Subgroups — C₂×C₄×#90805 #16832672

Order 3712.

1728 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×#90805	C₂	1	1	yes	maximal, characteristic
2	C₄×#90805	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₅₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₁₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂²⋊D₂₉	C₂	1	4	yes	maximal, complemented
6	C₂×#11745288	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11745234	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₂₉	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₄×C₂₉	C₂²	1	2	yes	—
11	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₂₉	C₂²	1	2	yes	—
13	C₂×#90805	C₄	1	16	yes	direct product factor
14	C₂×#90805	C₂²	1	8	yes	—
15	C₂²×D₁₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₂₉	C₂²	1	4	yes	—
17	C₂²×C₄×D₂₉	C₂²	1	2	yes	—
18	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₉	C₂²	1	8	yes	complemented
23	C₂×C₂²⋊Dic₂₉	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₂₉	C₂²	1	32	yes	complemented
25	C₂×#90781	C₂²	1	4	yes	—
26	#11745288	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
28	C₂×#90779	C₂²	1	4	yes	—
29	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
30	C₂×C₄×Dic₂₉	C₂²	1	4	yes	—
31	C₄×D₁₁₆	C₂²	1	16	yes	complemented
32	#11745234	C₂²	1	16	yes	complemented
33	C₄²×D₂₉	C₂²	1	16	yes	complemented
34	C₄×Dic₅₈	C₂²	1	16	yes	complemented
35	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
36	C₂³×D₂₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₉	—	2	8	no	—
38	C₂²×C₄×C₂₉	C₂×C₄	1	8	yes	complemented
39	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₉	C₂³	1	8	yes	—
41	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₂₉	C₂×C₄	1	16	yes	complemented
46	C₂×C₂²⋊D₂₉	C₂³	1	8	yes	—
47	C₂²×Dic₂₉	C₂³	1	4	yes	—
48	C₂²×Dic₂₉	C₂³	1	2	yes	—
49	#90805	C₂×C₄	1	64	yes	direct product factor
50	C₂×D₁₁₆	C₂×C₄	1	8	yes	complemented
51	C₂×D₁₁₆	C₂³	1	4	yes	—
52	C₂×C₄×D₂₉	—	2	16	no	—
53	C₂×C₄×D₂₉	C₂×C₄	1	16	yes	complemented
54	C₂×C₄×D₂₉	C₂³	1	16	yes	—
55	C₂×C₄×D₂₉	C₂³	1	8	yes	—
56	C₂×Dic₅₈	C₂×C₄	1	8	yes	complemented
57	C₂×Dic₅₈	C₂³	1	4	yes	—
58	C₄²×C₂₉	C₂³	1	8	yes	—
59	C₄²×C₂₉	C₂³	1	4	yes	—
60	C₄²×C₂₉	C₂³	1	4	yes	—
61	C₂²⋊Dic₂₉	C₂³	1	8	yes	—
62	#90781	C₂³	1	16	yes	—
63	C₄⋊Dic₂₉	C₂³	1	8	yes	—
64	#90779	C₂³	1	16	yes	—
65	C₄×Dic₂₉	C₂³	1	8	yes	—
66	C₄×Dic₂₉	C₂³	1	16	yes	—
67	C₂³×C₂₉	—	2	4	no	—
68	C₂³×C₂₉	C₂²×C₄	1	4	yes	—
69	C₂³×C₂₉	C₂⁴	1	2	yes	—
70	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
71	C₂²×D₂₉	—	2	8	no	—
72	C₂²×D₂₉	C₂²×C₄	1	8	yes	—
73	C₂²×D₂₉	C₂⁴	1	4	yes	—
74	C₂×C₄×C₂₉	—	2	4	no	—
75	C₂×C₄×C₂₉	—	2	16	no	—
76	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₂₉	C₂²×C₄	1	16	yes	complemented
79	C₂×C₄×C₂₉	#41	1	8	yes	complemented
80	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
81	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
82	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
83	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
84	C₂²⋊D₂₉	C₂²×C₄	1	32	yes	complemented
85	C₂×Dic₂₉	—	2	8	no	—
86	C₂×Dic₂₉	C₂²×C₄	1	8	yes	—
87	C₂×Dic₂₉	C₂⁴	1	8	yes	—
88	C₂×Dic₂₉	C₂⁴	1	4	yes	—
89	D₁₁₆	C₂²×C₄	1	16	yes	complemented
90	C₄×D₂₉	—	2	32	no	—
91	C₄×D₂₉	C₂²×C₄	1	32	yes	complemented
92	Dic₅₈	C₂²×C₄	1	16	yes	complemented
93	C₂×C₄×#41	—	29	1	no	maximal, Sylow
94	C₂²×C₂₉	—	2	8	no	—
95	C₂²×C₂₉	—	2	4	no	—
96	C₂²×C₂₉	C₂³×C₄	1	4	yes	—
97	C₂²×C₂₉	C₂³×C₄	1	2	yes	—
98	C₂²×C₂₉	C₂×#41	1	4	yes	—
99	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
100	C₂×D₂₉	—	2	16	no	—
101	C₂×D₂₉	—	2	8	no	—
102	C₂×D₂₉	C₂³×C₄	1	8	yes	—
103	C₄×C₂₉	—	2	4	no	—
104	C₄×C₂₉	C₂³×C₄	1	4	yes	—
105	C₄×C₂₉	C₂³×C₄	1	4	yes	—
106	C₄×C₂₉	C₂×#41	1	8	yes	complemented
107	Dic₂₉	—	2	8	no	—
108	Dic₂₉	C₂³×C₄	1	8	yes	—
109	C₂²×#41	—	29	1	no	—
110	C₄×#41	—	29	16	no	—
111	C₂×C₄×Q₈	—	29	2	no	—
112	C₂×C₄×D₄	—	29	4	no	—
113	C₂×C₄×D₄	—	29	2	no	—
114	C₂×#117	—	29	1	no	—
115	C₂×#117	—	29	2	no	—
116	C₂²×C₄²	—	29	2	no	—
117	C₂²×C₄²	D₂₉	1	1	yes	characteristic, complemented
118	C₂×C₂₉	—	2	4	no	—
119	C₂×C₂₉	C₄×#41	1	4	yes	complemented
120	C₂×C₂₉	C₂⁴×C₄	1	1	yes	characteristic
121	C₂×C₂₉	C₂²×#41	1	2	yes	—
122	D₂₉	—	2	8	no	—
123	C₂×#41	—	29	16	no	—
124	C₂×#41	—	29	8	no	—
125	C₂²×Q₈	—	29	1	no	—
126	C₂²×D₄	—	29	2	no	—
127	C₂²×D₄	—	29	1	no	—
128	C₂³×C₄	—	29	2	no	—
129	C₂³×C₄	—	29	4	no	—
130	C₂³×C₄	C₂×D₂₉	1	2	yes	—
131	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
132	C₄×Q₈	—	29	16	no	—
133	C₄×D₄	—	29	32	no	—
134	C₄×D₄	—	29	16	no	—
135	#117	—	29	8	no	—
136	#117	—	29	16	no	—
137	C₂×C₄⋊C₄	—	29	4	no	—
138	C₂×C₄⋊C₄	—	29	2	no	—
139	C₂×C₂²⋊C₄	—	29	2	no	—
140	C₂×C₂²⋊C₄	—	29	4	no	—
141	C₂×C₄²	—	29	2	no	—
142	C₂×C₄²	—	29	4	no	—
143	C₂×C₄²	—	29	16	no	—
144	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
145	C₂×C₄²	C₂×D₂₉	1	2	yes	—
146	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
147	C₂×C₄²	C₂×D₂₉	1	8	yes	complemented
148	C₂₉	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
149	C₂⁴	—	29	2	no	—
150	C₂⁴	C₂²×D₂₉	1	1	yes	characteristic
151	#41	—	29	64	no	—
152	C₂×Q₈	—	29	8	no	—
153	C₂×Q₈	—	29	4	no	—
154	C₂×D₄	—	29	16	no	—
155	C₂×D₄	—	29	8	no	—
156	C₂×D₄	—	29	8	no	—
157	C₂×D₄	—	29	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	29	2	no	—
160	C₂²×C₄	—	29	16	no	—
161	C₂²×C₄	—	29	8	no	—
162	C₂²×C₄	—	29	4	no	—
163	C₂²×C₄	—	29	16	no	—
164	C₂²×C₄	—	58	16	no	—
165	C₂²×C₄	C₄×D₂₉	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
167	C₂²×C₄	C₂²×D₂₉	1	8	yes	—
168	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₉	1	4	yes	—
171	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
172	C₄⋊C₄	—	29	16	no	—
173	C₄⋊C₄	—	29	8	no	—
174	C₂²⋊C₄	—	29	8	no	—
175	C₂²⋊C₄	—	29	16	no	—
176	C₄²	—	29	8	no	—
177	C₄²	—	29	16	no	—
178	C₄²	C₂²×D₂₉	1	4	yes	—
179	C₄²	C₂²×D₂₉	1	4	yes	—
180	C₄²	C₂²×D₂₉	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	29	8	no	—
183	C₂³	—	29	4	no	—
184	C₂³	—	58	8	no	—
185	C₂³	C₂×C₄×D₂₉	1	4	yes	—
186	C₂³	C₂³×D₂₉	1	2	yes	—
187	C₂³	C₂³×D₂₉	1	1	yes	characteristic
188	Q₈	—	29	16	no	—
189	D₄	—	29	32	no	—
190	D₄	—	29	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	29	8	no	—
194	C₂×C₄	—	29	4	no	—
195	C₂×C₄	—	29	32	no	—
196	C₂×C₄	—	29	8	no	—
197	C₂×C₄	—	58	8	no	—
198	C₂×C₄	—	58	32	no	—
199	C₂×C₄	C₂×C₄×D₂₉	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₂₉	1	4	yes	—
201	C₂×C₄	C₂×C₄×D₂₉	1	16	yes	complemented
202	C₂×C₄	#90805	1	8	yes	direct product factor
203	C₂×C₄	C₂³×D₂₉	1	4	yes	—
204	C₂×C₄	C₂³×D₂₉	1	2	yes	—
205	C₂×C₄	C₂³×D₂₉	1	2	yes	—
206	C₂×C₄	C₂³×D₂₉	1	4	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	29	8	no	—
210	C₂²	—	58	16	no	—
211	C₂²	—	58	8	no	—
212	C₂²	C₂²×C₄×D₂₉	1	4	yes	—
213	C₂²	C₂²×C₄×D₂₉	1	2	yes	—
214	C₂²	C₂×#90805	1	4	yes	—
215	C₂²	C₂⁴×D₂₉	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	29	8	no	—
218	C₄	—	58	8	no	—
219	C₄	C₂²×C₄×D₂₉	1	4	yes	—
220	C₄	C₂²×C₄×D₂₉	1	4	yes	—
221	C₄	C₂×#90805	1	8	yes	direct product factor
222	C₂	—	2	4	no	—
223	C₂	—	58	8	no	—
224	C₂	C₄×#90805	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₂₉	1	1	yes	characteristic
226	C₂	C₂²×#90805	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₂₉
commutator	C₂×C₂₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₂₉
Fitting	C₂²×C₄²×C₂₉	C₂
socle	C₂³×C₂₉	C₂⁴
prime radical	C₂²×#90805	C₂
cosocle	C₂²×C₂₉	C₂⁵
hypercenter	C₂²×C₄²	D₂₉
hypocenter	C₂₉	C₂×C₄×#41
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#90805	C₂	direct (N × H)
3. C₂×C₄×Dic₅₈	C₂	semidirect
4. C₂×C₄×D₁₁₆	C₂	semidirect
5. C₂×C₄×C₂²⋊D₂₉	C₂	semidirect
6. C₂×#11745288	C₂	semidirect
7. C₂×#11745234	C₂	semidirect
8. C₂×C₄²×D₂₉	C₂	semidirect
9. C₂²×C₄²×C₂₉	C₂	semidirect
13. C₂×#90805	C₄	direct (N × H)
22. C₂×C₄²×C₂₉	C₂²	semidirect
24. C₄×C₂²⋊D₂₉	C₂²	semidirect
26. #11745288	C₂²	semidirect
31. C₄×D₁₁₆	C₂²	semidirect
32. #11745234	C₂²	semidirect
33. C₄²×D₂₉	C₂²	semidirect
34. C₄×Dic₅₈	C₂²	semidirect
38. C₂²×C₄×C₂₉	C₂×C₄	semidirect
45. C₂×C₂²⋊D₂₉	C₂×C₄	semidirect
49. #90805	C₂×C₄	direct (N × H)
50. C₂×D₁₁₆	C₂×C₄	semidirect
53. C₂×C₄×D₂₉	C₂×C₄	semidirect
56. C₂×Dic₅₈	C₂×C₄	semidirect
78. C₂×C₄×C₂₉	C₂²×C₄	semidirect
79. C₂×C₄×C₂₉	#41	semidirect
84. C₂²⋊D₂₉	C₂²×C₄	semidirect
89. D₁₁₆	C₂²×C₄	semidirect
91. C₄×D₂₉	C₂²×C₄	semidirect
92. Dic₅₈	C₂²×C₄	semidirect
106. C₄×C₂₉	C₂×#41	semidirect
117. C₂²×C₄²	D₂₉	semidirect
119. C₂×C₂₉	C₄×#41	semidirect
147. C₂×C₄²	C₂×D₂₉	semidirect
148. C₂₉	C₂×C₄×#41	semidirect
165. C₂²×C₄	C₄×D₂₉	semidirect
201. C₂×C₄	C₂×C₄×D₂₉	semidirect
202. C₂×C₄	#90805	direct (N × H)
221. C₄	C₂×#90805	direct (N × H)
224. C₂	C₄×#90805	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×#90805	2
C₄×#90805	2
C₂×C₄×Dic₅₈	2
C₂×C₄×D₁₁₆	2
C₂×C₄×C₂²⋊D₂₉	2
C₂×#11745288	2
C₂×#11745234	2
C₂×C₄²×D₂₉	2
C₂²×C₄²×C₂₉	2
C₂×C₄×#41	29

Subgroups — C2×C4×#90805 #16832672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#90805 #16832672