Subgroups — #16828193

Order 3712.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#90772	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271670	C₂	1	1	yes	maximal, characteristic
3	#14271239	C₂	1	4	yes	maximal, complemented
4	#14271234	C₂	1	4	yes	maximal, complemented
5	#14271191	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271130	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271126	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271118	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
12	C₂₉×#131	—	2	4	no	—
13	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₉	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₉	C₂²	1	2	yes	complemented
17	#11745308	C₄	1	2	yes	—
18	#11745300	C₄	1	2	yes	—
19	#11745299	C₄	1	2	yes	—
20	#11745297	C₄	1	2	yes	—
21	C₄×C₂²⋊D₂₉	C₂²	1	4	yes	—
22	#11745288	C₂²	1	1	yes	characteristic
23	C₂×#90777	C₂²	1	1	yes	characteristic
24	C₂²×#6597	C₂²	1	1	yes	characteristic
25	C₂×#90772	—	2	4	no	—
26	C₂×#90772	C₂²	1	2	yes	—
27	C₂×#90772	C₂²	1	2	yes	—
28	C₂²⋊#6597	C₂²	1	4	yes	complemented
29	#11745168	C₂²	1	4	yes	—
30	#11745163	C₂²	1	2	yes	complemented
31	#11745162	C₂²	1	4	yes	complemented
32	C₈×Dic₂₉	C₂²	1	2	yes	complemented
33	C₂³×D₂₉	C₂×C₄	1	1	yes	characteristic
34	#41×C₂₉	—	2	2	no	—
35	#41×C₂₉	—	2	2	no	—
36	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₉	C₂×C₄	1	4	yes	—
42	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₂₉	—	2	2	no	—
44	C₂×C₄×D₂₉	—	2	2	no	—
45	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₉	C₂³	1	2	yes	—
47	M₁₆×C₂₉	—	2	2	no	—
48	M₁₆×C₂₉	—	2	4	no	—
49	C₂×C₈×C₂₉	—	2	4	no	—
50	C₂×C₈×C₂₉	D₄	1	4	yes	complemented
51	C₂×C₈×C₂₉	C₂³	1	2	yes	—
52	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
56	#90781	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
58	#90779	C₂×C₄	1	4	yes	—
59	C₄×Dic₂₉	C₂³	1	2	yes	—
60	#90777	—	2	2	no	—
61	C₂×#6597	—	2	2	no	—
62	C₂×#6597	C₂³	1	2	yes	—
63	C₂×#6597	C₂³	1	2	yes	—
64	#90772	—	2	4	no	—
65	#90772	—	2	4	no	—
66	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
67	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₂₉	—	2	2	no	—
69	C₂²×D₂₉	—	4	2	no	—
70	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
71	Q₈×C₂₉	—	2	2	no	—
72	D₄×C₂₉	—	2	4	no	—
73	D₄×C₂₉	—	2	2	no	—
74	C₂×C₄×C₂₉	—	2	2	no	—
75	C₂×C₄×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₉	#41	1	2	yes	—
80	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₂₉	—	4	4	no	—
82	C₂×Dic₂₉	—	2	2	no	—
83	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
84	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
85	C₄×D₂₉	—	2	4	no	—
86	C₄×D₂₉	—	2	2	no	—
87	C₄×D₂₉	—	2	2	no	—
88	C₈×C₂₉	—	2	2	no	—
89	C₈×C₂₉	C₂×D₄	1	2	yes	—
90	C₈×C₂₉	C₂×D₄	1	2	yes	—
91	#6597	—	2	2	no	—
92	#6597	—	2	2	no	—
93	#2930	—	29	1	no	maximal, Sylow
94	C₂²×C₂₉	—	2	2	no	—
95	C₂²×C₂₉	—	2	2	no	—
96	C₂²×C₂₉	—	2	4	no	—
97	C₂²×C₂₉	C₄×D₄	1	2	yes	—
98	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₂₉	—	2	2	no	—
100	C₂×D₂₉	—	2	2	no	—
101	C₂×D₂₉	—	2	2	no	—
102	C₂×D₂₉	—	4	4	no	—
103	C₄×C₂₉	—	2	2	no	—
104	C₄×C₂₉	C₄×D₄	1	2	yes	—
105	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
107	Dic₂₉	—	2	2	no	—
108	Dic₂₉	—	2	4	no	—
109	C₂×#131	D₂₉	1	1	yes	characteristic, complemented
110	C₂²×M₁₆	—	29	1	no	—
111	#525	—	29	1	no	—
112	#436	—	29	4	no	—
113	#435	—	29	4	no	—
114	#424	—	29	1	no	—
115	#408	—	29	1	no	—
116	#407	—	29	1	no	—
117	#405	—	29	1	no	—
118	C₂×C₂₉	—	2	2	no	—
119	C₂×C₂₉	—	4	2	no	—
120	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×C₂₉	#568	1	2	yes	—
122	D₂₉	—	4	2	no	—
123	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
124	C₂³×C₄	—	29	1	no	—
125	#131	—	2	4	no	—
126	C₂×M₁₆	—	29	1	no	—
127	C₂×M₁₆	—	29	2	no	—
128	C₂×M₁₆	—	29	2	no	—
129	C₂×M₁₆	—	58	4	no	—
130	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
131	C₂×M₁₆	C₂×D₂₉	1	2	yes	complemented
132	C₂²×C₈	—	29	1	no	—
133	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₂₉	1	2	yes	complemented
135	#123	—	29	2	no	—
136	C₂²⋊Q₈	—	29	2	no	—
137	#121	—	29	2	no	—
138	#120	—	29	2	no	—
139	C₄×D₄	—	29	4	no	—
140	#117	—	29	1	no	—
141	C₄⋊C₈	—	29	4	no	—
142	C₂²⋊C₈	—	29	4	no	—
143	C₂²⋊C₈	—	29	4	no	—
144	#97	—	29	2	no	—
145	C₄×C₈	—	29	2	no	—
146	C₂₉	#2930	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	—	29	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
151	C₂×D₄	—	29	4	no	—
152	C₂×D₄	C₄×D₂₉	1	2	yes	—
153	C₂×D₄	C₄×D₂₉	1	1	yes	characteristic
154	C₂²×C₄	—	29	1	no	—
155	C₂²×C₄	—	29	2	no	—
156	C₂²×C₄	—	29	2	no	—
157	C₂²×C₄	—	58	2	no	—
158	C₂²×C₄	—	58	2	no	—
159	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
160	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	58	2	no	—
164	M₁₆	—	58	4	no	—
165	M₁₆	—	58	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	29	2	no	—
168	C₂×C₈	—	29	2	no	—
169	C₂×C₈	—	58	2	no	—
170	C₂×C₈	C₂²⋊D₂₉	1	4	yes	complemented
171	C₂×C₈	C₂²×D₂₉	1	2	yes	—
172	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
174	C₄⋊C₄	—	29	2	no	—
175	C₄⋊C₄	—	29	4	no	—
176	C₂²⋊C₄	—	29	2	no	—
177	C₂²⋊C₄	—	29	4	no	—
178	C₂²⋊C₄	—	29	4	no	—
179	C₄²	—	29	2	no	—
180	C₂³	—	29	2	no	—
181	C₂³	—	58	2	no	—
182	C₂³	—	116	2	no	—
183	C₂³	C₂×C₄×D₂₉	1	2	yes	—
184	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	116	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	29	4	no	—
192	C₂×C₄	—	29	2	no	—
193	C₂×C₄	—	58	2	no	—
194	C₂×C₄	—	58	4	no	—
195	C₂×C₄	—	58	2	no	—
196	C₂×C₄	—	58	2	no	—
197	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
199	C₂×C₄	#90805	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
201	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	58	2	no	—
204	C₈	—	58	2	no	—
205	C₈	C₂×C₂²⋊D₂₉	1	2	yes	—
206	C₈	C₂×C₂²⋊D₂₉	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	58	2	no	—
211	C₂²	—	58	2	no	—
212	C₂²	—	58	2	no	—
213	C₂²	—	116	4	no	—
214	C₂²	C₄×C₂²⋊D₂₉	1	2	yes	—
215	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	58	2	no	—
218	C₄	—	58	4	no	—
219	C₄	C₄×C₂²⋊D₂₉	1	2	yes	—
220	C₄	C₂×#90805	1	1	yes	characteristic
221	C₄	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	116	2	no	—
225	C₂	C₂×C₄×C₂²⋊D₂₉	1	1	yes	characteristic
226	C₂	#14271928	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂×C₂₉×#131	C₂
prime radical	#14271670	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×#131	D₂₉
hypocenter	C₂₉	#2930
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#90772	C₂	semidirect
3. #14271239	C₂	semidirect
4. #14271234	C₂	semidirect
5. #14271191	C₂	semidirect
6. #14271130	C₂	semidirect
7. #14271126	C₂	semidirect
8. #14271118	C₂	semidirect
9. C₂×C₂₉×#131	C₂	semidirect
14. C₂×M₁₆×C₂₉	C₂²	semidirect
16. C₂²×C₈×C₂₉	C₂²	semidirect
28. C₂²⋊#6597	C₂²	semidirect
30. #11745163	C₂²	semidirect
31. #11745162	C₂²	semidirect
32. C₈×Dic₂₉	C₂²	semidirect
50. C₂×C₈×C₂₉	D₄	semidirect
109. C₂×#131	D₂₉	semidirect
131. C₂×M₁₆	C₂×D₂₉	semidirect
134. C₂²×C₈	C₂×D₂₉	semidirect
146. C₂₉	#2930	semidirect
170. C₂×C₈	C₂²⋊D₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#90772	2
#14271670	2
#14271239	2
#14271234	2
#14271191	2
#14271130	2
#14271126	2
#14271118	2
C₂×C₂₉×#131	2
#2930	29