Subgroups — C₃×#90276 #13060927

Order 1344.

472 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6510	C₂	1	2	yes	maximal, complemented
3	C₃×#6457	C₂	1	4	yes	maximal, complemented
4	C₃×#6456	C₂	1	2	yes	maximal, complemented
5	C₃×#6438	C₂	1	2	yes	maximal, complemented
6	C₃×#6437	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×D₂₈	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#90276	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₂₈	C₂²	1	2	yes	—
12	C₂×C₃×D₂₈	C₂²	1	2	yes	—
13	C₂×C₃×D₂₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₇	C₂²	1	4	yes	—
15	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
16	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₇	C₂²	1	2	yes	—
19	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
21	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
22	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
23	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
24	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄×Dic₇	C₂²	1	2	yes	—
26	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#6510	C₂×C₃	1	2	yes	complemented
28	#6457	C₂×C₃	1	4	yes	complemented
29	#6456	C₂×C₃	1	2	yes	complemented
30	#6438	C₂×C₃	1	2	yes	complemented
31	#6437	C₂×C₃	1	1	yes	characteristic
32	C₄×D₂₈	C₂×C₃	1	2	yes	complemented
33	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#559	—	7	1	no	maximal, Hall
35	C₂²×C₃×D₇	C₂³	1	4	yes	—
36	C₃×C₇×Q₈	—	2	2	no	—
37	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
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×Dic₇	C₂³	1	2	yes	—
43	C₂×C₃×Dic₇	C₂³	1	2	yes	—
44	C₃×D₂₈	—	2	4	no	—
45	C₃×D₂₈	—	2	4	no	—
46	C₃×D₂₈	—	4	2	no	—
47	C₃×C₄×D₇	—	4	4	no	—
48	C₃×Dic₁₄	—	2	2	no	—
49	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
50	C₂×D₂₈	C₂²×C₃	1	2	yes	—
51	C₂×D₂₈	C₂²×C₃	1	2	yes	—
52	C₂×D₂₈	C₂²×C₃	1	2	yes	—
53	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
54	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
55	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
58	C₄²×C₇	C₂²×C₃	1	2	yes	—
59	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
60	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
61	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
62	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
63	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
64	C₄×Dic₇	C₂²×C₃	1	2	yes	—
65	C₃×#127	—	7	1	no	—
66	C₃×#124	—	7	2	no	—
67	C₃×#124	—	7	2	no	—
68	C₃×#123	—	7	4	no	—
69	C₃×C₄×Q₈	—	7	1	no	—
70	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, complemented
71	C₃×C₄×D₄	—	7	2	no	—
72	C₃×C₄×D₄	—	7	2	no	—
73	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₃×D₇	—	2	4	no	—
75	C₂×C₃×D₇	—	2	4	no	—
76	C₂×C₃×D₇	—	2	4	no	—
77	C₃×C₄×C₇	—	2	2	no	—
78	C₃×C₄×C₇	—	2	2	no	—
79	C₃×C₄×C₇	—	2	1	no	—
80	C₃×C₄×C₇	#41	1	2	yes	—
81	C₃×C₄×C₇	#41	1	2	yes	—
82	C₃×Dic₇	—	2	2	no	—
83	C₃×Dic₇	—	2	2	no	—
84	#559	—	7	1	no	Sylow
85	C₂²×D₇	C₂³×C₃	1	4	yes	—
86	C₇×Q₈	—	2	2	no	—
87	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
89	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
90	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₂×Dic₇	C₂³×C₃	1	2	yes	—
93	C₂×Dic₇	C₂³×C₃	1	2	yes	—
94	D₂₈	—	2	4	no	—
95	D₂₈	—	2	4	no	—
96	D₂₈	—	4	2	no	—
97	C₄×D₇	—	4	4	no	—
98	Dic₁₄	—	2	2	no	—
99	C₂×C₃×Q₈	—	7	1	no	—
100	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
101	C₂×C₃×D₄	—	7	2	no	—
102	C₂×C₃×D₄	—	7	2	no	—
103	C₂×C₃×D₄	—	7	2	no	—
104	C₂²×C₃×C₄	—	7	4	no	—
105	C₃×C₄⋊C₄	—	7	2	no	—
106	C₃×C₄⋊C₄	—	7	1	no	—
107	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
108	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
109	C₃×C₂²⋊C₄	—	7	4	no	—
110	C₃×C₂²⋊C₄	—	7	4	no	—
111	C₃×C₂²⋊C₄	—	7	4	no	—
112	C₃×C₄²	—	7	2	no	—
113	C₃×C₄²	C₂×D₇	1	2	yes	—
114	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
115	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₃×D₇	—	4	4	no	—
119	#127	—	7	1	no	—
120	#124	—	7	2	no	—
121	#124	—	7	2	no	—
122	#123	—	7	4	no	—
123	C₄×Q₈	—	7	1	no	—
124	C₄×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
125	C₄×D₄	—	7	2	no	—
126	C₄×D₄	—	7	2	no	—
127	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
128	C₂×D₇	—	2	4	no	—
129	C₂×D₇	—	2	4	no	—
130	C₂×D₇	—	2	4	no	—
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	—	2	1	no	—
134	C₄×C₇	C₃×#41	1	2	yes	—
135	C₄×C₇	C₃×#41	1	2	yes	—
136	Dic₇	—	2	2	no	—
137	Dic₇	—	2	2	no	—
138	C₂³×C₃	—	7	4	no	—
139	C₃×Q₈	—	2	2	no	—
140	C₃×Q₈	—	14	2	no	—
141	C₃×D₄	—	14	4	no	—
142	C₃×D₄	—	14	4	no	—
143	C₃×D₄	—	28	2	no	—
144	C₂×C₃×C₄	—	7	2	no	—
145	C₂×C₃×C₄	—	7	2	no	—
146	C₂×C₃×C₄	—	28	4	no	—
147	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
148	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
149	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
152	C₃×C₇	#559	1	1	yes	characteristic, complemented, Hall
153	C₂×Q₈	—	7	1	no	—
154	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
155	C₂×D₄	—	7	2	no	—
156	C₂×D₄	—	7	2	no	—
157	C₂×D₄	—	7	2	no	—
158	C₂²×C₄	—	7	4	no	—
159	C₄⋊C₄	—	7	2	no	—
160	C₄⋊C₄	—	7	1	no	—
161	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
162	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
163	C₂²⋊C₄	—	7	4	no	—
164	C₂²⋊C₄	—	7	4	no	—
165	C₂²⋊C₄	—	7	4	no	—
166	C₄²	—	7	2	no	—
167	C₄²	C₂×C₃×D₇	1	2	yes	—
168	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
169	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
170	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
171	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
172	D₇	—	4	4	no	—
173	C₂²×C₃	—	14	4	no	—
174	C₂²×C₃	—	14	4	no	—
175	C₂²×C₃	—	14	4	no	—
176	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	14	2	no	—
181	C₃×C₄	—	14	2	no	—
182	C₃×C₄	#1157	1	2	yes	—
183	C₃×C₄	#1161	1	2	yes	—
184	C₂³	—	7	4	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	14	2	no	—
187	D₄	—	14	4	no	—
188	D₄	—	14	4	no	—
189	D₄	—	28	2	no	—
190	C₂×C₄	—	7	2	no	—
191	C₂×C₄	—	7	2	no	—
192	C₂×C₄	—	28	4	no	—
193	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
194	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
195	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
198	C₇	C₃×#559	1	1	yes	characteristic, complemented, Sylow
199	C₂×C₃	—	28	4	no	—
200	C₂×C₃	C₂×#1157	1	1	yes	characteristic
201	C₂×C₃	C₂×#1161	1	1	yes	characteristic
202	C₂×C₃	#6553	1	1	yes	characteristic
203	C₂²	—	14	4	no	—
204	C₂²	—	14	4	no	—
205	C₂²	—	14	4	no	—
206	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	14	2	no	—
211	C₄	—	14	2	no	—
212	C₄	C₃×#1157	1	2	yes	—
213	C₄	C₃×#1161	1	2	yes	—
214	C₃	#90276	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	28	4	no	—
216	C₂	C₂×C₃×#1157	1	1	yes	characteristic
217	C₂	C₂×C₃×#1161	1	1	yes	characteristic
218	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×Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₃×#6437	C₂
hypercenter	C₃×C₄×Q₈	D₇
hypocenter	C₇	C₃×#559
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₇×Q₈	C₂	semidirect
2. C₃×#6510	C₂	semidirect
3. C₃×#6457	C₂	semidirect
4. C₃×#6456	C₂	semidirect
5. C₃×#6438	C₂	semidirect
7. C₃×C₄×D₂₈	C₂	semidirect
8. C₃×C₄×Dic₁₄	C₂	semidirect
9. #90276	C₃	direct (N × H)
26. C₄×C₇×Q₈	C₂×C₃	semidirect
27. #6510	C₂×C₃	semidirect
28. #6457	C₂×C₃	semidirect
29. #6456	C₂×C₃	semidirect
30. #6438	C₂×C₃	semidirect
32. C₄×D₂₈	C₂×C₃	semidirect
33. C₄×Dic₁₄	C₂×C₃	semidirect
70. C₃×C₄×Q₈	D₇	semidirect
124. C₄×Q₈	C₃×D₇	semidirect
152. C₃×C₇	#559	semidirect
198. C₇	C₃×#559	semidirect
214. C₃	#90276	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×Q₈	2
C₃×#6510	2
C₃×#6457	2
C₃×#6456	2
C₃×#6438	2
C₃×#6437	2
C₃×C₄×D₂₈	2
C₃×C₄×Dic₁₄	2
#90276	3
C₃×#559	7

Subgroups — C3×#90276 #13060927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90276 #13060927