Subgroups — C₃×#90217 #13060871

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 212 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
2	C₂²×C₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6460	C₂	1	4	yes	maximal, complemented
5	C₃×#6449	C₂	1	4	yes	maximal, complemented
6	C₃×#6438	C₂	1	2	yes	maximal, complemented
7	C₃×#6432	C₂	1	2	yes	maximal, complemented
8	#90217	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
11	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
12	C₃×#1157	—	2	4	no	—
13	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₄	—	2	4	no	—
16	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
17	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
18	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₇	C₂²	1	2	yes	—
22	C₃×(C₂×D₇)⋊C₄	C₂²	1	8	yes	—
23	C₃×C₄⋊Dic₇	C₂²	1	8	yes	—
24	C₂×#1157	C₂×C₃	1	1	yes	characteristic
25	C₂²×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
26	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
27	#6460	C₂×C₃	1	4	yes	complemented
28	#6449	C₂×C₃	1	4	yes	complemented
29	#6438	C₂×C₃	1	2	yes	complemented
30	#6432	C₂×C₃	1	2	yes	complemented
31	C₃×#536	—	7	1	no	maximal, Hall
32	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
33	C₂²×C₃×D₇	C₂³	1	2	yes	—
34	C₂×C₃×C₄×C₇	D₄	1	4	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₃×C₂²⋊D₇	—	2	4	no	—
39	C₂×C₃×Dic₇	—	2	4	no	—
40	C₂×C₃×Dic₇	C₂³	1	2	yes	—
41	C₂×C₃×Dic₇	C₂³	1	4	yes	—
42	C₃×D₂₈	—	2	2	no	—
43	C₃×C₄×D₇	—	2	4	no	—
44	C₃×Dic₁₄	—	2	4	no	—
45	C₃×Dic₁₄	—	2	4	no	—
46	C₃×Dic₁₄	—	2	2	no	—
47	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
49	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
50	#1157	—	2	4	no	—
51	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
53	C₂×Dic₁₄	—	2	4	no	—
54	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
55	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
56	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
57	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
59	C₄²×C₇	C₂²×C₃	1	2	yes	—
60	(C₂×D₇)⋊C₄	C₂²×C₃	1	8	yes	—
61	C₄⋊Dic₇	C₂²×C₃	1	8	yes	—
62	C₂×C₃×#41	—	7	1	no	—
63	C₂²×C₃×Q₈	—	7	1	no	—
64	C₃×C₄⋊Q₈	—	7	2	no	—
65	C₃×#124	—	7	2	no	—
66	C₃×#123	—	7	4	no	—
67	C₃×C₂²⋊Q₈	—	7	4	no	—
68	C₃×#117	D₇	1	1	yes	characteristic, complemented
69	C₂²×C₃×C₇	—	2	2	no	—
70	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
71	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₃×D₇	—	2	2	no	—
73	C₂×C₃×D₇	—	2	4	no	—
74	C₃×C₄×C₇	—	2	4	no	—
75	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
76	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₃×Dic₇	—	2	4	no	—
78	C₃×Dic₇	—	2	2	no	—
79	#536	—	7	1	no	Sylow
80	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
81	C₂²×D₇	C₂³×C₃	1	2	yes	—
82	C₂×C₄×C₇	C₃×D₄	1	4	yes	—
83	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
84	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
86	C₂²⋊D₇	—	2	4	no	—
87	C₂×Dic₇	—	2	4	no	—
88	C₂×Dic₇	C₂³×C₃	1	2	yes	—
89	C₂×Dic₇	C₂³×C₃	1	4	yes	—
90	D₂₈	—	2	2	no	—
91	C₄×D₇	—	2	4	no	—
92	Dic₁₄	—	2	4	no	—
93	Dic₁₄	—	2	4	no	—
94	Dic₁₄	—	2	2	no	—
95	C₃×#41	—	14	4	no	—
96	C₂×C₃×Q₈	—	7	2	no	—
97	C₂×C₃×Q₈	—	7	2	no	—
98	C₂×C₃×Q₈	—	7	1	no	—
99	C₂×C₃×Q₈	—	14	4	no	—
100	C₂×C₃×D₄	—	7	2	no	—
101	C₂×C₃×D₄	—	7	1	no	—
102	C₂²×C₃×C₄	—	7	2	no	—
103	C₂²×C₃×C₄	—	7	2	no	—
104	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	7	8	no	—
106	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
107	C₃×C₂²⋊C₄	—	7	8	no	—
108	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
109	C₃×C₄²	C₂×D₇	1	2	yes	—
110	C₂×C₃×C₇	—	2	2	no	—
111	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
113	C₃×D₇	—	4	2	no	—
114	C₂×#41	—	7	1	no	—
115	C₂²×Q₈	—	7	1	no	—
116	C₄⋊Q₈	—	7	2	no	—
117	#124	—	7	2	no	—
118	#123	—	7	4	no	—
119	C₂²⋊Q₈	—	7	4	no	—
120	#117	C₃×D₇	1	1	yes	characteristic, complemented
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
123	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
124	C₂×D₇	—	2	2	no	—
125	C₂×D₇	—	2	4	no	—
126	C₄×C₇	—	2	4	no	—
127	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
128	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
129	Dic₇	—	2	4	no	—
130	Dic₇	—	2	2	no	—
131	C₂³×C₃	—	7	2	no	—
132	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
133	C₃×Q₈	—	14	4	no	—
134	C₃×Q₈	—	14	4	no	—
135	C₃×Q₈	—	14	2	no	—
136	C₃×D₄	—	14	4	no	—
137	C₃×D₄	—	14	2	no	—
138	C₂×C₃×C₄	—	7	4	no	—
139	C₂×C₃×C₄	—	7	2	no	—
140	C₂×C₃×C₄	—	14	4	no	—
141	C₂×C₃×C₄	—	14	4	no	—
142	C₂×C₃×C₄	D₂₈	1	4	yes	—
143	C₂×C₃×C₄	C₂²×D₇	1	4	yes	—
144	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
146	C₃×C₇	#536	1	1	yes	characteristic, complemented, Hall
147	#41	—	14	4	no	—
148	C₂×Q₈	—	7	2	no	—
149	C₂×Q₈	—	7	2	no	—
150	C₂×Q₈	—	7	1	no	—
151	C₂×Q₈	—	14	4	no	—
152	C₂×D₄	—	7	2	no	—
153	C₂×D₄	—	7	1	no	—
154	C₂²×C₄	—	7	2	no	—
155	C₂²×C₄	—	7	2	no	—
156	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
157	C₄⋊C₄	—	7	8	no	—
158	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
159	C₂²⋊C₄	—	7	8	no	—
160	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
161	C₄²	C₂×C₃×D₇	1	2	yes	—
162	C₂×C₇	—	2	2	no	—
163	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
164	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
165	D₇	—	4	2	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	14	2	no	—
168	C₂²×C₃	—	14	4	no	—
169	C₂²×C₃	C₂×D₂₈	1	2	yes	—
170	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	14	2	no	—
173	C₃×C₄	—	14	4	no	—
174	C₃×C₄	C₂×D₂₈	1	2	yes	—
175	C₃×C₄	C₂×D₂₈	1	2	yes	—
176	C₂³	—	7	2	no	—
177	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
178	Q₈	—	14	4	no	—
179	Q₈	—	14	4	no	—
180	Q₈	—	14	2	no	—
181	D₄	—	14	4	no	—
182	D₄	—	14	2	no	—
183	C₂×C₄	—	7	4	no	—
184	C₂×C₄	—	7	2	no	—
185	C₂×C₄	—	14	4	no	—
186	C₂×C₄	—	14	4	no	—
187	C₂×C₄	C₃×D₂₈	1	4	yes	—
188	C₂×C₄	C₂²×C₃×D₇	1	4	yes	—
189	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
191	C₇	C₃×#536	1	1	yes	characteristic, complemented, Sylow
192	C₂×C₃	—	2	2	no	—
193	C₂×C₃	—	28	2	no	—
194	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
195	C₂×C₃	#6554	1	2	yes	—
196	C₂²	—	2	2	no	—
197	C₂²	—	14	2	no	—
198	C₂²	—	14	4	no	—
199	C₂²	C₂×C₃×D₂₈	1	2	yes	—
200	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	14	2	no	—
203	C₄	—	14	4	no	—
204	C₄	C₂×C₃×D₂₈	1	2	yes	—
205	C₄	C₂×C₃×D₂₈	1	2	yes	—
206	C₃	#90217	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	—	2	2	no	—
208	C₂	—	28	2	no	—
209	C₂	C₂²×C₃×D₂₈	1	1	yes	characteristic
210	C₂	C₃×#6554	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
prime radical	C₂×C₃×#1157	C₂
hypercenter	C₃×#117	D₇
hypocenter	C₇	C₃×#536
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
2. C₂²×C₃×Dic₁₄	C₂	semidirect
3. C₃×C₇×#117	C₂	semidirect
4. C₃×#6460	C₂	semidirect
5. C₃×#6449	C₂	semidirect
6. C₃×#6438	C₂	semidirect
7. C₃×#6432	C₂	semidirect
8. #90217	C₃	direct (N × H)
25. C₂²×Dic₁₄	C₂×C₃	semidirect
26. C₇×#117	C₂×C₃	semidirect
27. #6460	C₂×C₃	semidirect
28. #6449	C₂×C₃	semidirect
29. #6438	C₂×C₃	semidirect
30. #6432	C₂×C₃	semidirect
68. C₃×#117	D₇	semidirect
120. #117	C₃×D₇	semidirect
146. C₃×C₇	#536	semidirect
191. C₇	C₃×#536	semidirect
206. C₃	#90217	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×Dic₁₄	2
C₃×C₇×#117	2
C₃×#6460	2
C₃×#6449	2
C₃×#6438	2
C₃×#6432	2
#90217	3
C₃×#536	7

Subgroups — C3×#90217 #13060871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90217 #13060871