Subgroups — #16828529

Order 3712.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₁₆	C₂	1	1	yes	maximal, characteristic
2	#14271240	C₂	1	2	yes	maximal, complemented
3	#14271236	C₂	1	4	yes	maximal, complemented
4	#14271228	C₂	1	2	yes	maximal, complemented
5	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271125	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₈×D₂₉	C₂	1	2	yes	maximal, complemented
8	C₂₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₂₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
12	C₂×#90781	C₄	1	2	yes	—
13	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
14	C₂×#90772	C₂²	1	4	yes	complemented
15	C₂×C₈×D₂₉	C₂²	1	4	yes	complemented
16	C₄×D₁₁₆	C₄	1	2	yes	—
17	C₄×D₁₁₆	C₄	1	1	yes	characteristic
18	C₄×D₁₁₆	C₄	1	1	yes	characteristic
19	C₄×D₁₁₆	C₂²	1	2	yes	—
20	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
21	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
22	C₂₉×C₄⋊C₈	C₂²	1	2	yes	complemented
23	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂₉×#97	C₂²	1	2	yes	complemented
25	C₂²⋊#6597	C₂²	1	1	yes	characteristic
26	C₂²⋊#6597	C₂²	1	1	yes	characteristic
27	#11745168	C₂²	1	4	yes	complemented
28	#11745168	C₂²	1	4	yes	complemented
29	C₄⋊#6597	C₂²	1	2	yes	complemented
30	C₄×#6597	C₂²	1	2	yes	complemented
31	C₂³×D₂₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
36	C₂×D₁₁₆	—	2	4	no	—
37	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
38	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₂₉	—	4	4	no	—
41	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₂₉	C₂³	1	2	yes	—
44	C₂×C₄×D₂₉	C₂³	1	2	yes	—
45	C₂×C₈×C₂₉	C₂³	1	4	yes	—
46	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₉	C₂³	1	1	yes	characteristic
49	C₄²×C₂₉	C₂³	1	1	yes	characteristic
50	#90781	C₂×C₄	1	2	yes	—
51	#90781	C₂×C₄	1	2	yes	—
52	#90781	C₂×C₄	1	2	yes	—
53	#90781	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
57	C₂×#6597	C₂³	1	2	yes	—
58	C₂×#6597	C₂³	1	2	yes	—
59	#90772	—	4	4	no	—
60	C₈×D₂₉	—	4	4	no	—
61	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₂₉	—	2	4	no	—
63	C₂²×D₂₉	—	2	4	no	—
64	C₂²×D₂₉	—	4	2	no	—
65	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
66	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₂₉	—	2	2	no	—
68	C₂×C₄×C₂₉	—	2	2	no	—
69	C₂×C₄×C₂₉	—	2	2	no	—
70	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₂₉	—	4	2	no	—
77	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
79	D₁₁₆	—	2	4	no	—
80	D₁₁₆	—	2	2	no	—
81	D₁₁₆	M₁₆	1	4	yes	complemented
82	C₄×D₂₉	—	2	4	no	—
83	C₄×D₂₉	—	2	4	no	—
84	C₄×D₂₉	—	4	2	no	—
85	C₄×D₂₉	—	4	2	no	—
86	C₈×C₂₉	—	2	4	no	—
87	#6597	—	2	2	no	—
88	#6597	—	2	2	no	—
89	#2974	—	29	1	no	maximal, Sylow
90	C₂²×C₂₉	—	2	1	no	—
91	C₂²×C₂₉	—	2	1	no	—
92	C₂²×C₂₉	#131	1	2	yes	—
93	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₂₉	—	2	2	no	—
95	C₂×D₂₉	—	2	2	no	—
96	C₂×D₂₉	—	2	2	no	—
97	C₂×D₂₉	—	2	2	no	—
98	C₂×D₂₉	—	2	2	no	—
99	C₂×D₂₉	—	4	8	no	—
100	C₂×D₂₉	C₂×M₁₆	1	4	yes	—
101	C₄×C₂₉	—	2	1	no	—
102	C₄×C₂₉	—	2	1	no	—
103	C₄×C₂₉	—	2	1	no	—
104	C₄×C₂₉	—	2	1	no	—
105	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
106	C₄×C₂₉	H(5,𝔽₂)	1	2	yes	—
107	Dic₂₉	—	2	2	no	—
108	Dic₂₉	—	2	2	no	—
109	C₂×C₄×D₄	—	29	1	no	—
110	#436	—	29	2	no	—
111	#435	—	29	4	no	—
112	C₈×D₄	—	29	2	no	—
113	#432	D₂₉	1	1	yes	characteristic, complemented
114	#431	—	29	1	no	—
115	#407	—	29	2	no	—
116	C₂×C₂²⋊C₈	—	29	2	no	—
117	C₂×C₂₉	—	2	2	no	—
118	C₂×C₂₉	#518	1	1	yes	characteristic
119	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
121	D₂₉	—	2	4	no	—
122	D₂₉	—	4	2	no	—
123	C₂²×D₄	—	29	1	no	—
124	C₂³×C₄	—	29	2	no	—
125	C₂×M₁₆	—	29	4	no	—
126	C₂²×C₈	—	29	4	no	—
127	C₄×D₄	—	29	1	no	—
128	C₄×D₄	—	29	2	no	—
129	C₄×D₄	—	29	1	no	—
130	C₄×D₄	—	29	1	no	—
131	C₄×D₄	—	29	2	no	—
132	C₄×D₄	—	29	1	no	—
133	C₂×C₄⋊C₄	—	29	1	no	—
134	C₂×C₂²⋊C₄	—	29	2	no	—
135	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
136	C₄⋊C₈	—	29	2	no	—
137	C₄⋊C₈	C₂×D₂₉	1	2	yes	complemented
138	C₂²⋊C₈	—	29	1	no	—
139	C₂²⋊C₈	—	29	1	no	—
140	C₂²⋊C₈	—	29	4	no	—
141	C₂²⋊C₈	—	29	4	no	—
142	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
143	#97	C₂×D₂₉	1	2	yes	complemented
144	C₄×C₈	—	29	2	no	—
145	C₂₉	#2974	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴	—	29	2	no	—
147	C₂×D₄	—	29	1	no	—
148	C₂×D₄	—	29	2	no	—
149	C₂×D₄	—	29	1	no	—
150	C₂×D₄	—	58	4	no	—
151	C₂²×C₄	—	29	2	no	—
152	C₂²×C₄	—	29	2	no	—
153	C₂²×C₄	—	29	2	no	—
154	C₂²×C₄	—	29	2	no	—
155	C₂²×C₄	—	29	2	no	—
156	C₂²×C₄	—	116	4	no	—
157	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
158	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
160	M₁₆	—	116	4	no	—
161	C₂×C₈	—	29	2	no	—
162	C₂×C₈	—	29	2	no	—
163	C₂×C₈	—	116	4	no	—
164	C₂×C₈	C₂²×D₂₉	1	4	yes	—
165	C₄⋊C₄	—	29	2	no	—
166	C₄⋊C₄	—	29	1	no	—
167	C₄⋊C₄	—	29	1	no	—
168	C₂²⋊C₄	—	29	2	no	—
169	C₂²⋊C₄	—	29	2	no	—
170	C₂²⋊C₄	—	29	2	no	—
171	C₂²⋊C₄	—	29	2	no	—
172	C₄²	C₄×D₂₉	1	1	yes	characteristic
173	C₄²	C₄×D₂₉	1	1	yes	characteristic
174	C₄²	C₂²×D₂₉	1	1	yes	characteristic
175	C₄²	C₂²×D₂₉	1	1	yes	characteristic
176	C₂³	—	29	2	no	—
177	C₂³	—	29	2	no	—
178	C₂³	—	58	4	no	—
179	C₂³	—	58	4	no	—
180	C₂³	—	116	2	no	—
181	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
182	D₄	—	29	4	no	—
183	D₄	—	58	4	no	—
184	D₄	—	58	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	29	2	no	—
189	C₂×C₄	—	29	2	no	—
190	C₂×C₄	—	58	4	no	—
191	C₂×C₄	—	58	4	no	—
192	C₂×C₄	—	116	2	no	—
193	C₂×C₄	—	116	2	no	—
194	C₂×C₄	—	116	2	no	—
195	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
201	C₈	—	2	4	no	—
202	C₈	—	58	2	no	—
203	C₈	—	58	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	29	4	no	—
207	C₂²	—	58	2	no	—
208	C₂²	—	58	2	no	—
209	C₂²	—	58	2	no	—
210	C₂²	—	58	2	no	—
211	C₂²	—	58	2	no	—
212	C₂²	—	116	8	no	—
213	C₂²	#11745269	1	2	yes	—
214	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	58	2	no	—
220	C₄	—	58	2	no	—
221	C₄	M₁₆×D₂₉	1	2	yes	—
222	C₄	#11745364	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	58	4	no	—
225	C₂	—	116	2	no	—
226	C₂	#14271637	1	1	yes	characteristic
227	C₂	C₂×M₁₆×D₂₉	1	1	yes	characteristic
228	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#432	C₂
prime radical	C₂²×D₁₁₆	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#432	D₂₉
hypocenter	C₂₉	#2974
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #14271240	C₂	semidirect
3. #14271236	C₂	semidirect
4. #14271228	C₂	semidirect
5. #14271216	C₂	semidirect
6. #14271125	C₂	semidirect
7. C₂²⋊C₈×D₂₉	C₂	semidirect
8. C₂₉×#432	C₂	semidirect
14. C₂×#90772	C₂²	semidirect
15. C₂×C₈×D₂₉	C₂²	semidirect
22. C₂₉×C₄⋊C₈	C₂²	semidirect
24. C₂₉×#97	C₂²	semidirect
27. #11745168	C₂²	semidirect
28. #11745168	C₂²	semidirect
29. C₄⋊#6597	C₂²	semidirect
30. C₄×#6597	C₂²	semidirect
81. D₁₁₆	M₁₆	semidirect
113. #432	D₂₉	semidirect
137. C₄⋊C₈	C₂×D₂₉	semidirect
143. #97	C₂×D₂₉	semidirect
145. C₂₉	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₁₆	2
#14271240	2
#14271236	2
#14271228	2
#14271216	2
#14271125	2
C₂²⋊C₈×D₂₉	2
C₂₉×#432	2
#2974	29