Subgroups — C₃×#90216 #13060870

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6458	C₂	1	4	yes	maximal, complemented
5	C₃×#6445	C₂	1	4	yes	maximal, complemented
6	C₃×#6438	C₂	1	2	yes	maximal, complemented
7	C₃×#6437	C₂	1	2	yes	maximal, complemented
8	#90216	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×D₇	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
12	C₃×#1157	—	2	4	no	—
13	C₂×C₃×D₂₈	—	2	4	no	—
14	C₂×C₃×D₂₈	C₂²	1	2	yes	—
15	C₂×C₃×D₂₈	C₂²	1	2	yes	—
16	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×D₂₈	C₂²	1	8	yes	complemented
18	C₂×C₃×C₄×D₇	C₂²	1	2	yes	complemented
19	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
20	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₇	C₂²	1	2	yes	complemented
23	C₃×(C₂×D₇)⋊C₄	C₂²	1	8	yes	complemented
24	C₂×#1157	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
27	#6458	C₂×C₃	1	4	yes	complemented
28	#6445	C₂×C₃	1	4	yes	complemented
29	#6438	C₂×C₃	1	2	yes	complemented
30	#6437	C₂×C₃	1	2	yes	complemented
31	C₃×#535	—	7	1	no	maximal, Hall
32	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
33	C₂²×C₃×D₇	—	2	4	no	—
34	C₂²×C₃×D₇	—	4	4	no	—
35	C₂²×C₃×D₇	C₂³	1	4	yes	—
36	C₂²×C₃×D₇	C₂³	1	2	yes	—
37	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
38	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
39	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₃×C₂²⋊D₇	—	2	4	no	—
42	C₂×C₃×Dic₇	C₂³	1	2	yes	—
43	C₃×D₂₈	—	2	4	no	—
44	C₃×D₂₈	—	2	4	no	—
45	C₃×D₂₈	—	2	2	no	—
46	C₃×D₂₈	—	4	8	no	—
47	C₃×C₄×D₇	—	2	4	no	—
48	C₃×Dic₁₄	—	2	2	no	—
49	C₂³×D₇	C₂²×C₃	1	2	yes	—
50	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
52	#1157	—	2	4	no	—
53	C₂×D₂₈	—	2	4	no	—
54	C₂×D₂₈	C₂²×C₃	1	2	yes	—
55	C₂×D₂₈	C₂²×C₃	1	2	yes	—
56	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic, complemented
57	C₂×D₂₈	C₂²×C₃	1	8	yes	complemented
58	C₂×C₄×D₇	C₂²×C₃	1	2	yes	complemented
59	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
60	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
61	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
62	C₄²×C₇	C₂²×C₃	1	2	yes	complemented
63	(C₂×D₇)⋊C₄	C₂²×C₃	1	8	yes	complemented
64	C₂×C₃×#41	—	7	1	no	—
65	C₂²×C₃×D₄	—	7	1	no	—
66	C₃×#127	—	7	2	no	—
67	C₃×#124	—	7	2	no	—
68	C₃×#121	—	7	4	no	—
69	C₃×#120	—	7	4	no	—
70	C₃×#117	D₇	1	1	yes	characteristic, complemented
71	C₂²×C₃×C₇	—	2	2	no	—
72	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
73	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₃×D₇	—	2	2	no	—
75	C₂×C₃×D₇	—	2	4	no	—
76	C₂×C₃×D₇	—	2	4	no	—
77	C₂×C₃×D₇	—	2	8	no	—
78	C₂×C₃×D₇	—	4	8	no	—
79	C₃×C₄×C₇	—	2	4	no	—
80	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
81	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₃×Dic₇	—	2	2	no	—
83	#535	—	7	1	no	Sylow
84	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
85	C₂²×D₇	—	2	4	no	—
86	C₂²×D₇	—	4	4	no	—
87	C₂²×D₇	C₂³×C₃	1	4	yes	—
88	C₂²×D₇	C₂³×C₃	1	2	yes	—
89	C₂×C₄×C₇	C₃×D₄	1	4	yes	complemented
90	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
91	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
93	C₂²⋊D₇	—	2	4	no	—
94	C₂×Dic₇	C₂³×C₃	1	2	yes	—
95	D₂₈	—	2	4	no	—
96	D₂₈	—	2	4	no	—
97	D₂₈	—	2	2	no	—
98	D₂₈	—	4	8	no	—
99	C₄×D₇	—	2	4	no	—
100	Dic₁₄	—	2	2	no	—
101	C₂⁴×C₃	—	7	2	no	—
102	C₃×#41	—	14	4	no	—
103	C₂×C₃×Q₈	—	7	1	no	—
104	C₂×C₃×D₄	—	7	2	no	—
105	C₂×C₃×D₄	—	7	2	no	—
106	C₂×C₃×D₄	—	7	2	no	—
107	C₂×C₃×D₄	—	7	1	no	—
108	C₂×C₃×D₄	—	7	8	no	—
109	C₂×C₃×D₄	—	14	4	no	—
110	C₂²×C₃×C₄	—	7	2	no	—
111	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
112	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	complemented
113	C₃×C₂²⋊C₄	—	7	8	no	—
114	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
115	C₃×C₄²	C₂×D₇	1	2	yes	complemented
116	C₂×C₃×C₇	—	2	2	no	—
117	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
119	C₃×D₇	—	4	2	no	—
120	C₃×D₇	—	4	4	no	—
121	C₂×#41	—	7	1	no	—
122	C₂²×D₄	—	7	1	no	—
123	#127	—	7	2	no	—
124	#124	—	7	2	no	—
125	#121	—	7	4	no	—
126	#120	—	7	4	no	—
127	#117	C₃×D₇	1	1	yes	characteristic, complemented
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
130	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
131	C₂×D₇	—	2	2	no	—
132	C₂×D₇	—	2	4	no	—
133	C₂×D₇	—	2	4	no	—
134	C₂×D₇	—	2	8	no	—
135	C₂×D₇	—	4	8	no	—
136	C₄×C₇	—	2	4	no	—
137	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
138	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
139	Dic₇	—	2	2	no	—
140	C₂³×C₃	—	7	4	no	—
141	C₂³×C₃	—	7	2	no	—
142	C₂³×C₃	—	14	4	no	—
143	C₂³×C₃	—	28	4	no	—
144	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
145	C₃×Q₈	—	14	2	no	—
146	C₃×D₄	—	14	4	no	—
147	C₃×D₄	—	14	4	no	—
148	C₃×D₄	—	14	4	no	—
149	C₃×D₄	—	14	2	no	—
150	C₃×D₄	—	28	8	no	—
151	C₂×C₃×C₄	—	7	2	no	—
152	C₂×C₃×C₄	—	14	4	no	—
153	C₂×C₃×C₄	D₂₈	1	4	yes	complemented
154	C₂×C₃×C₄	C₂²×D₇	1	4	yes	—
155	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
157	C₃×C₇	#535	1	1	yes	characteristic, complemented, Hall
158	C₂⁴	—	7	2	no	—
159	#41	—	14	4	no	—
160	C₂×Q₈	—	7	1	no	—
161	C₂×D₄	—	7	2	no	—
162	C₂×D₄	—	7	2	no	—
163	C₂×D₄	—	7	2	no	—
164	C₂×D₄	—	7	1	no	—
165	C₂×D₄	—	7	8	no	—
166	C₂×D₄	—	14	4	no	—
167	C₂²×C₄	—	7	2	no	—
168	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
169	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	complemented
170	C₂²⋊C₄	—	7	8	no	—
171	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
172	C₄²	C₂×C₃×D₇	1	2	yes	complemented
173	C₂×C₇	—	2	2	no	—
174	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
175	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
176	D₇	—	4	4	no	—
177	D₇	—	4	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	14	2	no	—
180	C₂²×C₃	—	14	4	no	—
181	C₂²×C₃	—	14	4	no	—
182	C₂²×C₃	—	14	8	no	—
183	C₂²×C₃	—	28	8	no	—
184	C₂²×C₃	C₂×D₂₈	1	2	yes	—
185	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
186	C₃×C₄	—	2	4	no	—
187	C₃×C₄	—	14	2	no	—
188	C₃×C₄	C₂×D₂₈	1	2	yes	—
189	C₃×C₄	C₂×D₂₈	1	2	yes	—
190	C₂³	—	7	4	no	—
191	C₂³	—	7	2	no	—
192	C₂³	—	14	4	no	—
193	C₂³	—	28	4	no	—
194	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
195	Q₈	—	14	2	no	—
196	D₄	—	14	4	no	—
197	D₄	—	14	4	no	—
198	D₄	—	14	4	no	—
199	D₄	—	14	2	no	—
200	D₄	—	28	8	no	—
201	C₂×C₄	—	7	2	no	—
202	C₂×C₄	—	14	4	no	—
203	C₂×C₄	C₃×D₂₈	1	4	yes	complemented
204	C₂×C₄	C₂²×C₃×D₇	1	4	yes	—
205	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
207	C₇	C₃×#535	1	1	yes	characteristic, complemented, Sylow
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	28	4	no	—
210	C₂×C₃	—	28	2	no	—
211	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
212	C₂×C₃	#6553	1	2	yes	—
213	C₂²	—	2	2	no	—
214	C₂²	—	14	2	no	—
215	C₂²	—	14	4	no	—
216	C₂²	—	14	4	no	—
217	C₂²	—	14	8	no	—
218	C₂²	—	28	8	no	—
219	C₂²	C₂×C₃×D₂₈	1	2	yes	—
220	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	14	2	no	—
223	C₄	C₂×C₃×D₂₈	1	2	yes	—
224	C₄	C₂×C₃×D₂₈	1	2	yes	—
225	C₃	#90216	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	28	4	no	—
228	C₂	—	28	2	no	—
229	C₂	C₂²×C₃×D₂₈	1	1	yes	characteristic
230	C₂	C₃×#6553	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#117	D₇
hypocenter	C₇	C₃×#535
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#1157	C₂	semidirect
2. C₂²×C₃×D₂₈	C₂	semidirect
3. C₃×C₇×#117	C₂	semidirect
4. C₃×#6458	C₂	semidirect
5. C₃×#6445	C₂	semidirect
6. C₃×#6438	C₂	semidirect
7. C₃×#6437	C₂	semidirect
8. #90216	C₃	direct (N × H)
16. C₂×C₃×D₂₈	C₂²	semidirect
17. C₂×C₃×D₂₈	C₂²	semidirect
18. C₂×C₃×C₄×D₇	C₂²	semidirect
19. C₂×C₃×Dic₁₄	C₂²	semidirect
20. C₃×C₇×C₄⋊C₄	C₂²	semidirect
22. C₃×C₄²×C₇	C₂²	semidirect
23. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
24. C₂×#1157	C₂×C₃	semidirect
25. C₂²×D₂₈	C₂×C₃	semidirect
26. C₇×#117	C₂×C₃	semidirect
27. #6458	C₂×C₃	semidirect
28. #6445	C₂×C₃	semidirect
29. #6438	C₂×C₃	semidirect
30. #6437	C₂×C₃	semidirect
37. C₂×C₃×C₄×C₇	D₄	semidirect
56. C₂×D₂₈	C₂²×C₃	semidirect
57. C₂×D₂₈	C₂²×C₃	semidirect
58. C₂×C₄×D₇	C₂²×C₃	semidirect
59. C₂×Dic₁₄	C₂²×C₃	semidirect
60. C₇×C₄⋊C₄	C₂²×C₃	semidirect
62. C₄²×C₇	C₂²×C₃	semidirect
63. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
70. C₃×#117	D₇	semidirect
89. C₂×C₄×C₇	C₃×D₄	semidirect
112. C₃×C₄⋊C₄	C₂×D₇	semidirect
115. C₃×C₄²	C₂×D₇	semidirect
127. #117	C₃×D₇	semidirect
153. C₂×C₃×C₄	D₂₈	semidirect
157. C₃×C₇	#535	semidirect
169. C₄⋊C₄	C₂×C₃×D₇	semidirect
172. C₄²	C₂×C₃×D₇	semidirect
203. C₂×C₄	C₃×D₂₈	semidirect
207. C₇	C₃×#535	semidirect
225. C₃	#90216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#1157	2
C₂²×C₃×D₂₈	2
C₃×C₇×#117	2
C₃×#6458	2
C₃×#6445	2
C₃×#6438	2
C₃×#6437	2
#90216	3
C₃×#535	7

Subgroups — C3×#90216 #13060870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90216 #13060870