Subgroups — C₂×C₂₉×#547 #16817178

Order 3712.

1632 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal, complemented
2	C₂₉×#547	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₉×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₉×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₉×#121	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄⋊C₄×C₂₉	C₂	1	2	yes	maximal, complemented
10	C₂×#41×C₂₉	—	2	16	no	—
11	C₂×#41×C₂₉	C₂²	1	16	yes	complemented
12	C₂²×Q₈×C₂₉	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₂₉	C₂²	1	4	yes	—
15	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₉	C₂²	1	2	yes	—
17	C₂³×C₄×C₂₉	C₂²	1	4	yes	—
18	C₂³×C₄×C₂₉	C₂²	1	2	yes	—
19	C₂₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
20	C₂₉×#123	C₂²	1	32	yes	complemented
21	C₂₉×C₂²⋊Q₈	C₂²	1	16	yes	complemented
22	C₂₉×#121	C₂²	1	16	yes	complemented
23	C₄×D₄×C₂₉	C₂²	1	8	yes	complemented
24	C₄×D₄×C₂₉	C₂²	1	8	yes	complemented
25	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₂₉	C₂²	1	4	yes	—
28	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₂₉	C₂²	1	16	yes	complemented
31	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₂₉	C₂²	1	4	yes	—
34	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
35	C₂⁴×C₂₉	C₂³	1	2	yes	—
36	C₂⁴×C₂₉	C₂³	1	2	yes	—
37	#41×C₂₉	—	2	64	no	—
38	C₂×Q₈×C₂₉	—	2	8	no	—
39	C₂×Q₈×C₂₉	C₂³	1	8	yes	—
40	C₂×D₄×C₂₉	—	2	16	no	—
41	C₂×D₄×C₂₉	—	2	4	no	—
42	C₂×D₄×C₂₉	D₄	1	8	yes	complemented
43	C₂×D₄×C₂₉	C₂³	1	4	yes	—
44	C₂×D₄×C₂₉	C₂³	1	16	yes	—
45	C₂×D₄×C₂₉	C₂³	1	4	yes	—
46	C₂²×C₄×C₂₉	—	2	16	no	—
47	C₂²×C₄×C₂₉	—	2	8	no	—
48	C₂²×C₄×C₂₉	—	2	8	no	—
49	C₂²×C₄×C₂₉	C₂³	1	8	yes	—
50	C₂²×C₄×C₂₉	C₂³	1	16	yes	—
51	C₂²×C₄×C₂₉	C₂³	1	8	yes	—
52	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
56	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
59	C₄⋊C₄×C₂₉	C₂³	1	8	yes	—
60	C₄⋊C₄×C₂₉	C₂³	1	16	yes	—
61	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
62	C₄⋊C₄×C₂₉	C₂³	1	8	yes	—
63	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
64	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
65	C₂²⋊C₄×C₂₉	C₂³	1	16	yes	—
66	C₄²×C₂₉	C₂³	1	4	yes	—
67	C₂³×C₂₉	—	2	4	no	—
68	C₂³×C₂₉	—	2	4	no	—
69	C₂³×C₂₉	—	2	4	no	—
70	C₂³×C₂₉	—	2	4	no	—
71	C₂³×C₂₉	—	2	4	no	—
72	C₂³×C₂₉	C₂×D₄	1	8	yes	—
73	C₂³×C₂₉	C₂⁴	1	4	yes	—
74	C₂³×C₂₉	C₂⁴	1	4	yes	—
75	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
76	Q₈×C₂₉	—	2	16	no	—
77	D₄×C₂₉	—	2	32	no	—
78	D₄×C₂₉	—	2	8	no	—
79	D₄×C₂₉	C₂×D₄	1	16	yes	complemented
80	C₂×C₄×C₂₉	—	2	4	no	—
81	C₂×C₄×C₂₉	—	2	8	no	—
82	C₂×C₄×C₂₉	—	2	32	no	—
83	C₂×C₄×C₂₉	—	2	4	no	—
84	C₂×C₄×C₂₉	—	2	2	no	—
85	C₂×C₄×C₂₉	—	2	16	no	—
86	C₂×C₄×C₂₉	—	2	16	no	—
87	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₂₉	#41	1	4	yes	complemented
89	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
91	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
92	C₂×C₄×C₂₉	C₂⁴	1	8	yes	—
93	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
94	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
95	C₂×#547	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₂₉	—	2	16	no	—
97	C₂²×C₂₉	—	2	4	no	—
98	C₂²×C₂₉	—	2	4	no	—
99	C₂²×C₂₉	—	2	4	no	—
100	C₂²×C₂₉	—	2	4	no	—
101	C₂²×C₂₉	—	2	4	no	—
102	C₂²×C₂₉	—	4	8	no	—
103	C₂²×C₂₉	C₂²×D₄	1	8	yes	—
104	C₂²×C₂₉	C₂²×D₄	1	2	yes	—
105	C₂²×C₂₉	C₂×#41	1	2	yes	—
106	C₂²×C₂₉	2₋¹⁺⁴	1	2	yes	—
107	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
108	C₄×C₂₉	—	2	4	no	—
109	C₄×C₂₉	—	2	8	no	—
110	C₄×C₂₉	—	2	4	no	—
111	C₄×C₂₉	—	2	2	no	—
112	C₄×C₂₉	C₂²×D₄	1	4	yes	—
113	C₄×C₂₉	C₂×#41	1	4	yes	complemented
114	C₂²×#41	C₂×C₂₉	1	2	yes	complemented
115	#547	C₂×C₂₉	1	16	yes	direct product factor
116	C₂×C₄⋊Q₈	C₂×C₂₉	1	1	yes	characteristic, complemented
117	C₂×#123	C₂×C₂₉	1	4	yes	complemented
118	C₂×C₂²⋊Q₈	C₂×C₂₉	1	2	yes	complemented
119	C₂×#121	C₂×C₂₉	1	2	yes	complemented
120	C₂×C₄×D₄	C₂×C₂₉	1	1	yes	characteristic, complemented
121	C₂×C₄×D₄	C₂×C₂₉	1	1	yes	characteristic, complemented
122	C₂²×C₄⋊C₄	C₂×C₂₉	1	2	yes	complemented
123	C₂×C₂₉	—	2	8	no	—
124	C₂×C₂₉	—	4	4	no	—
125	C₂×C₂₉	#547	1	4	yes	direct product factor
126	C₂×C₂₉	C₂³×D₄	1	1	yes	characteristic
127	C₂×C₂₉	C₂²×#41	1	1	yes	characteristic
128	C₂×C₂₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×#41	—	2	16	no	—
130	C₂×#41	C₂²×C₂₉	1	16	yes	complemented
131	C₂²×Q₈	C₂²×C₂₉	1	2	yes	—
132	C₂²×D₄	C₂²×C₂₉	1	1	yes	characteristic
133	C₂²×D₄	C₂²×C₂₉	1	4	yes	—
134	C₂²×D₄	C₂²×C₂₉	1	1	yes	characteristic
135	C₂³×C₄	C₂²×C₂₉	1	2	yes	—
136	C₂³×C₄	C₂²×C₂₉	1	4	yes	—
137	C₂³×C₄	C₂²×C₂₉	1	2	yes	—
138	C₄⋊Q₈	C₂²×C₂₉	1	8	yes	complemented
139	#123	C₂²×C₂₉	1	32	yes	complemented
140	C₂²⋊Q₈	C₂²×C₂₉	1	16	yes	complemented
141	#121	C₂²×C₂₉	1	16	yes	complemented
142	C₄×D₄	C₂²×C₂₉	1	8	yes	complemented
143	C₄×D₄	C₂²×C₂₉	1	8	yes	complemented
144	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₉	1	4	yes	—
147	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₂₉	1	16	yes	complemented
150	C₂×C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₂₉	1	4	yes	—
153	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
154	C₂₉	C₂×#547	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	C₂³×C₂₉	1	2	yes	—
156	C₂⁴	C₂³×C₂₉	1	2	yes	—
157	#41	—	2	64	no	—
158	C₂×Q₈	—	2	8	no	—
159	C₂×Q₈	C₂³×C₂₉	1	8	yes	—
160	C₂×D₄	—	2	16	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	D₄×C₂₉	1	8	yes	complemented
163	C₂×D₄	C₂³×C₂₉	1	4	yes	—
164	C₂×D₄	C₂³×C₂₉	1	16	yes	—
165	C₂×D₄	C₂³×C₂₉	1	4	yes	—
166	C₂²×C₄	—	2	16	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
171	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
172	C₂²×C₄	C₂³×C₂₉	1	4	yes	—
173	C₂²×C₄	C₂³×C₂₉	1	8	yes	—
174	C₂²×C₄	C₂³×C₂₉	1	16	yes	—
175	C₂²×C₄	C₂³×C₂₉	1	8	yes	—
176	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
179	C₄⋊C₄	C₂³×C₂₉	1	8	yes	—
180	C₄⋊C₄	C₂³×C₂₉	1	16	yes	—
181	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
182	C₄⋊C₄	C₂³×C₂₉	1	8	yes	—
183	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
184	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
185	C₂²⋊C₄	C₂³×C₂₉	1	16	yes	—
186	C₄²	C₂³×C₂₉	1	4	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×D₄×C₂₉	1	8	yes	—
193	C₂³	C₂⁴×C₂₉	1	4	yes	—
194	C₂³	C₂⁴×C₂₉	1	4	yes	—
195	C₂³	C₂⁴×C₂₉	1	1	yes	characteristic
196	Q₈	—	2	16	no	—
197	D₄	—	2	32	no	—
198	D₄	—	2	8	no	—
199	D₄	C₂×D₄×C₂₉	1	16	yes	complemented
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	32	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	C₂×D₄×C₂₉	1	4	yes	—
208	C₂×C₄	#41×C₂₉	1	4	yes	complemented
209	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
210	C₂×C₄	C₂⁴×C₂₉	1	4	yes	—
211	C₂×C₄	C₂⁴×C₂₉	1	4	yes	—
212	C₂×C₄	C₂⁴×C₂₉	1	8	yes	—
213	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
214	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
215	C₂²	—	2	16	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	4	8	no	—
222	C₂²	C₂²×D₄×C₂₉	1	8	yes	—
223	C₂²	C₂²×D₄×C₂₉	1	2	yes	—
224	C₂²	C₂×#41×C₂₉	1	2	yes	—
225	C₂²	C₂₉×2₋¹⁺⁴	1	2	yes	—
226	C₂²	C₂⁵×C₂₉	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	8	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	C₂²×D₄×C₂₉	1	4	yes	—
232	C₄	C₂×#41×C₂₉	1	4	yes	complemented
233	C₂	—	2	8	no	—
234	C₂	—	4	4	no	—
235	C₂	C₂₉×#547	1	4	yes	direct product factor
236	C₂	C₂³×D₄×C₂₉	1	1	yes	characteristic
237	C₂	C₂²×#41×C₂₉	1	1	yes	characteristic
238	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. C₂₉×#547	C₂	direct (N × H)
3. C₂×C₂₉×C₄⋊Q₈	C₂	semidirect
4. C₂×C₂₉×#123	C₂	semidirect
5. C₂×C₂₉×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₂₉×#121	C₂	semidirect
7. C₂×C₄×D₄×C₂₉	C₂	semidirect
8. C₂×C₄×D₄×C₂₉	C₂	semidirect
9. C₂²×C₄⋊C₄×C₂₉	C₂	semidirect
11. C₂×#41×C₂₉	C₂²	semidirect
19. C₂₉×C₄⋊Q₈	C₂²	semidirect
20. C₂₉×#123	C₂²	semidirect
21. C₂₉×C₂²⋊Q₈	C₂²	semidirect
22. C₂₉×#121	C₂²	semidirect
23. C₄×D₄×C₂₉	C₂²	semidirect
24. C₄×D₄×C₂₉	C₂²	semidirect
30. C₂×C₄⋊C₄×C₂₉	C₂²	semidirect
42. C₂×D₄×C₂₉	D₄	semidirect
79. D₄×C₂₉	C₂×D₄	semidirect
88. C₂×C₄×C₂₉	#41	semidirect
95. C₂×#547	C₂₉	direct (N × H)
113. C₄×C₂₉	C₂×#41	semidirect
114. C₂²×#41	C₂×C₂₉	semidirect
115. #547	C₂×C₂₉	direct (N × H)
116. C₂×C₄⋊Q₈	C₂×C₂₉	semidirect
117. C₂×#123	C₂×C₂₉	semidirect
118. C₂×C₂²⋊Q₈	C₂×C₂₉	semidirect
119. C₂×#121	C₂×C₂₉	semidirect
120. C₂×C₄×D₄	C₂×C₂₉	semidirect
121. C₂×C₄×D₄	C₂×C₂₉	semidirect
122. C₂²×C₄⋊C₄	C₂×C₂₉	semidirect
125. C₂×C₂₉	#547	direct (N × H)
130. C₂×#41	C₂²×C₂₉	semidirect
138. C₄⋊Q₈	C₂²×C₂₉	semidirect
139. #123	C₂²×C₂₉	semidirect
140. C₂²⋊Q₈	C₂²×C₂₉	semidirect
141. #121	C₂²×C₂₉	semidirect
142. C₄×D₄	C₂²×C₂₉	semidirect
143. C₄×D₄	C₂²×C₂₉	semidirect
149. C₂×C₄⋊C₄	C₂²×C₂₉	semidirect
154. C₂₉	C₂×#547	direct (N × H)
162. C₂×D₄	D₄×C₂₉	semidirect
199. D₄	C₂×D₄×C₂₉	semidirect
208. C₂×C₄	#41×C₂₉	semidirect
232. C₄	C₂×#41×C₂₉	semidirect
235. C₂	C₂₉×#547	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
C₂₉×#547	2
C₂×C₂₉×C₄⋊Q₈	2
C₂×C₂₉×#123	2
C₂×C₂₉×C₂²⋊Q₈	2
C₂×C₂₉×#121	2
C₂×C₄×D₄×C₂₉	2
C₂²×C₄⋊C₄×C₂₉	2
C₂×#547	29

Subgroups — C2×C29×#547 #16817178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#547 #16817178