q₈×C₂₉	C₂×D₄	semidirect
82. C₂×#548	C₂₉	direct (N × H)
92. C₂²×C₂₉	C₂×#41	semidirect
100. C₂²×#41	C₂×C₂₉	semidirect
101. C₂³×Q₈	C₂×C₂₉	semidirect
102. #548	C₂×C₂₉	direct (N × H)
103. C₂×#124	C₂×C₂₉	semidirect
104. C₂×C₂²⋊Q₈	C₂×C₂₉	semidirect
105. C₂×#121	C₂×C₂₉	semidirect
106. C₂×C₄×Q₈	C₂×C₂₉	semidirect
107. C₂×C₄×D₄	C₂×C₂₉	semidirect
110. C₂×C₂₉	#548	direct (N × H)
115. C₂×#41	C₂²×C₂₉	semidirect
119. C₂²×Q₈	C₂²×C₂₉	semidirect
124. #124	C₂²×C₂₉	semidirect
125. C₂²⋊Q₈	C₂²×C₂₉	semidirect
126. #121	C₂²×C₂₉	semidirect
127. C₄×Q₈	C₂²×C₂₉	semidirect
128. C₄×D₄	C₂²×C₂₉	semidirect
135. C₂₉	C₂×#548	direct (N × H)
141. C₂×Q₈	D₄×C₂₉	semidirect
168. C₂³	#41×C₂₉	semidirect
173. Q₈	C₂×D₄×C₂₉	semidirect
197. C₂²	C₂×#41×C₂₉	semidirect
207. C₂	C₂₉×#548	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#41×C₂₉	2
C₂³×Q₈×C₂₉	2
C₂₉×#548	2
C₂×C₂₉×#124	2
C₂×C₂₉×C₂²⋊Q₈	2
C₂×C₂₉×#121	2
C₂×C₄×Q₈×C₂₉	2
C₂×C₄×D₄×C₂₉	2
C₂×#548	29

Subgroups — C2×C29×#548 #16817179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#548 #16817179