Subgroups — C₃×#90237 #13060891

Order 1344.

588 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1159	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×Dic₇⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×(C₂²⋊D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₃×#6440	C₂	1	2	yes	maximal, complemented
10	C₃×#6435	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#90237	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
18	C₃×#1159	C₄	1	8	yes	complemented
19	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
22	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
32	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
34	C₃×C₄×Dic₇	C₂²	1	2	yes	—
35	C₃×C₄×Dic₇	C₂²	1	2	yes	—
36	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
37	C₂×#1159	C₂×C₃	1	1	yes	characteristic
38	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
39	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
40	C₄×C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
41	C₂×Dic₇⋊C₄	C₂×C₃	1	2	yes	complemented
42	D₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
43	#6450	C₂×C₃	1	1	yes	characteristic, complemented
44	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
45	#6440	C₂×C₃	1	2	yes	complemented
46	#6435	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
48	C₃×#520	—	7	1	no	maximal, Hall
49	C₂³×C₃×C₇	C₂³	1	2	yes	—
50	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
51	C₃×C₇×D₄	C₂×C₄	1	4	yes	complemented
52	C₂×C₃×C₄×C₇	—	4	2	no	—
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₃×C₂²⋊D₇	C₂×C₄	1	8	yes	complemented
58	C₂×C₃×Dic₇	—	4	2	no	—
59	C₂×C₃×Dic₇	C₂×C₄	1	8	yes	complemented
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₇	C₂³	1	2	yes	—
63	C₂×C₃×Dic₇	C₂³	1	2	yes	—
64	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
65	C₃×C₄×D₇	—	4	2	no	—
66	C₃×C₄×D₇	C₂×C₄	1	4	yes	complemented
67	C₃×Dic₁₄	C₂×C₄	1	4	yes	complemented
68	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
70	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
71	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
72	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
73	#1159	C₃×C₄	1	8	yes	complemented
74	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
75	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
76	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
77	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
80	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
81	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
85	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
88	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
89	C₄×Dic₇	C₂²×C₃	1	2	yes	—
90	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
91	C₄×Dic₇	C₂²×C₃	1	2	yes	—
92	C₂×C₃×#41	—	7	1	no	—
93	C₃×C₄×Q₈	—	7	1	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	C₃×C₄×D₄	—	7	1	no	—
96	C₃×C₄×D₄	—	7	2	no	—
97	C₃×C₄×D₄	—	7	2	no	—
98	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
99	C₃×#117	—	7	2	no	—
100	C₃×#117	—	7	1	no	—
101	C₂×C₃×C₄⋊C₄	—	7	2	no	—
102	C₂×C₃×C₄⋊C₄	—	7	1	no	—
103	C₂²×C₃×C₇	—	2	2	no	—
104	C₂²×C₃×C₇	—	2	2	no	—
105	C₂²×C₃×C₇	C₂²×C₄	1	4	yes	—
106	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
107	C₂×C₃×D₇	—	2	1	no	—
108	C₂×C₃×D₇	—	2	1	no	—
109	C₂×C₃×D₇	C₂²×C₄	1	2	yes	—
110	C₃×C₄×C₇	—	2	2	no	—
111	C₃×C₄×C₇	—	2	1	no	—
112	C₃×C₄×C₇	—	2	1	no	—
113	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
114	C₃×Dic₇	—	2	2	no	—
115	C₃×Dic₇	—	2	1	no	—
116	C₃×Dic₇	—	2	1	no	—
117	C₃×Dic₇	C₂²×C₄	1	4	yes	—
118	C₃×Dic₇	C₂²×C₄	1	2	yes	—
119	#520	—	7	1	no	Sylow
120	C₂³×C₇	C₂³×C₃	1	2	yes	—
121	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
122	C₇×D₄	C₂×C₃×C₄	1	4	yes	complemented
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
126	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
128	C₂²⋊D₇	C₂×C₃×C₄	1	8	yes	complemented
129	C₂×Dic₇	—	4	2	no	—
130	C₂×Dic₇	C₂×C₃×C₄	1	8	yes	complemented
131	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
132	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
133	C₂×Dic₇	C₂³×C₃	1	2	yes	—
134	C₂×Dic₇	C₂³×C₃	1	2	yes	—
135	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
136	C₄×D₇	—	4	2	no	—
137	C₄×D₇	C₂×C₃×C₄	1	4	yes	complemented
138	Dic₁₄	C₂×C₃×C₄	1	4	yes	complemented
139	C₃×#41	—	7	8	no	—
140	C₂×C₃×Q₈	—	7	1	no	—
141	C₂×C₃×D₄	—	7	2	no	—
142	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
143	C₂²×C₃×C₄	—	7	2	no	—
144	C₂²×C₃×C₄	—	7	1	no	—
145	C₂²×C₃×C₄	—	7	2	no	—
146	C₂²×C₃×C₄	—	7	2	no	—
147	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
148	C₃×C₄⋊C₄	—	7	2	no	—
149	C₃×C₄⋊C₄	—	7	1	no	—
150	C₃×C₄⋊C₄	—	7	2	no	—
151	C₃×C₄⋊C₄	—	7	2	no	—
152	C₃×C₄⋊C₄	—	7	1	no	—
153	C₃×C₄⋊C₄	—	7	1	no	—
154	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
155	C₃×C₂²⋊C₄	—	7	2	no	—
156	C₃×C₂²⋊C₄	—	7	1	no	—
157	C₃×C₂²⋊C₄	—	7	1	no	—
158	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
159	C₃×C₄²	—	7	2	no	—
160	C₃×C₄²	—	7	2	no	—
161	C₃×C₄²	—	7	1	no	—
162	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
163	C₂×C₃×C₇	—	2	4	no	—
164	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
167	C₃×D₇	—	2	2	no	—
168	C₂×#41	—	7	1	no	—
169	C₄×Q₈	—	7	1	no	—
170	C₄×Q₈	—	7	1	no	—
171	C₄×D₄	—	7	1	no	—
172	C₄×D₄	—	7	2	no	—
173	C₄×D₄	—	7	2	no	—
174	C₄×D₄	C₃×D₇	1	1	yes	characteristic, complemented
175	#117	—	7	2	no	—
176	#117	—	7	1	no	—
177	C₂×C₄⋊C₄	—	7	2	no	—
178	C₂×C₄⋊C₄	—	7	1	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	C₂²×C₃×C₄	1	4	yes	—
182	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
183	C₂×D₇	—	2	1	no	—
184	C₂×D₇	—	2	1	no	—
185	C₂×D₇	C₂²×C₃×C₄	1	2	yes	—
186	C₄×C₇	—	2	2	no	—
187	C₄×C₇	—	2	1	no	—
188	C₄×C₇	—	2	1	no	—
189	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
190	Dic₇	—	2	2	no	—
191	Dic₇	—	2	1	no	—
192	Dic₇	—	2	1	no	—
193	Dic₇	C₂²×C₃×C₄	1	4	yes	—
194	Dic₇	C₂²×C₃×C₄	1	2	yes	—
195	C₂³×C₃	—	7	1	no	—
196	C₂³×C₃	C₂²×D₇	1	2	yes	—
197	C₃×Q₈	—	7	4	no	—
198	C₃×D₄	—	7	8	no	—
199	C₃×D₄	C₄×D₇	1	4	yes	complemented
200	C₂×C₃×C₄	—	4	2	no	—
201	C₂×C₃×C₄	—	7	8	no	—
202	C₂×C₃×C₄	—	7	2	no	—
203	C₂×C₃×C₄	—	7	1	no	—
204	C₂×C₃×C₄	—	7	4	no	—
205	C₂×C₃×C₄	—	7	1	no	—
206	C₂×C₃×C₄	—	7	1	no	—
207	C₂×C₃×C₄	—	7	2	no	—
208	C₂×C₃×C₄	—	28	2	no	—
209	C₂×C₃×C₄	—	28	2	no	—
210	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
212	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
214	C₃×C₇	#520	1	1	yes	characteristic, complemented, Hall
215	#41	—	7	8	no	—
216	C₂×Q₈	—	7	1	no	—
217	C₂×D₄	—	7	2	no	—
218	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
219	C₂²×C₄	—	7	2	no	—
220	C₂²×C₄	—	7	1	no	—
221	C₂²×C₄	—	7	2	no	—
222	C₂²×C₄	—	7	2	no	—
223	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
224	C₄⋊C₄	—	7	2	no	—
225	C₄⋊C₄	—	7	1	no	—
226	C₄⋊C₄	—	7	2	no	—
227	C₄⋊C₄	—	7	2	no	—
228	C₄⋊C₄	—	7	1	no	—
229	C₄⋊C₄	—	7	1	no	—
230	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
231	C₂²⋊C₄	—	7	2	no	—
232	C₂²⋊C₄	—	7	1	no	—
233	C₂²⋊C₄	—	7	1	no	—
234	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
235	C₄²	—	7	2	no	—
236	C₄²	—	7	2	no	—
237	C₄²	—	7	1	no	—
238	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
239	C₂×C₇	—	2	4	no	—
240	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
241	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
243	D₇	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	7	2	no	—
247	C₂²×C₃	—	14	1	no	—
248	C₂²×C₃	—	14	1	no	—
249	C₂²×C₃	C₂×C₄×D₇	1	4	yes	—
250	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	7	2	no	—
255	C₃×C₄	—	7	4	no	—
256	C₃×C₄	—	14	2	no	—
257	C₃×C₄	—	14	1	no	—
258	C₃×C₄	—	14	1	no	—
259	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
260	C₂³	—	7	1	no	—
261	C₂³	C₂²×C₃×D₇	1	2	yes	—
262	Q₈	—	7	4	no	—
263	D₄	—	7	8	no	—
264	D₄	C₃×C₄×D₇	1	4	yes	complemented
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	7	8	no	—
267	C₂×C₄	—	7	2	no	—
268	C₂×C₄	—	7	1	no	—
269	C₂×C₄	—	7	4	no	—
270	C₂×C₄	—	7	1	no	—
271	C₂×C₄	—	7	2	no	—
272	C₂×C₄	—	7	1	no	—
273	C₂×C₄	—	28	2	no	—
274	C₂×C₄	—	28	2	no	—
275	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
277	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
279	C₇	C₃×#520	1	1	yes	characteristic, complemented, Sylow
280	C₂×C₃	—	2	4	no	—
281	C₂×C₃	—	14	2	no	—
282	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
283	C₂×C₃	#6548	1	1	yes	characteristic
284	C₂×C₃	#6554	1	1	yes	characteristic
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	7	2	no	—
288	C₂²	—	14	1	no	—
289	C₂²	—	14	1	no	—
290	C₂²	C₂×C₃×C₄×D₇	1	4	yes	—
291	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	7	2	no	—
296	C₄	—	7	4	no	—
297	C₄	—	14	2	no	—
298	C₄	—	14	1	no	—
299	C₄	—	14	1	no	—
300	C₄	C₂×C₃×C₄×D₇	1	2	yes	—
301	C₃	#90237	1	1	yes	characteristic, direct product factor, Sylow
302	C₂	—	2	4	no	—
303	C₂	—	14	2	no	—
304	C₂	C₂²×C₃×C₄×D₇	1	1	yes	characteristic
305	C₂	C₃×#6548	1	1	yes	characteristic
306	C₂	C₃×#6554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×#1159	C₂
cosocle	C₂²×C₇	C₂⁴×C₃
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×#520
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×D₄	C₂	semidirect
3. C₃×D₄×Dic₇	C₂	semidirect
4. C₃×C₄×C₂²⋊D₇	C₂	semidirect
5. C₂×C₃×Dic₇⋊C₄	C₂	semidirect
6. C₃×D₇×C₄⋊C₄	C₂	semidirect
7. C₃×#6450	C₂	semidirect
8. C₃×(C₂²⋊D₇)⋊C₄	C₂	semidirect
9. C₃×#6440	C₂	semidirect
10. C₃×#6435	C₂	semidirect
11. C₃×C₄×Dic₁₄	C₂	semidirect
12. #90237	C₃	direct (N × H)
18. C₃×#1159	C₄	semidirect
38. C₄×C₇×D₄	C₂×C₃	semidirect
39. D₄×Dic₇	C₂×C₃	semidirect
40. C₄×C₂²⋊D₇	C₂×C₃	semidirect
41. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
42. D₇×C₄⋊C₄	C₂×C₃	semidirect
43. #6450	C₂×C₃	semidirect
44. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
45. #6440	C₂×C₃	semidirect
46. #6435	C₂×C₃	semidirect
47. C₄×Dic₁₄	C₂×C₃	semidirect
51. C₃×C₇×D₄	C₂×C₄	semidirect
57. C₃×C₂²⋊D₇	C₂×C₄	semidirect
59. C₂×C₃×Dic₇	C₂×C₄	semidirect
66. C₃×C₄×D₇	C₂×C₄	semidirect
67. C₃×Dic₁₄	C₂×C₄	semidirect
73. #1159	C₃×C₄	semidirect
98. C₃×C₄×D₄	D₇	semidirect
122. C₇×D₄	C₂×C₃×C₄	semidirect
128. C₂²⋊D₇	C₂×C₃×C₄	semidirect
130. C₂×Dic₇	C₂×C₃×C₄	semidirect
137. C₄×D₇	C₂×C₃×C₄	semidirect
138. Dic₁₄	C₂×C₃×C₄	semidirect
174. C₄×D₄	C₃×D₇	semidirect
199. C₃×D₄	C₄×D₇	semidirect
214. C₃×C₇	#520	semidirect
264. D₄	C₃×C₄×D₇	semidirect
279. C₇	C₃×#520	semidirect
301. C₃	#90237	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×#1159	2
C₃×C₄×C₇×D₄	2
C₃×D₄×Dic₇	2
C₃×C₄×C₂²⋊D₇	2
C₂×C₃×Dic₇⋊C₄	2
C₃×D₇×C₄⋊C₄	2
C₃×#6450	2
C₃×(C₂²⋊D₇)⋊C₄	2
C₃×#6440	2
C₃×#6435	2
C₃×C₄×Dic₁₄	2
#90237	3
C₃×#520	7

Subgroups — C3×#90237 #13060891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90237 #13060891