Subgroups — C₃×#90336 #13060987

Order 1344.

436 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6509	C₂	1	2	yes	maximal, complemented
3	C₃×#6493	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6461	C₂	1	2	yes	maximal, complemented
6	C₃×#6457	C₂	1	2	yes	maximal, complemented
7	C₃×#6453	C₂	1	2	yes	maximal, complemented
8	C₃×#6449	C₂	1	2	yes	maximal, complemented
9	C₃×(C₂²⋊D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
10	#90336	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	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
14	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
18	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
22	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
23	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×C₄×Dic₇	C₂²	1	2	yes	—
32	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#6509	C₂×C₃	1	2	yes	complemented
34	#6493	C₂×C₃	1	1	yes	characteristic
35	C₂×Dic₇⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
36	#6461	C₂×C₃	1	2	yes	complemented
37	#6457	C₂×C₃	1	2	yes	complemented
38	#6453	C₂×C₃	1	2	yes	complemented
39	#6449	C₂×C₃	1	2	yes	complemented
40	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
41	C₃×#539	—	7	1	no	maximal, Hall
42	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₃×D₇	C₂³	1	2	yes	—
44	C₃×C₇×Q₈	—	4	1	no	—
45	C₂×C₃×C₄×C₇	—	4	1	no	—
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×C₄×C₇	C₂³	1	2	yes	—
50	C₃×C₂²⋊D₇	—	2	4	no	—
51	C₂×C₃×Dic₇	—	2	2	no	—
52	C₂×C₃×Dic₇	—	4	1	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	—	4	2	no	—
58	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
61	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
62	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
63	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
65	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
68	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
69	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
70	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
71	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
72	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
73	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
74	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
75	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
76	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
77	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄×Dic₇	C₂²×C₃	1	2	yes	—
79	C₃×#126	—	7	2	no	—
80	C₃×#125	—	7	2	no	—
81	C₃×#123	—	7	2	no	—
82	C₃×#123	—	7	2	no	—
83	C₃×C₂²⋊Q₈	—	7	2	no	—
84	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
85	C₃×#121	—	7	1	no	—
86	C₃×C₄×D₄	—	7	2	no	—
87	C₂×C₃×C₄⋊C₄	—	7	1	no	—
88	C₂²×C₃×C₇	—	2	1	no	—
89	C₂²×C₃×C₇	—	2	1	no	—
90	C₂²×C₃×C₇	#41	1	2	yes	—
91	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₃×D₇	—	2	2	no	—
93	C₂×C₃×D₇	—	2	2	no	—
94	C₂×C₃×D₇	—	2	2	no	—
95	C₃×C₄×C₇	—	2	2	no	—
96	C₃×C₄×C₇	—	2	1	no	—
97	C₃×C₄×C₇	—	2	1	no	—
98	C₃×C₄×C₇	—	2	2	no	—
99	C₃×Dic₇	—	2	2	no	—
100	C₃×Dic₇	—	2	2	no	—
101	C₃×Dic₇	—	2	2	no	—
102	#539	—	7	1	no	Sylow
103	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
104	C₂²×D₇	C₂³×C₃	1	2	yes	—
105	C₇×Q₈	—	4	1	no	—
106	C₂×C₄×C₇	—	4	1	no	—
107	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
108	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
111	C₂²⋊D₇	—	2	4	no	—
112	C₂×Dic₇	—	2	2	no	—
113	C₂×Dic₇	—	4	1	no	—
114	C₂×Dic₇	C₂³×C₃	1	2	yes	—
115	C₂×Dic₇	C₂³×C₃	1	2	yes	—
116	C₂×Dic₇	C₂³×C₃	1	2	yes	—
117	D₂₈	—	4	1	no	—
118	C₄×D₇	—	4	2	no	—
119	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
120	C₂×C₃×D₄	—	7	2	no	—
121	C₂×C₃×D₄	—	7	1	no	—
122	C₂²×C₃×C₄	—	7	1	no	—
123	C₂²×C₃×C₄	—	7	1	no	—
124	C₂²×C₃×C₄	—	7	2	no	—
125	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	7	2	no	—
127	C₃×C₄⋊C₄	—	7	2	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	1	no	—
130	C₃×C₄⋊C₄	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
133	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
134	C₃×C₂²⋊C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	2	no	—
136	C₃×C₂²⋊C₄	—	7	2	no	—
137	C₃×C₂²⋊C₄	—	7	2	no	—
138	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
139	C₃×C₄²	—	7	2	no	—
140	C₂×C₃×C₇	—	2	2	no	—
141	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₃×D₇	—	4	2	no	—
145	#126	—	7	2	no	—
146	#125	—	7	2	no	—
147	#123	—	7	2	no	—
148	#123	—	7	2	no	—
149	C₂²⋊Q₈	—	7	2	no	—
150	C₂²⋊Q₈	C₃×D₇	1	1	yes	characteristic, complemented
151	#121	—	7	1	no	—
152	C₄×D₄	—	7	2	no	—
153	C₂×C₄⋊C₄	—	7	1	no	—
154	C₂²×C₇	—	2	1	no	—
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	C₃×#41	1	2	yes	—
157	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
158	C₂×D₇	—	2	2	no	—
159	C₂×D₇	—	2	2	no	—
160	C₂×D₇	—	2	2	no	—
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	2	2	no	—
165	Dic₇	—	2	2	no	—
166	Dic₇	—	2	2	no	—
167	Dic₇	—	2	2	no	—
168	C₂³×C₃	—	7	2	no	—
169	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
170	C₃×Q₈	—	4	1	no	—
171	C₃×D₄	—	14	4	no	—
172	C₃×D₄	—	28	1	no	—
173	C₂×C₃×C₄	—	4	1	no	—
174	C₂×C₃×C₄	—	7	2	no	—
175	C₂×C₃×C₄	—	7	2	no	—
176	C₂×C₃×C₄	—	7	2	no	—
177	C₂×C₃×C₄	—	14	2	no	—
178	C₂×C₃×C₄	—	28	2	no	—
179	C₂×C₃×C₄	—	28	1	no	—
180	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
183	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
184	C₃×C₇	#539	1	1	yes	characteristic, complemented, Hall
185	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
186	C₂×D₄	—	7	2	no	—
187	C₂×D₄	—	7	1	no	—
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	7	1	no	—
190	C₂²×C₄	—	7	2	no	—
191	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
192	C₄⋊C₄	—	7	2	no	—
193	C₄⋊C₄	—	7	2	no	—
194	C₄⋊C₄	—	7	2	no	—
195	C₄⋊C₄	—	7	1	no	—
196	C₄⋊C₄	—	7	2	no	—
197	C₄⋊C₄	—	7	2	no	—
198	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
199	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
200	C₂²⋊C₄	—	7	2	no	—
201	C₂²⋊C₄	—	7	2	no	—
202	C₂²⋊C₄	—	7	2	no	—
203	C₂²⋊C₄	—	7	2	no	—
204	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
205	C₄²	—	7	2	no	—
206	C₂×C₇	—	2	2	no	—
207	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
208	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
210	D₇	—	4	2	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	14	2	no	—
214	C₂²×C₃	—	14	2	no	—
215	C₂²×C₃	—	14	2	no	—
216	C₂²×C₃	#1161	1	2	yes	—
217	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	14	2	no	—
223	C₃×C₄	—	14	2	no	—
224	C₃×C₄	—	14	2	no	—
225	C₂³	—	7	2	no	—
226	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
227	Q₈	—	4	1	no	—
228	D₄	—	14	4	no	—
229	D₄	—	28	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	7	2	no	—
232	C₂×C₄	—	7	2	no	—
233	C₂×C₄	—	7	2	no	—
234	C₂×C₄	—	14	2	no	—
235	C₂×C₄	—	28	2	no	—
236	C₂×C₄	—	28	1	no	—
237	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
238	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
240	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
241	C₇	C₃×#539	1	1	yes	characteristic, complemented, Sylow
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	28	2	no	—
244	C₂×C₃	#6548	1	1	yes	characteristic
245	C₂×C₃	C₂×#1161	1	1	yes	characteristic
246	C₂×C₃	#6554	1	1	yes	characteristic
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	14	2	no	—
250	C₂²	—	14	2	no	—
251	C₂²	—	14	2	no	—
252	C₂²	C₃×#1161	1	2	yes	—
253	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	14	2	no	—
259	C₄	—	14	2	no	—
260	C₄	—	14	2	no	—
261	C₃	#90336	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	—	2	2	no	—
263	C₂	—	28	2	no	—
264	C₂	C₃×#6548	1	1	yes	characteristic
265	C₂	C₂×C₃×#1161	1	1	yes	characteristic
266	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	C₃×#6493	C₂
hypercenter	C₃×C₂²⋊Q₈	D₇
hypocenter	C₇	C₃×#539
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
1. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
2. C₃×#6509	C₂	semidirect
4. C₂×C₃×Dic₇⋊C₄	C₂	semidirect
5. C₃×#6461	C₂	semidirect
6. C₃×#6457	C₂	semidirect
7. C₃×#6453	C₂	semidirect
8. C₃×#6449	C₂	semidirect
9. C₃×(C₂²⋊D₇)⋊C₄	C₂	semidirect
10. #90336	C₃	direct (N × H)
32. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
33. #6509	C₂×C₃	semidirect
35. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
36. #6461	C₂×C₃	semidirect
37. #6457	C₂×C₃	semidirect
38. #6453	C₂×C₃	semidirect
39. #6449	C₂×C₃	semidirect
40. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
84. C₃×C₂²⋊Q₈	D₇	semidirect
150. C₂²⋊Q₈	C₃×D₇	semidirect
184. C₃×C₇	#539	semidirect
241. C₇	C₃×#539	semidirect
261. C₃	#90336	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×#6509	2
C₃×#6493	2
C₂×C₃×Dic₇⋊C₄	2
C₃×#6461	2
C₃×#6457	2
C₃×#6453	2
C₃×#6449	2
C₃×(C₂²⋊D₇)⋊C₄	2
#90336	3
C₃×#539	7

Subgroups — C3×#90336 #13060987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90336 #13060987