Subgroups — #16828196

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	C₂×#11745310	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#90777	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271241	C₂	1	4	yes	maximal, complemented
4	#14271235	C₂	1	4	yes	maximal, complemented
5	#14271190	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271130	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271127	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271119	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#525	C₂	1	1	yes	maximal, characteristic
10	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
12	C₂₉×#123	C₄	1	2	yes	—
13	C₂₉×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₂₉×#121	C₄	1	2	yes	—
15	C₂₉×#120	C₄	1	2	yes	—
16	C₄×D₄×C₂₉	C₂²	1	4	yes	—
17	C₂₉×#117	C₂²	1	1	yes	characteristic
18	#11745310	—	2	4	no	—
19	C₂×#90777	—	2	4	no	—
20	C₂×#90777	C₂²	1	2	yes	—
21	C₂×#90777	C₂²	1	2	yes	—
22	C₂×#90777	C₂²	1	1	yes	characteristic
23	C₂×#90777	C₂²	1	1	yes	characteristic
24	C₂×#90777	C₂²	1	2	yes	complemented
25	C₂²×#6597	C₂²	1	1	yes	characteristic
26	C₂²×#6597	C₂²	1	1	yes	characteristic
27	C₂²×#6597	C₂²	1	2	yes	complemented
28	C₂²⋊#6597	C₂²	1	4	yes	—
29	C₂²⋊#6597	C₂²	1	4	yes	complemented
30	C₄⋊#6597	C₂²	1	4	yes	complemented
31	#11745151	C₂²	1	2	yes	complemented
32	C₄×#6597	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₉	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	C₂×D₄×C₂₉	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₂₉	—	2	2	no	—
41	C₂²×C₄×C₂₉	—	2	2	no	—
42	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
49	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
50	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
51	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
52	C₄²×C₂₉	C₂³	1	2	yes	—
53	#90777	—	2	2	no	—
54	#90777	—	2	2	no	—
55	#90777	—	2	4	no	—
56	#90777	—	2	4	no	—
57	#90777	—	2	4	no	—
58	C₂×#6597	—	2	2	no	—
59	C₂×#6597	—	2	4	no	—
60	C₂×#6597	D₄	1	4	yes	complemented
61	C₂×#6597	C₂³	1	2	yes	—
62	C₂×#6597	C₂³	1	2	yes	—
63	C₂×#6597	C₂³	1	2	yes	—
64	C₂×#6597	C₂³	1	1	yes	characteristic
65	C₂×#6597	C₂³	1	1	yes	characteristic
66	C₂³×C₂₉	—	2	2	no	—
67	C₂³×C₂₉	—	4	2	no	—
68	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
69	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
70	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₂₉	—	2	2	no	—
72	D₄×C₂₉	—	2	4	no	—
73	D₄×C₂₉	—	2	2	no	—
74	D₄×C₂₉	—	4	4	no	—
75	C₂×C₄×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	—	2	4	no	—
77	C₂×C₄×C₂₉	—	2	2	no	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	2	no	—
80	C₂×C₄×C₂₉	—	2	2	no	—
81	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
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	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₉	#41	1	2	yes	—
87	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
88	#6597	—	2	2	no	—
89	#6597	—	2	2	no	—
90	#6597	—	2	2	no	—
91	#6597	C₂×D₄	1	2	yes	—
92	#6597	C₂×D₄	1	2	yes	—
93	#2930	—	29	1	no	maximal, Sylow
94	C₂²×C₂₉	—	2	2	no	—
95	C₂²×C₂₉	—	2	2	no	—
96	C₂²×C₂₉	—	2	2	no	—
97	C₂²×C₂₉	—	2	2	no	—
98	C₂²×C₂₉	—	2	4	no	—
99	C₂²×C₂₉	—	2	2	no	—
100	C₂²×C₂₉	—	4	4	no	—
101	C₂²×C₂₉	C₄×D₄	1	2	yes	—
102	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
103	C₄×C₂₉	—	2	2	no	—
104	C₄×C₂₉	—	2	2	no	—
105	C₄×C₂₉	—	2	4	no	—
106	C₄×C₂₉	C₄×D₄	1	2	yes	—
107	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
109	C₂×#131	—	29	1	no	—
110	C₂²×M₁₆	—	29	1	no	—
111	#525	D₂₉	1	1	yes	characteristic
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	—	4	2	no	—
121	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂×C₂₉	#568	1	2	yes	—
123	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
124	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
125	#131	—	58	4	no	—
126	C₂×M₁₆	—	29	2	no	—
127	C₂×M₁₆	—	29	2	no	—
128	C₂×M₁₆	—	29	1	no	—
129	C₂×M₁₆	—	29	1	no	—
130	C₂×M₁₆	—	29	2	no	—
131	C₂×M₁₆	—	58	4	no	—
132	C₂²×C₈	—	29	1	no	—
133	C₂²×C₈	—	29	1	no	—
134	C₂²×C₈	—	29	2	no	—
135	#123	Dic₂₉	1	2	yes	—
136	C₂²⋊Q₈	Dic₂₉	1	2	yes	—
137	#121	Dic₂₉	1	2	yes	—
138	#120	Dic₂₉	1	2	yes	—
139	C₄×D₄	C₂×D₂₉	1	4	yes	—
140	#117	C₂×D₂₉	1	1	yes	characteristic
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	C₂×Dic₂₉	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×Dic₂₉	1	1	yes	characteristic
151	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
152	C₂×D₄	C₂×Dic₂₉	1	1	yes	characteristic
153	C₂×D₄	C₂×Dic₂₉	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
157	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
159	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
160	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
161	M₁₆	—	58	4	no	—
162	M₁₆	—	58	4	no	—
163	M₁₆	—	58	4	no	—
164	M₁₆	—	58	2	no	—
165	M₁₆	—	58	2	no	—
166	C₂×C₈	—	29	4	no	—
167	C₂×C₈	—	29	1	no	—
168	C₂×C₈	—	29	1	no	—
169	C₂×C₈	—	29	2	no	—
170	C₂×C₈	—	29	2	no	—
171	C₂×C₈	—	29	2	no	—
172	C₂×C₈	—	58	4	no	—
173	C₂×C₈	—	58	2	no	—
174	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₂₉	1	4	yes	—
176	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₂₉	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₂₉	1	4	yes	—
179	C₄²	C₂²×D₂₉	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	C₂²×Dic₂₉	1	2	yes	—
183	C₂³	C₂²×Dic₂₉	1	2	yes	—
184	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	4	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	D₄×D₂₉	1	2	yes	—
196	C₂×C₄	#90807	1	2	yes	—
197	C₂×C₄	C₂²×Dic₂₉	1	4	yes	—
198	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
199	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
200	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
202	C₈	—	29	2	no	—
203	C₈	—	29	2	no	—
204	C₈	—	58	2	no	—
205	C₈	—	58	2	no	—
206	C₈	—	58	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	D₄×Dic₂₉	1	2	yes	—
215	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	D₄×Dic₂₉	1	2	yes	—
220	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
221	C₄	C₂×#90807	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₂₉	1	1	yes	characteristic
226	C₂	#14271930	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting, prime radical	C₂₉×#525	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#525	D₂₉
hypocenter	C₂₉	#2930
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#11745310	C₂	semidirect
2. C₂²×#90777	C₂	semidirect
3. #14271241	C₂	semidirect
4. #14271235	C₂	semidirect
5. #14271190	C₂	semidirect
6. #14271130	C₂	semidirect
7. #14271127	C₂	semidirect
8. #14271119	C₂	semidirect
24. C₂×#90777	C₂²	semidirect
27. C₂²×#6597	C₂²	semidirect
29. C₂²⋊#6597	C₂²	semidirect
30. C₄⋊#6597	C₂²	semidirect
31. #11745151	C₂²	semidirect
32. C₄×#6597	C₂²	semidirect
60. C₂×#6597	D₄	semidirect
146. C₂₉	#2930	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745310	2
C₂²×#90777	2
#14271241	2
#14271235	2
#14271190	2
#14271130	2
#14271127	2
#14271119	2
C₂₉×#525	2
#2930	29