Subgroups — #16828527

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	#14271241	C₂	1	2	yes	maximal, complemented
2	#14271235	C₂	1	4	yes	maximal, complemented
3	D₄×#6597	C₂	1	2	yes	maximal, complemented
4	#14271220	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271127	C₂	1	2	yes	maximal, complemented
7	C₂×C₂²⋊#6597	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₂₉	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₂₉	C₂²	1	2	yes	—
11	C₄×D₄×C₂₉	C₄	1	2	yes	—
12	C₄×D₄×C₂₉	C₄	1	1	yes	characteristic
13	C₄×D₄×C₂₉	C₄	1	1	yes	characteristic
14	C₄×D₄×C₂₉	C₂²	1	2	yes	—
15	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₉	C₄	1	2	yes	—
19	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×#90777	C₂²	1	4	yes	complemented
21	C₂²×#6597	C₂²	1	4	yes	complemented
22	C₂²⋊#6597	C₂²	1	1	yes	characteristic
23	C₂²⋊#6597	C₂²	1	2	yes	—
24	C₂²⋊#6597	C₂²	1	1	yes	characteristic
25	C₂²⋊#6597	C₂²	1	4	yes	complemented
26	C₂²⋊#6597	C₂²	1	4	yes	complemented
27	C₄⋊#6597	C₂²	1	2	yes	complemented
28	C₄⋊#6597	C₂²	1	2	yes	complemented
29	#11745151	C₂²	1	2	yes	complemented
30	C₄×#6597	C₂²	1	2	yes	complemented
31	C₂⁴×C₂₉	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₂₉	—	2	4	no	—
33	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₉	—	4	4	no	—
37	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
46	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
49	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
50	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
52	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₂₉	C₂³	1	1	yes	characteristic
55	C₄²×C₂₉	C₂³	1	1	yes	characteristic
56	#90777	—	4	4	no	—
57	C₂×#6597	—	4	4	no	—
58	C₂×#6597	C₂³	1	2	yes	—
59	C₂×#6597	C₂³	1	4	yes	—
60	C₂×#6597	C₂³	1	2	yes	—
61	C₂³×C₂₉	—	2	4	no	—
62	C₂³×C₂₉	—	2	4	no	—
63	C₂³×C₂₉	—	4	2	no	—
64	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
65	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
66	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	D₄×C₂₉	—	2	4	no	—
68	D₄×C₂₉	—	2	2	no	—
69	D₄×C₂₉	M₁₆	1	4	yes	complemented
70	C₂×C₄×C₂₉	—	2	2	no	—
71	C₂×C₄×C₂₉	—	2	2	no	—
72	C₂×C₄×C₂₉	—	2	4	no	—
73	C₂×C₄×C₂₉	—	2	4	no	—
74	C₂×C₄×C₂₉	—	2	2	no	—
75	C₂×C₄×C₂₉	—	4	2	no	—
76	C₂×C₄×C₂₉	—	4	2	no	—
77	C₂×C₄×C₂₉	—	4	2	no	—
78	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
86	#6597	—	2	2	no	—
87	#6597	—	2	4	no	—
88	#6597	—	2	2	no	—
89	#2974	—	29	1	no	maximal, Sylow
90	C₂²×C₂₉	—	2	1	no	—
91	C₂²×C₂₉	—	2	2	no	—
92	C₂²×C₂₉	—	2	1	no	—
93	C₂²×C₂₉	—	2	2	no	—
94	C₂²×C₂₉	—	2	2	no	—
95	C₂²×C₂₉	—	2	2	no	—
96	C₂²×C₂₉	—	2	2	no	—
97	C₂²×C₂₉	—	4	8	no	—
98	C₂²×C₂₉	C₂×M₁₆	1	4	yes	—
99	C₂²×C₂₉	#131	1	2	yes	—
100	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₂₉	—	2	1	no	—
102	C₄×C₂₉	—	2	2	no	—
103	C₄×C₂₉	—	2	2	no	—
104	C₄×C₂₉	—	2	1	no	—
105	C₄×C₂₉	—	2	1	no	—
106	C₄×C₂₉	—	2	1	no	—
107	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
108	C₄×C₂₉	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×D₄	D₂₉	1	1	yes	characteristic
110	#436	—	29	2	no	—
111	#435	—	29	4	no	—
112	C₈×D₄	—	29	2	no	—
113	#432	—	29	1	no	—
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	—	2	4	no	—
119	C₂×C₂₉	—	4	2	no	—
120	C₂×C₂₉	#518	1	1	yes	characteristic
121	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
123	C₂²×D₄	Dic₂₉	1	1	yes	characteristic
124	C₂³×C₄	C₂×D₂₉	1	2	yes	—
125	C₂×M₁₆	—	29	4	no	—
126	C₂²×C₈	—	29	4	no	—
127	C₄×D₄	Dic₂₉	1	2	yes	—
128	C₄×D₄	Dic₂₉	1	1	yes	characteristic
129	C₄×D₄	Dic₂₉	1	1	yes	characteristic
130	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
131	C₄×D₄	C₂×D₂₉	1	2	yes	—
132	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
133	C₂×C₄⋊C₄	Dic₂₉	1	1	yes	characteristic
134	C₂×C₂²⋊C₄	Dic₂₉	1	2	yes	—
135	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
136	C₄⋊C₈	—	29	2	no	—
137	C₄⋊C₈	—	29	2	no	—
138	C₂²⋊C₈	—	29	1	no	—
139	C₂²⋊C₈	—	29	2	no	—
140	C₂²⋊C₈	—	29	4	no	—
141	C₂²⋊C₈	—	29	1	no	—
142	C₂²⋊C₈	—	29	4	no	—
143	#97	—	29	2	no	—
144	C₄×C₈	—	29	2	no	—
145	C₂₉	#2974	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴	C₂×Dic₂₉	1	2	yes	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
149	C₂×D₄	C₂×Dic₂₉	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₂₉	1	1	yes	characteristic
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
155	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
156	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
157	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
159	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
160	M₁₆	—	116	4	no	—
161	C₂×C₈	—	29	2	no	—
162	C₂×C₈	—	29	2	no	—
163	C₂×C₈	—	29	4	no	—
164	C₂×C₈	—	116	4	no	—
165	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
166	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
167	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
168	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
172	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
173	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
174	C₄²	C₂²×D₂₉	1	1	yes	characteristic
175	C₄²	C₂²×D₂₉	1	1	yes	characteristic
176	C₂³	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	4	2	no	—
179	C₂³	C₂²×Dic₂₉	1	2	yes	—
180	C₂³	C₂²×Dic₂₉	1	2	yes	—
181	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
182	D₄	—	2	4	no	—
183	D₄	—	2	2	no	—
184	D₄	#90777	1	4	yes	complemented
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
198	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
199	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
201	C₈	—	58	2	no	—
202	C₈	—	58	4	no	—
203	C₈	—	58	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	8	no	—
212	C₂²	C₂×#90777	1	4	yes	—
213	C₂²	#11745310	1	2	yes	—
214	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	C₂×#90777	1	2	yes	—
222	C₄	#11745359	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	#14271638	1	1	yes	characteristic
227	C₂	C₂²×#90777	1	1	yes	characteristic
228	C₂	C₂×#11745310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂×C₄×D₄×C₂₉	C₂
prime radical	C₂²×D₄×C₂₉	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₉
hypocenter	C₂₉	#2974
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #14271241	C₂	semidirect
2. #14271235	C₂	semidirect
3. D₄×#6597	C₂	semidirect
4. #14271220	C₂	semidirect
5. #14271216	C₂	semidirect
6. #14271127	C₂	semidirect
7. C₂×C₂²⋊#6597	C₂	semidirect
20. C₂×#90777	C₂²	semidirect
21. C₂²×#6597	C₂²	semidirect
25. C₂²⋊#6597	C₂²	semidirect
26. C₂²⋊#6597	C₂²	semidirect
27. C₄⋊#6597	C₂²	semidirect
28. C₄⋊#6597	C₂²	semidirect
29. #11745151	C₂²	semidirect
30. C₄×#6597	C₂²	semidirect
69. D₄×C₂₉	M₁₆	semidirect
145. C₂₉	#2974	semidirect
184. D₄	#90777	semidirect

Maximal subgroups

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

subgroup	index
#14271241	2
#14271235	2
D₄×#6597	2
#14271220	2
#14271216	2
#14271127	2
C₂×C₂²⋊#6597	2
C₂×C₄×D₄×C₂₉	2
#2974	29