Subgroups — C₃×#90521 #13061171

Order 1344.

668 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	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6510	C₂	1	2	yes	maximal, complemented
4	C₃×#6503	C₂	1	2	yes	maximal, complemented
5	C₃×#6500	C₂	1	4	yes	maximal, complemented
6	C₃×#6493	C₂	1	4	yes	maximal, complemented
7	C₃×#6489	C₂	1	1	yes	maximal, characteristic, complemented
8	#90521	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×D₇	C₂²	1	2	yes	—
10	C₃×C₇×#41	—	2	4	no	—
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	complemented
16	C₂×C₃×C₂²⋊D₇	C₂²	1	8	yes	complemented
17	C₂×C₃×D₂₈	—	2	4	no	—
18	C₂×C₃×D₂₈	C₂²	1	2	yes	—
19	C₂×C₃×D₂₈	C₂²	1	2	yes	—
20	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
21	C₃×(C₂×D₇)⋊C₄	C₂²	1	8	yes	complemented
22	C₃×C₄⋊Dic₇	C₂²	1	2	yes	complemented
23	C₃×C₄×Dic₇	C₂²	1	2	yes	complemented
24	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
26	#6510	C₂×C₃	1	2	yes	complemented
27	#6503	C₂×C₃	1	2	yes	complemented
28	#6500	C₂×C₃	1	4	yes	complemented
29	#6493	C₂×C₃	1	4	yes	complemented
30	#6489	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#535	—	7	1	no	maximal, Hall
32	C₂³×C₃×C₇	C₂³	1	2	yes	—
33	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₃×D₇	—	2	4	no	—
35	C₂²×C₃×D₇	—	4	4	no	—
36	C₂²×C₃×D₇	C₂³	1	4	yes	—
37	C₃×C₇×Q₈	—	2	2	no	—
38	C₃×C₇×D₄	—	2	4	no	—
39	C₃×C₇×D₄	—	2	2	no	—
40	C₂×C₃×C₄×C₇	—	2	4	no	—
41	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
42	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₃×C₂²⋊D₇	—	4	8	no	—
46	C₂×C₃×Dic₇	C₂³	1	4	yes	—
47	C₃×D₂₈	—	2	4	no	—
48	C₃×D₂₈	—	2	4	no	—
49	C₂³×D₇	C₂²×C₃	1	2	yes	—
50	C₇×#41	—	2	4	no	—
51	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
52	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
53	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
54	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	complemented
56	C₂×C₂²⋊D₇	C₂²×C₃	1	8	yes	complemented
57	C₂×D₂₈	—	2	4	no	—
58	C₂×D₂₈	C₂²×C₃	1	2	yes	—
59	C₂×D₂₈	C₂²×C₃	1	2	yes	—
60	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
61	(C₂×D₇)⋊C₄	C₂²×C₃	1	8	yes	complemented
62	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
63	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
64	C₂×C₃×#41	D₇	1	1	yes	characteristic, complemented
65	C₂²×C₃×D₄	—	7	1	no	—
66	C₃×#127	—	7	2	no	—
67	C₃×#124	—	7	2	no	—
68	C₃×#121	—	7	4	no	—
69	C₃×#120	—	7	4	no	—
70	C₃×#117	—	7	1	no	—
71	C₂²×C₃×C₇	—	2	2	no	—
72	C₂²×C₃×C₇	—	2	2	no	—
73	C₂²×C₃×C₇	—	2	4	no	—
74	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
75	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₃×D₇	—	2	4	no	—
77	C₂×C₃×D₇	—	2	8	no	—
78	C₂×C₃×D₇	—	4	8	no	—
79	C₃×C₄×C₇	—	2	2	no	—
80	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
81	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₃×Dic₇	—	2	4	no	—
83	#535	—	7	1	no	Sylow
84	C₂³×C₇	C₂³×C₃	1	2	yes	—
85	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
86	C₂²×D₇	—	2	4	no	—
87	C₂²×D₇	—	4	4	no	—
88	C₂²×D₇	C₂³×C₃	1	4	yes	—
89	C₇×Q₈	—	2	2	no	—
90	C₇×D₄	—	2	4	no	—
91	C₇×D₄	—	2	2	no	—
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	C₃×D₄	1	4	yes	complemented
94	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
95	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
97	C₂²⋊D₇	—	4	8	no	—
98	C₂×Dic₇	C₂³×C₃	1	4	yes	—
99	D₂₈	—	2	4	no	—
100	D₂₈	—	2	4	no	—
101	C₂⁴×C₃	—	7	2	no	—
102	C₃×#41	—	2	4	no	—
103	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
104	C₂×C₃×D₄	—	7	8	no	—
105	C₂×C₃×D₄	—	7	2	no	—
106	C₂×C₃×D₄	—	7	2	no	—
107	C₂×C₃×D₄	—	14	4	no	—
108	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
109	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic, complemented
110	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×D₇	1	2	yes	complemented
112	C₃×C₄⋊C₄	—	7	2	no	—
113	C₃×C₂²⋊C₄	—	7	2	no	—
114	C₃×C₂²⋊C₄	—	7	8	no	—
115	C₃×C₄²	—	7	2	no	—
116	C₂×C₃×C₇	—	2	2	no	—
117	C₂×C₃×C₇	—	4	2	no	—
118	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
120	C₃×D₇	—	4	4	no	—
121	C₂×#41	C₃×D₇	1	1	yes	characteristic, complemented
122	C₂²×D₄	—	7	1	no	—
123	#127	—	7	2	no	—
124	#124	—	7	2	no	—
125	#121	—	7	4	no	—
126	#120	—	7	4	no	—
127	#117	—	7	1	no	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	2	2	no	—
130	C₂²×C₇	—	2	4	no	—
131	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
132	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
133	C₂×D₇	—	2	4	no	—
134	C₂×D₇	—	2	8	no	—
135	C₂×D₇	—	4	8	no	—
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
138	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
139	Dic₇	—	2	4	no	—
140	C₂³×C₃	—	7	4	no	—
141	C₂³×C₃	—	14	4	no	—
142	C₂³×C₃	—	28	4	no	—
143	C₂³×C₃	C₂²×D₇	1	2	yes	—
144	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
145	C₃×Q₈	—	2	2	no	—
146	C₃×D₄	—	2	4	no	—
147	C₃×D₄	—	2	2	no	—
148	C₃×D₄	—	14	4	no	—
149	C₃×D₄	—	14	4	no	—
150	C₃×D₄	—	28	8	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	7	4	no	—
153	C₂×C₃×C₄	C₂²⋊D₇	1	4	yes	complemented
154	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
155	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
157	C₃×C₇	#535	1	1	yes	characteristic, complemented, Hall
158	C₂⁴	—	7	2	no	—
159	#41	—	2	4	no	—
160	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic, complemented
161	C₂×D₄	—	7	8	no	—
162	C₂×D₄	—	7	2	no	—
163	C₂×D₄	—	7	2	no	—
164	C₂×D₄	—	14	4	no	—
165	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
166	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic, complemented
167	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₃×D₇	1	2	yes	complemented
169	C₄⋊C₄	—	7	2	no	—
170	C₂²⋊C₄	—	7	2	no	—
171	C₂²⋊C₄	—	7	8	no	—
172	C₄²	—	7	2	no	—
173	C₂×C₇	—	2	2	no	—
174	C₂×C₇	—	4	2	no	—
175	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
176	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
177	D₇	—	4	4	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	14	8	no	—
182	C₂²×C₃	—	14	4	no	—
183	C₂²×C₃	—	28	8	no	—
184	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
185	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	14	4	no	—
188	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
189	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
190	C₂³	—	7	4	no	—
191	C₂³	—	14	4	no	—
192	C₂³	—	28	4	no	—
193	C₂³	C₂²×C₃×D₇	1	2	yes	—
194	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
195	Q₈	—	2	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	—	14	4	no	—
199	D₄	—	14	4	no	—
200	D₄	—	28	8	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	7	4	no	—
203	C₂×C₄	C₃×C₂²⋊D₇	1	4	yes	complemented
204	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
205	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
207	C₇	C₃×#535	1	1	yes	characteristic, complemented, Sylow
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	4	2	no	—
210	C₂×C₃	—	28	4	no	—
211	C₂×C₃	#6553	1	2	yes	—
212	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	14	8	no	—
217	C₂²	—	14	4	no	—
218	C₂²	—	28	8	no	—
219	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
220	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	14	4	no	—
223	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
224	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
225	C₃	#90521	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	28	4	no	—
229	C₂	C₃×#6553	1	2	yes	—
230	C₂	C₂²×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₃×#41	D₇
hypocenter	C₇	C₃×#535
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×#41	C₂	semidirect
2. C₂²×C₃×D₂₈	C₂	semidirect
3. C₃×#6510	C₂	semidirect
4. C₃×#6503	C₂	semidirect
5. C₃×#6500	C₂	semidirect
6. C₃×#6493	C₂	semidirect
7. C₃×#6489	C₂	semidirect
8. #90521	C₃	direct (N × H)
11. C₂×C₃×C₇×Q₈	C₂²	semidirect
13. C₂×C₃×C₇×D₄	C₂²	semidirect
15. C₂²×C₃×C₄×C₇	C₂²	semidirect
16. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
21. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
22. C₃×C₄⋊Dic₇	C₂²	semidirect
23. C₃×C₄×Dic₇	C₂²	semidirect
24. C₂×C₇×#41	C₂×C₃	semidirect
25. C₂²×D₂₈	C₂×C₃	semidirect
26. #6510	C₂×C₃	semidirect
27. #6503	C₂×C₃	semidirect
28. #6500	C₂×C₃	semidirect
29. #6493	C₂×C₃	semidirect
30. #6489	C₂×C₃	semidirect
41. C₂×C₃×C₄×C₇	D₄	semidirect
51. C₂×C₇×Q₈	C₂²×C₃	semidirect
53. C₂×C₇×D₄	C₂²×C₃	semidirect
55. C₂²×C₄×C₇	C₂²×C₃	semidirect
56. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
61. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
62. C₄⋊Dic₇	C₂²×C₃	semidirect
63. C₄×Dic₇	C₂²×C₃	semidirect
64. C₂×C₃×#41	D₇	semidirect
93. C₂×C₄×C₇	C₃×D₄	semidirect
103. C₂×C₃×Q₈	C₂×D₇	semidirect
109. C₂×C₃×D₄	C₂×D₇	semidirect
111. C₂²×C₃×C₄	C₂×D₇	semidirect
121. C₂×#41	C₃×D₇	semidirect
153. C₂×C₃×C₄	C₂²⋊D₇	semidirect
157. C₃×C₇	#535	semidirect
160. C₂×Q₈	C₂×C₃×D₇	semidirect
166. C₂×D₄	C₂×C₃×D₇	semidirect
168. C₂²×C₄	C₂×C₃×D₇	semidirect
203. C₂×C₄	C₃×C₂²⋊D₇	semidirect
207. C₇	C₃×#535	semidirect
225. C₃	#90521	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×C₇×#41	2
C₂²×C₃×D₂₈	2
C₃×#6510	2
C₃×#6503	2
C₃×#6500	2
C₃×#6493	2
C₃×#6489	2
#90521	3
C₃×#535	7

Subgroups — C3×#90521 #13061171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90521 #13061171