Subgroups — C₃×#90361 #13061012

Order 1344.

472 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6507	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6502	C₂	1	2	yes	maximal, complemented
5	C₃×#6448	C₂	1	2	yes	maximal, complemented
6	C₃×#6447	C₂	1	2	yes	maximal, complemented
7	C₃×(C₂²⋊D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₃×#6440	C₂	1	2	yes	maximal, complemented
9	C₃×#6435	C₂	1	2	yes	maximal, complemented
10	#90361	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
14	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
16	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
18	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
26	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
27	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₃×C₄×Dic₇	C₂²	1	2	yes	—
31	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
32	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	#6510	C₂×C₃	1	1	yes	characteristic, complemented
34	#6507	C₂×C₃	1	1	yes	characteristic, complemented
35	#6502	C₂×C₃	1	2	yes	complemented
36	#6448	C₂×C₃	1	2	yes	complemented
37	#6447	C₂×C₃	1	2	yes	complemented
38	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
39	#6440	C₂×C₃	1	2	yes	complemented
40	#6435	C₂×C₃	1	2	yes	complemented
41	C₃×#555	—	7	1	no	maximal, Hall
42	C₂³×C₃×C₇	C₂³	1	2	yes	—
43	C₂²×C₃×D₇	C₂³	1	2	yes	—
44	C₃×C₇×Q₈	—	4	1	no	—
45	C₃×C₇×D₄	—	4	1	no	—
46	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₃×C₂²⋊D₇	—	2	4	no	—
51	C₃×C₂²⋊D₇	—	4	2	no	—
52	C₂×C₃×Dic₇	—	2	4	no	—
53	C₂×C₃×Dic₇	C₂³	1	2	yes	—
54	C₂×C₃×Dic₇	C₂³	1	2	yes	—
55	C₂×C₃×Dic₇	C₂³	1	2	yes	—
56	C₃×D₂₈	—	4	1	no	—
57	C₃×C₄×D₇	—	2	4	no	—
58	C₃×Dic₁₄	—	4	1	no	—
59	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
62	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
63	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
64	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
66	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
67	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
68	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
69	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
70	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
71	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
73	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
76	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
77	C₄×Dic₇	C₂²×C₃	1	2	yes	—
78	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
80	C₃×C₄⋊Q₈	—	7	1	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#124	—	7	2	no	—
83	C₃×#124	D₇	1	1	yes	characteristic, complemented
84	C₃×#121	—	7	2	no	—
85	C₃×#121	—	7	2	no	—
86	C₃×C₄×D₄	—	7	2	no	—
87	C₃×#117	—	7	2	no	—
88	C₃×#117	—	7	2	no	—
89	C₂²×C₃×C₇	—	2	2	no	—
90	C₂²×C₃×C₇	—	2	2	no	—
91	C₂²×C₃×C₇	—	2	2	no	—
92	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
93	C₂×C₃×D₇	—	2	2	no	—
94	C₂×C₃×D₇	—	2	2	no	—
95	C₂×C₃×D₇	—	2	2	no	—
96	C₃×C₄×C₇	—	2	2	no	—
97	C₃×C₄×C₇	—	2	1	no	—
98	C₃×C₄×C₇	—	2	1	no	—
99	C₃×C₄×C₇	—	2	1	no	—
100	C₃×Dic₇	—	2	2	no	—
101	C₃×Dic₇	—	2	2	no	—
102	C₃×Dic₇	#41	1	4	yes	—
103	#555	—	7	1	no	Sylow
104	C₂³×C₇	C₂³×C₃	1	2	yes	—
105	C₂²×D₇	C₂³×C₃	1	2	yes	—
106	C₇×Q₈	—	4	1	no	—
107	C₇×D₄	—	4	1	no	—
108	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
110	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
112	C₂²⋊D₇	—	2	4	no	—
113	C₂²⋊D₇	—	4	2	no	—
114	C₂×Dic₇	—	2	4	no	—
115	C₂×Dic₇	C₂³×C₃	1	2	yes	—
116	C₂×Dic₇	C₂³×C₃	1	2	yes	—
117	C₂×Dic₇	C₂³×C₃	1	2	yes	—
118	D₂₈	—	4	1	no	—
119	C₄×D₇	—	2	4	no	—
120	Dic₁₄	—	4	1	no	—
121	C₂×C₃×Q₈	—	7	1	no	—
122	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
123	C₂×C₃×D₄	—	7	2	no	—
124	C₂×C₃×D₄	—	7	2	no	—
125	C₂×C₃×D₄	—	7	1	no	—
126	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
127	C₂²×C₃×C₄	—	7	2	no	—
128	C₂²×C₃×C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	2	no	—
130	C₃×C₄⋊C₄	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₂²⋊C₄	—	7	2	no	—
133	C₃×C₂²⋊C₄	—	7	2	no	—
134	C₃×C₂²⋊C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	2	no	—
136	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
137	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
138	C₃×C₄²	—	7	1	no	—
139	C₃×C₄²	—	7	2	no	—
140	C₃×C₄²	—	7	1	no	—
141	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
142	C₂×C₃×C₇	—	4	2	no	—
143	C₂×C₃×C₇	C₂×#41	1	2	yes	—
144	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
145	C₃×D₇	—	4	2	no	—
146	C₄⋊Q₈	—	7	1	no	—
147	#124	—	7	1	no	—
148	#124	—	7	2	no	—
149	#124	C₃×D₇	1	1	yes	characteristic, complemented
150	#121	—	7	2	no	—
151	#121	—	7	2	no	—
152	C₄×D₄	—	7	2	no	—
153	#117	—	7	2	no	—
154	#117	—	7	2	no	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
159	C₂×D₇	—	2	2	no	—
160	C₂×D₇	—	2	2	no	—
161	C₂×D₇	—	2	2	no	—
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	2	1	no	—
166	Dic₇	—	2	2	no	—
167	Dic₇	—	2	2	no	—
168	Dic₇	C₃×#41	1	4	yes	—
169	C₂³×C₃	—	7	2	no	—
170	C₂³×C₃	C₂²×D₇	1	2	yes	—
171	C₃×Q₈	—	4	1	no	—
172	C₃×Q₈	—	28	1	no	—
173	C₃×D₄	—	4	1	no	—
174	C₃×D₄	—	14	4	no	—
175	C₃×D₄	—	28	1	no	—
176	C₃×D₄	—	28	2	no	—
177	C₂×C₃×C₄	—	7	2	no	—
178	C₂×C₃×C₄	—	7	2	no	—
179	C₂×C₃×C₄	—	7	2	no	—
180	C₂×C₃×C₄	—	14	4	no	—
181	C₂×C₃×C₄	—	14	4	no	—
182	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
183	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
186	C₃×C₇	#555	1	1	yes	characteristic, complemented, Hall
187	C₂×Q₈	—	7	1	no	—
188	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
189	C₂×D₄	—	7	2	no	—
190	C₂×D₄	—	7	2	no	—
191	C₂×D₄	—	7	1	no	—
192	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
193	C₂²×C₄	—	7	2	no	—
194	C₂²×C₄	—	7	2	no	—
195	C₄⋊C₄	—	7	2	no	—
196	C₄⋊C₄	—	7	2	no	—
197	C₄⋊C₄	—	7	2	no	—
198	C₂²⋊C₄	—	7	2	no	—
199	C₂²⋊C₄	—	7	2	no	—
200	C₂²⋊C₄	—	7	2	no	—
201	C₂²⋊C₄	—	7	2	no	—
202	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
203	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
204	C₄²	—	7	1	no	—
205	C₄²	—	7	2	no	—
206	C₄²	—	7	1	no	—
207	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
208	C₂×C₇	—	4	2	no	—
209	C₂×C₇	C₂×C₃×#41	1	2	yes	—
210	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
211	D₇	—	4	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	14	2	no	—
216	C₂²×C₃	—	14	2	no	—
217	C₂²×C₃	—	14	2	no	—
218	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	7	4	no	—
224	C₃×C₄	—	14	2	no	—
225	C₃×C₄	—	14	2	no	—
226	C₂³	—	7	2	no	—
227	C₂³	C₂²×C₃×D₇	1	2	yes	—
228	Q₈	—	4	1	no	—
229	Q₈	—	28	1	no	—
230	D₄	—	4	1	no	—
231	D₄	—	14	4	no	—
232	D₄	—	28	1	no	—
233	D₄	—	28	2	no	—
234	C₂×C₄	—	7	2	no	—
235	C₂×C₄	—	7	2	no	—
236	C₂×C₄	—	7	2	no	—
237	C₂×C₄	—	14	4	no	—
238	C₂×C₄	—	14	4	no	—
239	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
240	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
243	C₇	C₃×#555	1	1	yes	characteristic, complemented, Sylow
244	C₂×C₃	—	4	2	no	—
245	C₂×C₃	—	28	2	no	—
246	C₂×C₃	#6548	1	1	yes	characteristic
247	C₂×C₃	D₇×#41	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	14	2	no	—
252	C₂²	—	14	2	no	—
253	C₂²	—	14	2	no	—
254	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	7	4	no	—
260	C₄	—	14	2	no	—
261	C₄	—	14	2	no	—
262	C₃	#90361	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	4	2	no	—
264	C₂	—	28	2	no	—
265	C₂	C₃×#6548	1	1	yes	characteristic
266	C₂	C₃×D₇×#41	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×#124	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#124	D₇
hypocenter	C₇	C₃×#555
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#124	C₂	semidirect
2. C₃×#6510	C₂	semidirect
3. C₃×#6507	C₂	semidirect
4. C₃×#6502	C₂	semidirect
5. C₃×#6448	C₂	semidirect
6. C₃×#6447	C₂	semidirect
7. C₃×(C₂²⋊D₇)⋊C₄	C₂	semidirect
8. C₃×#6440	C₂	semidirect
9. C₃×#6435	C₂	semidirect
10. #90361	C₃	direct (N × H)
32. C₇×#124	C₂×C₃	semidirect
33. #6510	C₂×C₃	semidirect
34. #6507	C₂×C₃	semidirect
35. #6502	C₂×C₃	semidirect
36. #6448	C₂×C₃	semidirect
37. #6447	C₂×C₃	semidirect
38. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
39. #6440	C₂×C₃	semidirect
40. #6435	C₂×C₃	semidirect
83. C₃×#124	D₇	semidirect
149. #124	C₃×D₇	semidirect
186. C₃×C₇	#555	semidirect
243. C₇	C₃×#555	semidirect
262. C₃	#90361	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×C₇×#124	2
C₃×#6510	2
C₃×#6507	2
C₃×#6502	2
C₃×#6448	2
C₃×#6447	2
C₃×(C₂²⋊D₇)⋊C₄	2
C₃×#6440	2
C₃×#6435	2
#90361	3
C₃×#555	7

Subgroups — C3×#90361 #13061012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90361 #13061012