Subgroups — C₃×#90254 #13060905

Order 1344.

472 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	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6501	C₂	1	2	yes	maximal, complemented
3	C₃×#6493	C₂	1	2	yes	maximal, complemented
4	C₃×#6489	C₂	1	2	yes	maximal, complemented
5	C₃×#6455	C₂	1	2	yes	maximal, complemented
6	C₃×#6448	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6432	C₂	1	1	yes	maximal, characteristic, complemented
9	#90254	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	—
13	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
19	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	—
20	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
21	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
22	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
25	C₃×C₄×Dic₇	C₂²	1	4	yes	—
26	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
27	#6501	C₂×C₃	1	2	yes	complemented
28	#6493	C₂×C₃	1	2	yes	complemented
29	#6489	C₂×C₃	1	2	yes	complemented
30	#6455	C₂×C₃	1	2	yes	complemented
31	#6448	C₂×C₃	1	4	yes	complemented
32	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#6432	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#555	—	7	1	no	maximal, Hall
35	C₂³×C₃×C₇	C₂³	1	2	yes	—
36	C₂²×C₃×D₇	C₂³	1	2	yes	—
37	C₃×C₇×D₄	—	2	2	no	—
38	C₂×C₃×C₄×C₇	—	2	4	no	—
39	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊D₇	—	4	4	no	—
44	C₂×C₃×Dic₇	C₂³	1	2	yes	—
45	C₂×C₃×Dic₇	C₂³	1	4	yes	—
46	C₃×D₂₈	—	2	2	no	—
47	C₃×C₄×D₇	—	2	4	no	—
48	C₃×Dic₁₄	—	4	2	no	—
49	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
51	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
52	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
54	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
55	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
57	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
58	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
59	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
60	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
61	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
62	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
63	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
64	C₄×Dic₇	C₂²×C₃	1	4	yes	—
65	C₃×C₄⋊Q₈	—	7	1	no	—
66	C₃×#124	—	7	4	no	—
67	C₃×#121	—	7	2	no	—
68	C₃×#121	—	7	2	no	—
69	C₃×C₄×D₄	—	7	1	no	—
70	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
71	C₃×#117	—	7	2	no	—
72	C₃×#117	—	7	2	no	—
73	C₂²×C₃×C₇	—	2	2	no	—
74	C₂²×C₃×C₇	—	2	2	no	—
75	C₂²×C₃×C₇	—	2	2	no	—
76	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₃×D₇	—	2	2	no	—
78	C₂×C₃×D₇	—	2	2	no	—
79	C₂×C₃×D₇	—	2	2	no	—
80	C₃×C₄×C₇	—	2	2	no	—
81	C₃×C₄×C₇	—	2	1	no	—
82	C₃×C₄×C₇	#41	1	2	yes	—
83	C₃×C₄×C₇	#41	1	2	yes	—
84	C₃×Dic₇	—	2	2	no	—
85	C₃×Dic₇	—	2	4	no	—
86	#555	—	7	1	no	Sylow
87	C₂³×C₇	C₂³×C₃	1	2	yes	—
88	C₂²×D₇	C₂³×C₃	1	2	yes	—
89	C₇×D₄	—	2	2	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
92	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
95	C₂²⋊D₇	—	4	4	no	—
96	C₂×Dic₇	C₂³×C₃	1	2	yes	—
97	C₂×Dic₇	C₂³×C₃	1	4	yes	—
98	D₂₈	—	2	2	no	—
99	C₄×D₇	—	2	4	no	—
100	Dic₁₄	—	4	2	no	—
101	C₂×C₃×Q₈	—	7	2	no	—
102	C₂×C₃×D₄	—	7	4	no	—
103	C₂×C₃×D₄	—	7	1	no	—
104	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
105	C₂²×C₃×C₄	—	7	2	no	—
106	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
107	C₃×C₄⋊C₄	—	7	1	no	—
108	C₃×C₄⋊C₄	—	7	2	no	—
109	C₃×C₄⋊C₄	—	7	2	no	—
110	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
111	C₃×C₂²⋊C₄	—	7	4	no	—
112	C₃×C₂²⋊C₄	—	7	4	no	—
113	C₃×C₂²⋊C₄	—	7	2	no	—
114	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
115	C₃×C₄²	—	7	4	no	—
116	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
117	C₂×C₃×C₇	—	4	2	no	—
118	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
121	C₃×D₇	—	4	2	no	—
122	C₄⋊Q₈	—	7	1	no	—
123	#124	—	7	4	no	—
124	#121	—	7	2	no	—
125	#121	—	7	2	no	—
126	C₄×D₄	—	7	1	no	—
127	C₄×D₄	C₃×D₇	1	1	yes	characteristic, complemented
128	#117	—	7	2	no	—
129	#117	—	7	2	no	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	—	2	2	no	—
133	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
134	C₂×D₇	—	2	2	no	—
135	C₂×D₇	—	2	2	no	—
136	C₂×D₇	—	2	2	no	—
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	1	no	—
139	C₄×C₇	C₃×#41	1	2	yes	—
140	C₄×C₇	C₃×#41	1	2	yes	—
141	Dic₇	—	2	2	no	—
142	Dic₇	—	2	4	no	—
143	C₂³×C₃	—	7	2	no	—
144	C₂³×C₃	C₂²×D₇	1	2	yes	—
145	C₃×Q₈	—	28	2	no	—
146	C₃×D₄	—	2	2	no	—
147	C₃×D₄	—	14	2	no	—
148	C₃×D₄	—	28	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	7	2	no	—
151	C₂×C₃×C₄	—	7	4	no	—
152	C₂×C₃×C₄	—	14	4	no	—
153	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
154	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
157	C₃×C₇	#555	1	1	yes	characteristic, complemented, Hall
158	C₂×Q₈	—	7	2	no	—
159	C₂×D₄	—	7	4	no	—
160	C₂×D₄	—	7	1	no	—
161	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
162	C₂²×C₄	—	7	2	no	—
163	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
164	C₄⋊C₄	—	7	1	no	—
165	C₄⋊C₄	—	7	2	no	—
166	C₄⋊C₄	—	7	2	no	—
167	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
168	C₂²⋊C₄	—	7	4	no	—
169	C₂²⋊C₄	—	7	4	no	—
170	C₂²⋊C₄	—	7	2	no	—
171	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
172	C₄²	—	7	4	no	—
173	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
174	C₂×C₇	—	4	2	no	—
175	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
176	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
177	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
178	D₇	—	4	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	14	2	no	—
183	C₂²×C₃	—	14	2	no	—
184	C₂²×C₃	—	14	2	no	—
185	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	14	2	no	—
189	C₃×C₄	—	14	4	no	—
190	C₃×C₄	#1157	1	2	yes	—
191	C₃×C₄	#1159	1	2	yes	—
192	C₂³	—	7	2	no	—
193	C₂³	C₂²×C₃×D₇	1	2	yes	—
194	Q₈	—	28	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	14	2	no	—
197	D₄	—	28	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	7	2	no	—
200	C₂×C₄	—	7	4	no	—
201	C₂×C₄	—	14	4	no	—
202	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
203	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
206	C₇	C₃×#555	1	1	yes	characteristic, complemented, Sylow
207	C₂×C₃	—	4	2	no	—
208	C₂×C₃	—	28	2	no	—
209	C₂×C₃	C₂×#1157	1	1	yes	characteristic
210	C₂×C₃	C₂×#1159	1	1	yes	characteristic
211	C₂×C₃	#6553	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	14	2	no	—
216	C₂²	—	14	2	no	—
217	C₂²	—	14	2	no	—
218	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	14	2	no	—
222	C₄	—	14	4	no	—
223	C₄	C₃×#1157	1	2	yes	—
224	C₄	C₃×#1159	1	2	yes	—
225	C₃	#90254	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	4	2	no	—
227	C₂	—	28	2	no	—
228	C₂	C₂×C₃×#1157	1	1	yes	characteristic
229	C₂	C₂×C₃×#1159	1	1	yes	characteristic
230	C₂	C₃×#6553	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×#555
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₇×D₄	C₂	semidirect
2. C₃×#6501	C₂	semidirect
3. C₃×#6493	C₂	semidirect
4. C₃×#6489	C₂	semidirect
5. C₃×#6455	C₂	semidirect
6. C₃×#6448	C₂	semidirect
7. C₃×C₄×D₂₈	C₂	semidirect
8. C₃×#6432	C₂	semidirect
9. #90254	C₃	direct (N × H)
26. C₄×C₇×D₄	C₂×C₃	semidirect
27. #6501	C₂×C₃	semidirect
28. #6493	C₂×C₃	semidirect
29. #6489	C₂×C₃	semidirect
30. #6455	C₂×C₃	semidirect
31. #6448	C₂×C₃	semidirect
32. C₄×D₂₈	C₂×C₃	semidirect
33. #6432	C₂×C₃	semidirect
70. C₃×C₄×D₄	D₇	semidirect
127. C₄×D₄	C₃×D₇	semidirect
157. C₃×C₇	#555	semidirect
206. C₇	C₃×#555	semidirect
225. C₃	#90254	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇×D₄	2
C₃×#6501	2
C₃×#6493	2
C₃×#6489	2
C₃×#6455	2
C₃×#6448	2
C₃×C₄×D₂₈	2
C₃×#6432	2
#90254	3
C₃×#555	7

Subgroups — C3×#90254 #13060905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90254 #13060905