Subgroups — C₃×#90314 #13060967

Order 1344.

412 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#6507	C₂	1	2	yes	maximal, complemented
3	C₃×#6487	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₃×#6452	C₂	1	2	yes	maximal, complemented
6	C₃×#6452	C₂	1	2	yes	maximal, complemented
7	C₃×#6451	C₂	1	2	yes	maximal, complemented
8	C₃×#6441	C₂	1	2	yes	maximal
9	C₃×#6440	C₂	1	2	yes	maximal
10	#90314	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
16	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
17	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₂²⋊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	2	yes	—
25	C₃×Dic₇⋊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×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄×Dic₇	C₂²	1	2	yes	—
31	C₃×C₄×Dic₇	C₂²	1	2	yes	—
32	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
33	#6507	C₂×C₃	1	2	yes	complemented
34	#6487	C₂×C₃	1	1	yes	characteristic
35	C₂×Dic₇⋊C₄	C₂×C₃	1	1	yes	characteristic
36	#6452	C₂×C₃	1	2	yes	complemented
37	#6452	C₂×C₃	1	2	yes	complemented
38	#6451	C₂×C₃	1	2	yes	complemented
39	#6441	C₂×C₃	1	2	yes	—
40	#6440	C₂×C₃	1	2	yes	—
41	C₃×#544	—	7	1	no	maximal, Hall
42	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
43	C₃×C₇×Q₈	—	4	1	no	—
44	C₂×C₃×C₄×C₇	—	4	1	no	—
45	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×Dic₇	—	4	1	no	—
50	C₂×C₃×Dic₇	Q₈	1	4	yes	complemented
51	C₂×C₃×Dic₇	C₂³	1	2	yes	—
52	C₂×C₃×Dic₇	C₂³	1	2	yes	—
53	C₂×C₃×Dic₇	C₂³	1	2	yes	—
54	C₂×C₃×Dic₇	C₂³	1	2	yes	—
55	C₃×Dic₁₄	—	4	1	no	—
56	C₃×Dic₁₄	—	4	2	no	—
57	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
59	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
60	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
61	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
62	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
63	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
67	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
68	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
69	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
70	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
71	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
72	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
73	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
76	C₄×Dic₇	C₂²×C₃	1	2	yes	—
77	C₄×Dic₇	C₂²×C₃	1	2	yes	—
78	C₃×C₄⋊Q₈	—	7	2	no	—
79	C₃×C₄⋊Q₈	—	7	2	no	—
80	C₃×#125	—	7	2	no	—
81	C₃×#125	—	7	2	no	—
82	C₃×C₂²⋊Q₈	—	7	1	no	—
83	C₃×C₂²⋊Q₈	—	7	2	no	—
84	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic
85	C₃×#117	—	7	2	no	—
86	C₂×C₃×C₄⋊C₄	—	7	1	no	—
87	C₂²×C₃×C₇	—	2	1	no	—
88	C₂²×C₃×C₇	—	2	1	no	—
89	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
91	C₃×C₄×C₇	—	2	1	no	—
92	C₃×C₄×C₇	—	2	2	no	—
93	C₃×C₄×C₇	—	2	1	no	—
94	C₃×C₄×C₇	—	2	2	no	—
95	C₃×Dic₇	—	2	2	no	—
96	C₃×Dic₇	—	2	2	no	—
97	C₃×Dic₇	—	2	2	no	—
98	C₃×Dic₇	C₂×Q₈	1	4	yes	—
99	#544	—	7	1	no	Sylow
100	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
101	C₇×Q₈	—	4	1	no	—
102	C₂×C₄×C₇	—	4	1	no	—
103	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
104	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
105	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
107	C₂×Dic₇	—	4	1	no	—
108	C₂×Dic₇	C₃×Q₈	1	4	yes	complemented
109	C₂×Dic₇	C₂³×C₃	1	2	yes	—
110	C₂×Dic₇	C₂³×C₃	1	2	yes	—
111	C₂×Dic₇	C₂³×C₃	1	2	yes	—
112	C₂×Dic₇	C₂³×C₃	1	2	yes	—
113	Dic₁₄	—	4	1	no	—
114	Dic₁₄	—	4	2	no	—
115	C₂×C₃×Q₈	—	7	1	no	—
116	C₂×C₃×Q₈	—	7	2	no	—
117	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
118	C₂²×C₃×C₄	—	7	1	no	—
119	C₂²×C₃×C₄	—	7	1	no	—
120	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	7	2	no	—
122	C₃×C₄⋊C₄	—	7	2	no	—
123	C₃×C₄⋊C₄	—	7	2	no	—
124	C₃×C₄⋊C₄	—	7	2	no	—
125	C₃×C₄⋊C₄	—	7	2	no	—
126	C₃×C₄⋊C₄	—	7	2	no	—
127	C₃×C₄⋊C₄	—	7	2	no	—
128	C₃×C₄⋊C₄	—	7	1	no	—
129	C₃×C₄⋊C₄	—	7	2	no	—
130	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
131	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
132	C₃×C₂²⋊C₄	—	7	2	no	—
133	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
134	C₃×C₄²	—	7	2	no	—
135	C₃×C₄²	—	7	2	no	—
136	C₂×C₃×C₇	—	2	2	no	—
137	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₄⋊Q₈	—	7	2	no	—
141	C₄⋊Q₈	—	7	2	no	—
142	#125	—	7	2	no	—
143	#125	—	7	2	no	—
144	C₂²⋊Q₈	—	7	1	no	—
145	C₂²⋊Q₈	—	7	2	no	—
146	C₂²⋊Q₈	C₃×D₇	1	1	yes	characteristic
147	#117	—	7	2	no	—
148	C₂×C₄⋊C₄	—	7	1	no	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	—	2	1	no	—
151	C₂²×C₇	C₂×C₃×Q₈	1	2	yes	—
152	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
153	C₄×C₇	—	2	1	no	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	1	no	—
156	C₄×C₇	—	2	2	no	—
157	Dic₇	—	2	2	no	—
158	Dic₇	—	2	2	no	—
159	Dic₇	—	2	2	no	—
160	Dic₇	C₂×C₃×Q₈	1	4	yes	—
161	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
162	C₃×Q₈	—	4	1	no	—
163	C₃×Q₈	—	28	1	no	—
164	C₃×Q₈	—	28	2	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	—	7	4	no	—
167	C₂×C₃×C₄	—	7	2	no	—
168	C₂×C₃×C₄	—	7	2	no	—
169	C₂×C₃×C₄	—	7	2	no	—
170	C₂×C₃×C₄	—	7	2	no	—
171	C₂×C₃×C₄	—	28	1	no	—
172	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
173	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
174	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
176	C₃×C₇	#544	1	1	yes	characteristic, complemented, Hall
177	C₂×Q₈	—	7	1	no	—
178	C₂×Q₈	—	7	2	no	—
179	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
180	C₂²×C₄	—	7	1	no	—
181	C₂²×C₄	—	7	1	no	—
182	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
183	C₄⋊C₄	—	7	2	no	—
184	C₄⋊C₄	—	7	2	no	—
185	C₄⋊C₄	—	7	2	no	—
186	C₄⋊C₄	—	7	2	no	—
187	C₄⋊C₄	—	7	2	no	—
188	C₄⋊C₄	—	7	2	no	—
189	C₄⋊C₄	—	7	2	no	—
190	C₄⋊C₄	—	7	1	no	—
191	C₄⋊C₄	—	7	2	no	—
192	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
193	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
194	C₂²⋊C₄	—	7	2	no	—
195	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
196	C₄²	—	7	2	no	—
197	C₄²	—	7	2	no	—
198	C₂×C₇	—	2	2	no	—
199	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
200	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	Q₈×D₇	1	2	yes	—
205	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	7	4	no	—
211	C₃×C₄	—	14	2	no	—
212	C₃×C₄	—	14	2	no	—
213	C₃×C₄	—	14	2	no	—
214	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
215	Q₈	—	4	1	no	—
216	Q₈	—	28	1	no	—
217	Q₈	—	28	2	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	7	4	no	—
220	C₂×C₄	—	7	2	no	—
221	C₂×C₄	—	7	2	no	—
222	C₂×C₄	—	7	2	no	—
223	C₂×C₄	—	7	2	no	—
224	C₂×C₄	—	28	1	no	—
225	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
226	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
227	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
229	C₇	C₃×#544	1	1	yes	characteristic, complemented, Sylow
230	C₂×C₃	—	2	2	no	—
231	C₂×C₃	#6548	1	1	yes	characteristic
232	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
233	C₂×C₃	#6554	1	1	yes	characteristic
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	C₃×Q₈×D₇	1	2	yes	—
237	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	7	4	no	—
243	C₄	—	14	2	no	—
244	C₄	—	14	2	no	—
245	C₄	—	14	2	no	—
246	C₃	#90314	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	2	2	no	—
248	C₂	C₃×#6548	1	1	yes	characteristic
249	C₂	C₂×C₃×Q₈×D₇	1	1	yes	characteristic
250	C₂	C₃×#6554	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×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂³×C₃×C₇	C₂³
hypercenter	C₃×C₂²⋊Q₈	D₇
hypocenter	C₇	C₃×#544
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×#6507	C₂	semidirect
5. C₃×#6452	C₂	semidirect
6. C₃×#6452	C₂	semidirect
7. C₃×#6451	C₂	semidirect
10. #90314	C₃	direct (N × H)
33. #6507	C₂×C₃	semidirect
36. #6452	C₂×C₃	semidirect
37. #6452	C₂×C₃	semidirect
38. #6451	C₂×C₃	semidirect
50. C₂×C₃×Dic₇	Q₈	semidirect
108. C₂×Dic₇	C₃×Q₈	semidirect
176. C₃×C₇	#544	semidirect
229. C₇	C₃×#544	semidirect
246. C₃	#90314	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×C₂²⋊Q₈	2
C₃×#6507	2
C₃×#6487	2
C₂×C₃×Dic₇⋊C₄	2
C₃×#6452	2
C₃×#6451	2
C₃×#6441	2
C₃×#6440	2
#90314	3
C₃×#544	7

Subgroups — C3×#90314 #13060967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90314 #13060967