Subgroups — C₃×#90234 #13060887

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 216 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
2	C₃×D₄×Dic₇	C₂	1	2	yes	maximal
3	C₂×C₃×C₄⋊Dic₇	C₂	1	2	yes	maximal, complemented
4	C₃×#6487	C₂	1	2	yes	maximal, complemented
5	C₃×#6453	C₂	1	2	yes	maximal, complemented
6	C₃×#6441	C₂	1	4	yes	maximal, complemented
7	C₃×#6432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#90234	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	C₂²×C₃×Dic₇	C₂²	1	4	yes	—
13	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
15	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
18	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	—
19	C₃×C₄⋊Dic₇	C₂²	1	4	yes	—
20	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
21	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
22	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
23	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
24	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄×Dic₇	C₂²	1	2	yes	—
26	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
27	D₄×Dic₇	C₂×C₃	1	2	yes	—
28	C₂×C₄⋊Dic₇	C₂×C₃	1	2	yes	complemented
29	#6487	C₂×C₃	1	2	yes	complemented
30	#6453	C₂×C₃	1	2	yes	complemented
31	#6441	C₂×C₃	1	4	yes	complemented
32	#6432	C₂×C₃	1	1	yes	characteristic, complemented
33	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#554	—	7	1	no	maximal, Hall
35	C₂³×C₃×C₇	C₂³	1	2	yes	—
36	C₃×C₇×D₄	Q₈	1	4	yes	complemented
37	C₂×C₃×C₄×C₇	—	2	4	no	—
38	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₇	—	4	4	no	—
43	C₂×C₃×Dic₇	C₂³	1	2	yes	—
44	C₂×C₃×Dic₇	C₂³	1	2	yes	—
45	C₂×C₃×Dic₇	C₂³	1	4	yes	—
46	C₃×Dic₁₄	—	2	2	no	—
47	C₃×Dic₁₄	—	4	2	no	—
48	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
50	C₂²×Dic₇	C₂²×C₃	1	4	yes	—
51	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
53	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
54	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
55	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
56	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
57	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
58	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
59	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
60	C₄⋊Dic₇	C₂²×C₃	1	4	yes	—
61	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
62	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
63	C₄×Dic₇	C₂²×C₃	1	2	yes	—
64	C₃×C₄⋊Q₈	—	7	1	no	—
65	C₃×#125	—	7	2	no	—
66	C₃×C₂²⋊Q₈	—	7	2	no	—
67	C₃×C₂²⋊Q₈	—	7	4	no	—
68	C₃×C₄×Q₈	—	7	1	no	—
69	C₃×C₄×D₄	—	7	2	no	—
70	C₃×C₄×D₄	D₇	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	—	7	2	no	—
72	C₂²×C₃×C₇	—	2	2	no	—
73	C₂²×C₃×C₇	—	2	2	no	—
74	C₂²×C₃×C₇	C₂×Q₈	1	4	yes	—
75	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
76	C₃×C₄×C₇	—	2	2	no	—
77	C₃×C₄×C₇	—	2	1	no	—
78	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
79	C₃×C₄×C₇	#41	1	2	yes	—
80	C₃×Dic₇	—	2	2	no	—
81	C₃×Dic₇	—	2	4	no	—
82	C₃×Dic₇	—	2	2	no	—
83	#554	—	7	1	no	Sylow
84	C₂³×C₇	C₂³×C₃	1	2	yes	—
85	C₇×D₄	C₃×Q₈	1	4	yes	complemented
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
88	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
91	C₂×Dic₇	—	4	4	no	—
92	C₂×Dic₇	C₂³×C₃	1	2	yes	—
93	C₂×Dic₇	C₂³×C₃	1	2	yes	—
94	C₂×Dic₇	C₂³×C₃	1	4	yes	—
95	Dic₁₄	—	2	2	no	—
96	Dic₁₄	—	4	2	no	—
97	C₂×C₃×Q₈	—	7	1	no	—
98	C₂×C₃×Q₈	—	7	2	no	—
99	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	7	4	no	—
101	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
102	C₃×C₄⋊C₄	—	7	2	no	—
103	C₃×C₄⋊C₄	—	7	4	no	—
104	C₃×C₄⋊C₄	—	7	1	no	—
105	C₃×C₄⋊C₄	—	7	4	no	—
106	C₃×C₄⋊C₄	—	7	2	no	—
107	C₃×C₄⋊C₄	—	7	2	no	—
108	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
109	C₃×C₂²⋊C₄	—	7	4	no	—
110	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
111	C₃×C₄²	—	7	2	no	—
112	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
113	C₂×C₃×C₇	—	2	4	no	—
114	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₄⋊Q₈	—	7	1	no	—
118	#125	—	7	2	no	—
119	C₂²⋊Q₈	—	7	2	no	—
120	C₂²⋊Q₈	—	7	4	no	—
121	C₄×Q₈	—	7	1	no	—
122	C₄×D₄	—	7	2	no	—
123	C₄×D₄	C₃×D₇	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	7	2	no	—
125	C₂²×C₇	—	2	2	no	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	C₂×C₃×Q₈	1	4	yes	—
128	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	2	1	no	—
131	C₄×C₇	C₂×C₃×Q₈	1	2	yes	—
132	C₄×C₇	C₃×#41	1	2	yes	—
133	Dic₇	—	2	2	no	—
134	Dic₇	—	2	4	no	—
135	Dic₇	—	2	2	no	—
136	C₂³×C₃	C₂²×D₇	1	2	yes	—
137	C₃×Q₈	—	14	2	no	—
138	C₃×Q₈	—	28	2	no	—
139	C₃×D₄	Dic₁₄	1	4	yes	complemented
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	7	2	no	—
142	C₂×C₃×C₄	—	7	2	no	—
143	C₂×C₃×C₄	—	7	4	no	—
144	C₂×C₃×C₄	—	28	4	no	—
145	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
147	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
149	C₃×C₇	#554	1	1	yes	characteristic, complemented, Hall
150	C₂×Q₈	—	7	1	no	—
151	C₂×Q₈	—	7	2	no	—
152	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
153	C₂²×C₄	—	7	4	no	—
154	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
155	C₄⋊C₄	—	7	2	no	—
156	C₄⋊C₄	—	7	4	no	—
157	C₄⋊C₄	—	7	1	no	—
158	C₄⋊C₄	—	7	4	no	—
159	C₄⋊C₄	—	7	2	no	—
160	C₄⋊C₄	—	7	2	no	—
161	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
162	C₂²⋊C₄	—	7	4	no	—
163	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
164	C₄²	—	7	2	no	—
165	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
166	C₂×C₇	—	2	4	no	—
167	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
168	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
169	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	C₂×Dic₁₄	1	4	yes	—
173	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	14	2	no	—
177	C₃×C₄	—	14	2	no	—
178	C₃×C₄	—	14	4	no	—
179	C₃×C₄	C₂×Dic₁₄	1	2	yes	—
180	C₃×C₄	#1159	1	2	yes	—
181	C₂³	C₂²×C₃×D₇	1	2	yes	—
182	Q₈	—	14	2	no	—
183	Q₈	—	28	2	no	—
184	D₄	C₃×Dic₁₄	1	4	yes	complemented
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	7	2	no	—
187	C₂×C₄	—	7	2	no	—
188	C₂×C₄	—	7	4	no	—
189	C₂×C₄	—	28	4	no	—
190	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
192	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
194	C₇	C₃×#554	1	1	yes	characteristic, complemented, Sylow
195	C₂×C₃	—	2	4	no	—
196	C₂×C₃	C₂²×Dic₁₄	1	1	yes	characteristic
197	C₂×C₃	C₂×#1159	1	1	yes	characteristic
198	C₂×C₃	#6553	1	1	yes	characteristic
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	C₂×C₃×Dic₁₄	1	4	yes	—
202	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	1	no	—
205	C₄	—	14	2	no	—
206	C₄	—	14	2	no	—
207	C₄	—	14	4	no	—
208	C₄	C₂×C₃×Dic₁₄	1	2	yes	—
209	C₄	C₃×#1159	1	2	yes	—
210	C₃	#90234	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	4	no	—
212	C₂	C₂²×C₃×Dic₁₄	1	1	yes	characteristic
213	C₂	C₂×C₃×#1159	1	1	yes	characteristic
214	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
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×#554
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
3. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
4. C₃×#6487	C₂	semidirect
5. C₃×#6453	C₂	semidirect
6. C₃×#6441	C₂	semidirect
7. C₃×#6432	C₂	semidirect
8. C₃×C₄×Dic₁₄	C₂	semidirect
9. #90234	C₃	direct (N × H)
28. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
29. #6487	C₂×C₃	semidirect
30. #6453	C₂×C₃	semidirect
31. #6441	C₂×C₃	semidirect
32. #6432	C₂×C₃	semidirect
33. C₄×Dic₁₄	C₂×C₃	semidirect
36. C₃×C₇×D₄	Q₈	semidirect
85. C₇×D₄	C₃×Q₈	semidirect
139. C₃×D₄	Dic₁₄	semidirect
149. C₃×C₇	#554	semidirect
184. D₄	C₃×Dic₁₄	semidirect
194. C₇	C₃×#554	semidirect
210. C₃	#90234	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×D₄×Dic₇	2
C₂×C₃×C₄⋊Dic₇	2
C₃×#6487	2
C₃×#6453	2
C₃×#6441	2
C₃×#6432	2
C₃×C₄×Dic₁₄	2
#90234	3
C₃×#554	7

Subgroups — C3×#90234 #13060887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90234 #13060887