Subgroups — #13057797

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621353	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6438	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621018	C₂	1	1	yes	maximal, characteristic
5	#10621014	C₂	1	2	yes	maximal, complemented
6	#10621003	C₂	1	2	yes	maximal, complemented
7	Dic₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#10620955	C₂	1	2	yes	maximal, complemented
9	#10620931	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620925	C₂	1	2	yes	maximal, complemented
11	Dic₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#90170	—	3	1	no	maximal, Hall
13	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
18	C₂×#4245	C₂²	1	2	yes	—
19	C₂×Dic₃×D₇	C₂²	1	2	yes	—
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
23	#66629	C₂²	1	2	yes	—
24	#66629	C₂²	1	2	yes	—
25	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
26	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
27	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
28	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
29	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
30	#66577	C₂²	1	2	yes	—
31	Dic₇⋊Dic₃	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
33	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
34	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₇	—	3	1	no	—
37	#6493	—	3	1	no	—
38	#6492	—	3	2	no	—
39	C₄×C₂²⋊D₇	—	3	2	no	—
40	#6487	—	3	1	no	—
41	#6439	—	3	2	no	—
42	#6438	D₃	1	1	yes	characteristic, complemented
43	C₄×D₂₈	—	3	1	no	—
44	#6435	—	3	2	no	—
45	#6433	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5680	—	7	1	no	maximal, Hall
48	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	2	yes	—
54	C₂×Dic₂₁	C₂³	1	2	yes	—
55	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
58	C₄×D₃×C₇	—	2	2	no	—
59	C₄×D₃×C₇	—	2	2	no	—
60	C₄×D₃×C₇	—	2	2	no	—
61	C₂×C₃×Dic₇	C₂³	1	2	yes	—
62	C₃×D₂₈	—	2	2	no	—
63	C₃×Dic₁₄	—	2	2	no	—
64	#4245	—	2	4	no	—
65	Dic₃×D₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	3	1	no	—
67	C₂²×C₄×C₇	—	3	1	no	—
68	C₂²×C₄×C₇	—	3	1	no	—
69	C₂×C₂²⋊D₇	—	3	2	no	—
70	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₄×D₇	—	3	2	no	—
72	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
73	C₄²×C₇	—	3	1	no	—
74	C₄²×C₇	—	3	1	no	—
75	C₄²×C₇	—	3	1	no	—
76	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
77	C₂²⋊Dic₇	—	3	2	no	—
78	(C₂×D₇)⋊C₄	—	3	2	no	—
79	(C₂×D₇)⋊C₄	—	3	2	no	—
80	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	—
81	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	—
82	C₄⋊Dic₇	—	3	1	no	—
83	C₄⋊Dic₇	—	3	1	no	—
84	Dic₇⋊C₄	—	3	2	no	—
85	Dic₇⋊C₄	—	3	2	no	—
86	Dic₇⋊C₄	—	3	2	no	—
87	Dic₇⋊C₄	—	3	2	no	—
88	C₄×Dic₇	—	3	2	no	—
89	C₃×#124	—	7	1	no	—
90	#946	—	7	1	no	—
91	Q₈×Dic₃	—	7	1	no	—
92	#938	—	7	1	no	—
93	D₄×Dic₃	—	7	1	no	—
94	#883	—	7	2	no	—
95	#67⋊C₄	—	7	2	no	—
96	#879	—	7	2	no	—
97	#877	—	7	2	no	—
98	C₄²×D₃	D₇	1	1	yes	characteristic, complemented
99	#870	—	7	1	no	—
100	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
101	C₂×D₃×C₇	—	2	1	no	—
102	C₂×D₃×C₇	—	2	1	no	—
103	C₂×D₃×C₇	#41	1	2	yes	—
104	C₂×C₃×D₇	—	2	2	no	—
105	C₂×C₃×D₇	—	2	2	no	—
106	C₂×C₃×D₇	—	2	2	no	—
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×C₄×C₇	—	2	1	no	—
109	C₃×C₄×C₇	#41	1	2	yes	—
110	Dic₂₁	—	2	2	no	—
111	Dic₂₁	—	2	2	no	—
112	C₃×Dic₇	—	2	2	no	—
113	C₇×Dic₃	—	2	1	no	—
114	C₇×Dic₃	—	2	1	no	—
115	C₇×Dic₃	#41	1	2	yes	—
116	#529	—	21	1	no	Sylow
117	C₂³×C₇	—	3	1	no	—
118	C₂²×D₇	C₂²×D₃	1	2	yes	—
119	C₂×C₄×C₇	—	3	1	no	—
120	C₂×C₄×C₇	—	3	1	no	—
121	C₂×C₄×C₇	—	3	1	no	—
122	C₂×C₄×C₇	—	6	2	no	—
123	C₂×C₄×C₇	—	6	2	no	—
124	C₂×C₄×C₇	—	6	2	no	—
125	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
128	C₂²⋊D₇	—	6	4	no	—
129	C₂×Dic₇	—	3	2	no	—
130	C₂×Dic₇	—	3	2	no	—
131	C₂×Dic₇	C₂²×D₃	1	2	yes	—
132	D₂₈	—	2	2	no	—
133	C₄×D₇	—	6	4	no	—
134	Dic₁₄	—	2	2	no	—
135	C₂×C₃×Q₈	—	7	1	no	—
136	C₂×C₃×D₄	—	7	1	no	—
137	C₂×#67	—	7	2	no	—
138	C₂²×Dic₃	—	7	2	no	—
139	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
140	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
141	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
142	C₃×C₂²⋊C₄	—	7	2	no	—
143	C₃×C₂²⋊C₄	—	7	2	no	—
144	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
145	#217	—	7	2	no	—
146	#217	—	7	2	no	—
147	(C₂×D₃)⋊C₄	—	7	2	no	—
148	C₄⋊Dic₃	—	7	1	no	—
149	C₄⋊Dic₃	—	7	1	no	—
150	C₄⋊Dic₃	—	7	2	no	—
151	Dic₃⋊C₄	—	7	2	no	—
152	Dic₃⋊C₄	—	7	2	no	—
153	Dic₃⋊C₄	—	7	2	no	—
154	C₄×Dic₃	—	7	2	no	—
155	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
156	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
157	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
158	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
159	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
160	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
161	C₃×D₇	—	4	2	no	—
162	D₃×C₇	—	2	2	no	—
163	#126	—	21	2	no	—
164	#125	—	21	1	no	—
165	#124	—	7	1	no	—
166	#123	—	21	2	no	—
167	C₂²⋊Q₈	—	21	1	no	—
168	#121	—	21	1	no	—
169	C₄×Q₈	—	21	1	no	—
170	C₄×D₄	—	21	1	no	—
171	C₄×D₄	—	21	2	no	—
172	#117	—	21	2	no	—
173	C₂×C₄²	—	3	1	no	—
174	C₂²×C₇	—	3	2	no	—
175	C₂²×C₇	—	6	1	no	—
176	C₂²×C₇	—	6	1	no	—
177	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
178	C₂×D₇	—	2	2	no	—
179	C₂×D₇	—	2	2	no	—
180	C₂×D₇	—	2	2	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	3	2	no	—
184	C₄×C₇	—	6	1	no	—
185	C₄×C₇	—	6	1	no	—
186	C₄×C₇	#237	1	2	yes	—
187	Dic₇	—	2	2	no	—
188	Dic₇	—	6	2	no	—
189	Dic₇	—	6	2	no	—
190	C₂³×C₃	—	7	2	no	—
191	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
192	C₃×Q₈	—	14	2	no	—
193	C₃×D₄	—	14	2	no	—
194	C₂×C₃×C₄	—	7	2	no	—
195	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
198	#67	—	14	4	no	—
199	C₂×Dic₃	—	7	2	no	—
200	C₂×Dic₃	—	7	2	no	—
201	C₂×Dic₃	—	14	4	no	—
202	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
203	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
204	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
205	C₄×D₃	—	2	2	no	—
206	C₄×D₃	—	2	2	no	—
207	C₄×D₃	—	2	2	no	—
208	C₃×C₇	#529	1	1	yes	characteristic, complemented, Hall
209	C₂×Q₈	—	7	1	no	—
210	C₂×D₄	—	7	1	no	—
211	C₂×D₄	—	21	2	no	—
212	C₂²×C₄	—	3	1	no	—
213	C₂²×C₄	—	3	1	no	—
214	C₂²×C₄	—	3	1	no	—
215	C₂²×C₄	—	21	2	no	—
216	C₄⋊C₄	—	21	2	no	—
217	C₄⋊C₄	—	21	2	no	—
218	C₄⋊C₄	—	21	2	no	—
219	C₄⋊C₄	—	21	2	no	—
220	C₄⋊C₄	—	21	1	no	—
221	C₄⋊C₄	—	21	1	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	21	2	no	—
225	C₂²⋊C₄	—	21	2	no	—
226	C₂²⋊C₄	—	21	2	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	1	no	—
229	C₄²	—	3	1	no	—
230	C₄²	—	21	2	no	—
231	C₄²	D₃×D₇	1	1	yes	characteristic
232	C₂×C₇	—	6	2	no	—
233	C₂×C₇	C₂×#237	1	1	yes	characteristic
234	C₂×C₇	D₃×#41	1	1	yes	characteristic
235	C₂×C₇	D₃×#41	1	1	yes	characteristic
236	D₇	—	4	2	no	—
237	C₂²×C₃	—	14	2	no	—
238	C₂²×C₃	—	14	2	no	—
239	C₂²×C₃	—	14	2	no	—
240	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
241	C₂×D₃	—	2	1	no	—
242	C₂×D₃	—	2	1	no	—
243	C₂×D₃	#1157	1	2	yes	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	14	2	no	—
247	C₃×C₄	#1157	1	2	yes	—
248	Dic₃	—	2	1	no	—
249	Dic₃	—	2	1	no	—
250	Dic₃	—	14	2	no	—
251	Dic₃	—	14	2	no	—
252	Dic₃	#1157	1	2	yes	—
253	C₂³	—	3	1	no	—
254	C₂³	—	7	2	no	—
255	Q₈	—	14	2	no	—
256	D₄	—	14	2	no	—
257	D₄	—	42	4	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	6	2	no	—
262	C₂×C₄	—	6	2	no	—
263	C₂×C₄	—	6	2	no	—
264	C₂×C₄	—	7	2	no	—
265	C₂×C₄	—	21	2	no	—
266	C₂×C₄	—	21	2	no	—
267	C₂×C₄	—	42	4	no	—
268	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
269	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
270	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
271	C₇	#5680	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	—	28	2	no	—
273	C₂×C₃	C₂×#1157	1	1	yes	characteristic
274	C₂×C₃	C₂×#1157	1	1	yes	characteristic
275	C₂×C₃	C₂×#1157	1	1	yes	characteristic
276	D₃	—	2	2	no	—
277	C₂²	—	3	2	no	—
278	C₂²	—	6	1	no	—
279	C₂²	—	6	1	no	—
280	C₂²	—	14	2	no	—
281	C₂²	—	14	2	no	—
282	C₂²	—	14	2	no	—
283	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	3	2	no	—
287	C₄	—	6	1	no	—
288	C₄	—	6	1	no	—
289	C₄	—	14	2	no	—
290	C₄	—	42	2	no	—
291	C₄	—	42	2	no	—
292	C₄	#66669	1	2	yes	—
293	C₃	#90170	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	6	2	no	—
295	C₂	—	28	2	no	—
296	C₂	C₂×#66669	1	1	yes	characteristic
297	C₂	D₃×#1157	1	1	yes	characteristic
298	C₂	D₃×#1157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₄²×C₇	C₂²
prime radical	#10621018	C₂
hypercenter	C₄²	D₃×D₇
hypocenter	C₃×C₇	#529
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. #10621353	C₂	semidirect
2. C₄²×D₃×C₇	C₂	semidirect
3. C₃×#6438	C₂	semidirect
5. #10621014	C₂	semidirect
6. #10621003	C₂	semidirect
7. Dic₃×D₂₈	C₂	semidirect
8. #10620955	C₂	semidirect
9. #10620931	C₂	semidirect
10. #10620925	C₂	semidirect
11. Dic₃×Dic₁₄	C₂	semidirect
42. #6438	D₃	semidirect
98. C₄²×D₃	D₇	semidirect
208. C₃×C₇	#529	semidirect
271. C₇	#5680	semidirect
293. C₃	#90170	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10621353	2
C₄²×D₃×C₇	2
C₃×#6438	2
#10621018	2
#10621014	2
#10621003	2
Dic₃×D₂₈	2
#10620955	2
#10620931	2
#10620925	2
Dic₃×Dic₁₄	2
#90170	3
#5680	7