Subgroups — #13059831

Order 1344.

444 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621407	C₂	1	1	yes	maximal, characteristic
2	C₇×#924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
4	#10621136	C₂	1	2	yes	maximal
5	#10621090	C₂	1	2	yes	maximal
6	#10621030	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620940	C₂	1	2	yes	maximal, complemented
8	#10620938	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620895	C₂	1	1	yes	maximal, characteristic, complemented
10	Dic₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620885	C₂	1	2	yes	maximal, complemented
12	#90313	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
15	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×#4248	C₂²	1	2	yes	—
19	#66636	C₂²	1	2	yes	—
20	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
21	#66629	C₂²	1	2	yes	—
22	C₇×#217	C₂²	1	2	yes	—
23	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
26	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
27	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	#66577	C₂²	1	2	yes	—
31	Dic₇⋊Dic₃	C₂²	1	2	yes	—
32	Dic₇⋊Dic₃	C₂²	1	2	yes	—
33	Dic₇⋊Dic₃	C₂²	1	2	yes	—
34	Dic₃×Dic₇	C₂²	1	2	yes	—
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	C₇×C₂²⋊Q₈	—	3	1	no	—
37	Q₈×Dic₇	—	3	1	no	—
38	#6507	—	3	1	no	—
39	#6487	—	3	1	no	—
40	C₂×C₄×Dic₇	D₃	1	1	yes	characteristic
41	#6452	—	3	2	no	—
42	#6451	—	3	1	no	—
43	#6450	—	3	2	no	—
44	#6450	—	3	1	no	—
45	#6441	—	3	2	no	—
46	#6440	—	3	2	no	—
47	#5470	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	—	2	2	no	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	2	yes	—
53	C₂×Dic₂₁	C₂³	1	2	yes	—
54	Dic₄₂	—	2	2	no	—
55	C₂×C₇×Dic₃	C₂³	1	2	yes	—
56	C₂×C₇×Dic₃	C₂³	1	2	yes	—
57	C₇×Dic₆	—	2	2	no	—
58	C₂×C₃×Dic₇	—	2	2	no	—
59	C₂×C₃×Dic₇	Q₈	1	4	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	1	yes	characteristic
63	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
64	#4248	—	2	4	no	—
65	C₂×C₇×Q₈	—	3	1	no	—
66	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
67	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
68	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
69	C₂×Dic₁₄	—	3	1	no	—
70	C₂×Dic₁₄	—	3	2	no	—
71	C₇×C₄⋊C₄	—	3	1	no	—
72	C₇×C₄⋊C₄	—	3	2	no	—
73	C₇×C₂²⋊C₄	—	3	2	no	—
74	C₂²⋊Dic₇	—	3	2	no	—
75	C₄⋊Dic₇	—	3	1	no	—
76	C₄⋊Dic₇	—	3	2	no	—
77	Dic₇⋊C₄	—	3	2	no	—
78	Dic₇⋊C₄	—	3	2	no	—
79	Dic₇⋊C₄	—	3	2	no	—
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	C₄×Dic₇	—	3	2	no	—
83	C₄×Dic₇	—	3	2	no	—
84	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
85	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
86	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
88	C₂×C₃×C₄²	—	7	1	no	—
89	#926	—	7	2	no	—
90	#924	—	7	1	no	—
91	#924	—	7	2	no	—
92	#924	D₇	1	1	yes	characteristic
93	#870	—	7	2	no	—
94	#869	—	7	1	no	—
95	#869	—	7	1	no	—
96	C₄×Dic₆	—	7	1	no	—
97	C₄×Dic₆	—	7	1	no	—
98	C₄×Dic₆	—	7	2	no	—
99	C₂²×C₃×C₇	—	2	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	—
102	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
103	C₃×C₄×C₇	—	2	1	no	—
104	C₃×C₄×C₇	#41	1	2	yes	—
105	Dic₂₁	—	2	2	no	—
106	Dic₂₁	—	2	2	no	—
107	C₃×Dic₇	—	2	1	no	—
108	C₃×Dic₇	C₂×Q₈	1	2	yes	—
109	C₃×Dic₇	C₂×Q₈	1	2	yes	—
110	C₃×Dic₇	#41	1	2	yes	—
111	C₇×Dic₃	—	2	2	no	—
112	C₇×Dic₃	—	2	2	no	—
113	#533	—	21	1	no	Sylow
114	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
115	C₇×Q₈	—	6	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	3	2	no	—
118	C₂×C₄×C₇	—	3	2	no	—
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
121	C₂×Dic₇	—	2	2	no	—
122	C₂×Dic₇	—	3	2	no	—
123	C₂×Dic₇	—	3	2	no	—
124	C₂×Dic₇	Dic₆	1	4	yes	complemented
125	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
127	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
128	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
129	Dic₁₄	—	6	2	no	—
130	Dic₁₄	—	6	4	no	—
131	C₂²×C₃×C₄	—	7	1	no	—
132	C₂²×C₃×C₄	—	7	1	no	—
133	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
134	C₂×Dic₆	—	7	1	no	—
135	C₂×Dic₆	—	7	2	no	—
136	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
137	C₃×C₄²	—	7	1	no	—
138	C₃×C₄²	—	7	1	no	—
139	C₃×C₄²	—	7	1	no	—
140	C₃×C₄²	—	7	1	no	—
141	#217	—	7	2	no	—
142	#217	C₂×D₇	1	2	yes	—
143	C₄⋊Dic₃	—	7	1	no	—
144	C₄⋊Dic₃	—	7	2	no	—
145	C₄⋊Dic₃	—	7	2	no	—
146	C₄⋊Dic₃	—	7	2	no	—
147	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
148	Dic₃⋊C₄	—	7	2	no	—
149	Dic₃⋊C₄	—	7	2	no	—
150	Dic₃⋊C₄	—	7	2	no	—
151	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
152	C₄×Dic₃	—	7	2	no	—
153	C₄×Dic₃	—	7	2	no	—
154	C₂×C₃×C₇	—	2	2	no	—
155	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
158	C₄⋊Q₈	—	21	1	no	—
159	C₄⋊Q₈	—	21	1	no	—
160	#125	—	21	2	no	—
161	C₂²⋊Q₈	—	3	1	no	—
162	C₂²⋊Q₈	—	21	2	no	—
163	C₂²⋊Q₈	—	21	1	no	—
164	C₄×Q₈	—	21	2	no	—
165	C₄×Q₈	—	21	1	no	—
166	C₄×Q₈	—	21	1	no	—
167	#117	—	21	2	no	—
168	C₂×C₄²	—	7	1	no	—
169	C₂²×C₇	—	2	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	C₂×Dic₆	1	2	yes	—
172	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	6	2	no	—
175	C₄×C₇	—	6	2	no	—
176	C₄×C₇	#235	1	2	yes	—
177	Dic₇	—	2	1	no	—
178	Dic₇	—	6	2	no	—
179	Dic₇	—	6	2	no	—
180	Dic₇	C₂×Dic₆	1	2	yes	—
181	Dic₇	C₂×Dic₆	1	2	yes	—
182	Dic₇	#235	1	2	yes	—
183	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	7	4	no	—
186	C₂×C₃×C₄	—	7	1	no	—
187	C₂×C₃×C₄	—	7	1	no	—
188	C₂×C₃×C₄	—	7	1	no	—
189	C₂×C₃×C₄	—	7	1	no	—
190	C₂×C₃×C₄	—	14	2	no	—
191	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
193	C₂×Dic₃	—	7	2	no	—
194	C₂×Dic₃	—	7	2	no	—
195	C₂×Dic₃	C₂²×D₇	1	2	yes	—
196	C₂×Dic₃	C₂²×D₇	1	2	yes	—
197	Dic₆	—	2	2	no	—
198	Dic₆	—	14	4	no	—
199	Dic₆	—	14	2	no	—
200	C₃×C₇	#533	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	3	1	no	—
202	C₂×Q₈	—	21	2	no	—
203	C₂×Q₈	—	21	1	no	—
204	C₂²×C₄	—	7	1	no	—
205	C₂²×C₄	—	7	1	no	—
206	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	2	no	—
211	C₄⋊C₄	—	21	2	no	—
212	C₄⋊C₄	—	21	2	no	—
213	C₄⋊C₄	—	21	2	no	—
214	C₄⋊C₄	—	21	2	no	—
215	C₄⋊C₄	—	21	1	no	—
216	C₂²⋊C₄	—	3	2	no	—
217	C₂²⋊C₄	—	21	2	no	—
218	C₄²	—	7	1	no	—
219	C₄²	—	7	1	no	—
220	C₄²	—	7	1	no	—
221	C₄²	—	7	1	no	—
222	C₄²	—	21	2	no	—
223	C₄²	—	21	2	no	—
224	C₂×C₇	—	2	2	no	—
225	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
226	C₂×C₇	C₂×#235	1	1	yes	characteristic
227	C₂×C₇	C₂×#235	1	1	yes	characteristic
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	Q₈×D₇	1	2	yes	—
231	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	7	2	no	—
234	C₃×C₄	—	7	2	no	—
235	C₃×C₄	—	7	2	no	—
236	C₃×C₄	—	14	1	no	—
237	C₃×C₄	#1159	1	2	yes	—
238	Dic₃	—	2	2	no	—
239	Dic₃	—	2	2	no	—
240	Dic₃	—	14	2	no	—
241	Dic₃	—	14	2	no	—
242	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
243	Q₈	—	6	2	no	—
244	Q₈	—	42	4	no	—
245	Q₈	—	42	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	7	4	no	—
250	C₂×C₄	—	7	1	no	—
251	C₂×C₄	—	7	1	no	—
252	C₂×C₄	—	7	1	no	—
253	C₂×C₄	—	7	1	no	—
254	C₂×C₄	—	14	2	no	—
255	C₂×C₄	—	21	2	no	—
256	C₂×C₄	—	21	2	no	—
257	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
258	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
259	C₇	#5470	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	C₂×#1159	1	1	yes	characteristic
262	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
263	C₂×C₃	D₇×#41	1	1	yes	characteristic
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	D₇×Dic₆	1	2	yes	—
267	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	6	2	no	—
270	C₄	—	6	2	no	—
271	C₄	—	7	2	no	—
272	C₄	—	7	2	no	—
273	C₄	—	7	2	no	—
274	C₄	—	14	1	no	—
275	C₄	—	42	2	no	—
276	C₄	—	42	2	no	—
277	C₄	#66686	1	2	yes	—
278	C₃	#90313	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	2	2	no	—
280	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
281	C₂	D₇×#235	1	1	yes	characteristic
282	C₂	C₂×#66686	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₂²×C₃×C₄×C₇	C₂²
prime radical	C₂³×C₃×C₇	C₂³
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#533
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
6. #10621030	C₂	semidirect
7. #10620940	C₂	semidirect
8. #10620938	C₂	semidirect
9. #10620895	C₂	semidirect
10. Dic₆×Dic₇	C₂	semidirect
11. #10620885	C₂	semidirect
59. C₂×C₃×Dic₇	Q₈	semidirect
124. C₂×Dic₇	Dic₆	semidirect
200. C₃×C₇	#533	semidirect
259. C₇	#5470	semidirect
278. C₃	#90313	semidirect

Maximal subgroups

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

subgroup	index
#10621407	2
C₇×#924	2
C₂×C₃×C₄×Dic₇	2
#10621136	2
#10621090	2
#10621030	2
#10620940	2
#10620938	2
#10620895	2
Dic₆×Dic₇	2
#10620885	2
#90313	3
#5470	7