Subgroups — #13058736

Order 1344.

444 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621371	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#892	C₂	1	1	yes	maximal, characteristic
3	C₃×#6453	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621032	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620998	C₂	1	2	yes	maximal
6	#10620994	C₂	1	2	yes	maximal
7	C₄×D₃×Dic₇	C₂	1	1	yes	maximal, characteristic
8	#10620950	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620902	C₂	1	2	yes	maximal, complemented
10	#10620900	C₂	1	1	yes	maximal, characteristic
11	#10620886	C₂	1	2	yes	maximal, complemented
12	#90208	—	3	1	no	maximal, Hall
13	C₂×Dic₄₂	C₂²	1	2	yes	—
14	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
15	C₂×#4248	C₂²	1	2	yes	—
16	C₂×D₃×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×D₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
20	#66629	C₂²	1	2	yes	—
21	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
22	C₇×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₄×C₇×Dic₃	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
28	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	#66577	C₂²	1	2	yes	—
31	#66577	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
33	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
34	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	C₇×#117	—	3	1	no	—
37	#6489	—	3	1	no	—
38	C₂×C₄×Dic₇	—	3	1	no	—
39	#6453	—	3	1	no	—
40	#6453	D₃	1	1	yes	characteristic, complemented
41	#6451	—	3	1	no	—
42	#6451	—	3	1	no	—
43	#6441	—	3	2	no	—
44	#6441	—	3	2	no	—
45	C₄×Dic₁₄	—	3	2	no	—
46	C₄×Dic₁₄	—	3	2	no	—
47	#5688	—	7	1	no	maximal, Hall
48	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₁	C₂³	1	2	yes	—
52	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
54	Dic₄₂	—	2	4	no	—
55	C₂×C₇×Dic₃	C₂³	1	2	yes	—
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₄×D₃×C₇	Q₈	1	4	yes	complemented
58	C₂×C₃×Dic₇	C₂³	1	2	yes	—
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
61	#4248	—	2	4	no	—
62	D₃×Dic₇	—	2	2	no	—
63	D₃×Dic₇	—	2	2	no	—
64	C₂²×C₄×C₇	—	3	1	no	—
65	C₂²×Dic₇	—	3	1	no	—
66	C₂²×Dic₇	—	3	1	no	—
67	C₂×Dic₁₄	—	3	2	no	—
68	C₂×Dic₁₄	—	3	2	no	—
69	C₇×C₄⋊C₄	—	3	1	no	—
70	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₇×C₂²⋊C₄	—	3	2	no	—
72	C₄²×C₇	—	3	2	no	—
73	C₂²⋊Dic₇	—	3	2	no	—
74	C₄⋊Dic₇	—	3	1	no	—
75	C₄⋊Dic₇	—	3	2	no	—
76	C₄⋊Dic₇	C₂×D₃	1	2	yes	—
77	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
78	Dic₇⋊C₄	—	3	2	no	—
79	Dic₇⋊C₄	—	3	2	no	—
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
82	C₄×Dic₇	—	3	1	no	—
83	C₄×Dic₇	—	3	1	no	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	—	3	2	no	—
86	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
87	C₃×#125	—	7	1	no	—
88	#896	—	7	2	no	—
89	#896	—	7	2	no	—
90	#892	—	7	1	no	—
91	#892	D₇	1	1	yes	characteristic
92	#888	—	7	1	no	—
93	#888	—	7	1	no	—
94	#887	—	7	2	no	—
95	#887	—	7	2	no	—
96	C₄²×D₃	—	7	1	no	—
97	#870	—	7	1	no	—
98	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
99	C₂×D₃×C₇	—	2	1	no	—
100	C₂×D₃×C₇	—	2	1	no	—
101	C₂×D₃×C₇	C₂×Q₈	1	2	yes	—
102	C₃×C₄×C₇	—	2	2	no	—
103	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
104	Dic₂₁	—	2	2	no	—
105	Dic₂₁	#41	1	2	yes	—
106	Dic₂₁	#41	1	2	yes	—
107	C₃×Dic₇	—	2	1	no	—
108	C₃×Dic₇	—	2	1	no	—
109	C₃×Dic₇	—	2	2	no	—
110	C₇×Dic₃	—	2	2	no	—
111	C₇×Dic₃	C₂×Q₈	1	2	yes	—
112	#533	—	21	1	no	Sylow
113	C₂³×C₇	—	3	1	no	—
114	C₂×C₄×C₇	—	3	1	no	—
115	C₂×C₄×C₇	—	3	4	no	—
116	C₂×C₄×C₇	—	3	2	no	—
117	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
118	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
119	C₂×Dic₇	—	3	1	no	—
120	C₂×Dic₇	—	3	1	no	—
121	C₂×Dic₇	—	3	2	no	—
122	C₂×Dic₇	—	6	2	no	—
123	C₂×Dic₇	—	6	2	no	—
124	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
125	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂²×D₃	1	2	yes	—
127	Dic₁₄	—	6	4	no	—
128	Dic₁₄	—	6	4	no	—
129	C₂×C₄×D₃	—	7	1	no	—
130	C₂×C₄×D₃	—	7	1	no	—
131	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
132	C₂×Dic₆	—	7	2	no	—
133	C₂×Dic₆	—	7	2	no	—
134	C₃×C₄⋊C₄	—	7	2	no	—
135	C₃×C₄⋊C₄	—	7	2	no	—
136	C₃×C₄⋊C₄	—	7	1	no	—
137	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
138	C₃×C₄²	—	7	1	no	—
139	(C₂×D₃)⋊C₄	—	7	2	no	—
140	(C₂×D₃)⋊C₄	C₂×D₇	1	2	yes	—
141	C₄⋊Dic₃	—	7	1	no	—
142	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
143	Dic₃⋊C₄	—	7	2	no	—
144	Dic₃⋊C₄	—	7	2	no	—
145	Dic₃⋊C₄	—	7	2	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	C₄×Dic₃	—	7	1	no	—
148	C₄×Dic₃	—	7	1	no	—
149	C₄×Dic₃	—	7	1	no	—
150	C₄×Dic₃	—	7	2	no	—
151	C₄×Dic₃	C₂×D₇	1	2	yes	—
152	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
155	D₃×C₇	—	2	2	no	—
156	C₄⋊Q₈	—	21	1	no	—
157	C₄⋊Q₈	—	21	1	no	—
158	#125	—	7	1	no	—
159	#125	—	21	1	no	—
160	C₂²⋊Q₈	—	21	2	no	—
161	C₂²⋊Q₈	—	21	2	no	—
162	C₄×Q₈	—	21	2	no	—
163	C₄×Q₈	—	21	2	no	—
164	#117	—	3	1	no	—
165	#117	—	21	1	no	—
166	C₂×C₄²	—	21	1	no	—
167	C₂²×C₇	—	3	2	no	—
168	C₂²×C₇	—	6	1	no	—
169	C₂²×C₇	—	6	1	no	—
170	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	3	2	no	—
173	C₄×C₇	—	6	2	no	—
174	C₄×C₇	D₃×Q₈	1	2	yes	—
175	Dic₇	—	2	2	no	—
176	Dic₇	—	2	1	no	—
177	Dic₇	—	2	1	no	—
178	Dic₇	—	3	2	no	—
179	Dic₇	—	3	2	no	—
180	Dic₇	—	6	2	no	—
181	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
182	C₂×C₃×C₄	—	7	1	no	—
183	C₂×C₃×C₄	—	7	1	no	—
184	C₂×C₃×C₄	—	7	2	no	—
185	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
187	C₂×Dic₃	—	7	1	no	—
188	C₂×Dic₃	—	7	1	no	—
189	C₂×Dic₃	—	7	2	no	—
190	C₂×Dic₃	C₂²×D₇	1	2	yes	—
191	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
192	C₄×D₃	—	14	2	no	—
193	C₄×D₃	—	14	2	no	—
194	C₄×D₃	Dic₁₄	1	4	yes	complemented
195	Dic₆	—	14	4	no	—
196	Dic₆	—	14	4	no	—
197	C₃×C₇	#533	1	1	yes	characteristic, complemented, Hall
198	C₂×Q₈	—	21	2	no	—
199	C₂×Q₈	—	21	2	no	—
200	C₂²×C₄	—	3	1	no	—
201	C₂²×C₄	—	21	1	no	—
202	C₂²×C₄	—	21	1	no	—
203	C₄⋊C₄	—	3	1	no	—
204	C₄⋊C₄	—	7	2	no	—
205	C₄⋊C₄	—	7	1	no	—
206	C₄⋊C₄	—	7	2	no	—
207	C₄⋊C₄	—	21	2	no	—
208	C₄⋊C₄	—	21	2	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	2	no	—
211	C₄⋊C₄	—	21	1	no	—
212	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
213	C₂²⋊C₄	—	3	2	no	—
214	C₂²⋊C₄	—	21	2	no	—
215	C₄²	—	3	2	no	—
216	C₄²	—	7	1	no	—
217	C₄²	—	21	1	no	—
218	C₄²	—	21	1	no	—
219	C₄²	—	21	2	no	—
220	C₄²	—	21	1	no	—
221	C₂×C₇	—	6	2	no	—
222	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
223	C₂×C₇	D₃×#41	1	1	yes	characteristic
224	C₂×C₇	D₃×#41	1	1	yes	characteristic
225	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
226	C₂×D₃	—	2	1	no	—
227	C₂×D₃	—	2	1	no	—
228	C₂×D₃	C₂×Dic₁₄	1	2	yes	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	14	2	no	—
231	C₃×C₄	—	14	1	no	—
232	C₃×C₄	—	14	1	no	—
233	C₃×C₄	C₂×Dic₁₄	1	2	yes	—
234	Dic₃	—	2	2	no	—
235	Dic₃	—	7	2	no	—
236	Dic₃	—	7	2	no	—
237	Dic₃	—	14	2	no	—
238	Dic₃	C₂×Dic₁₄	1	2	yes	—
239	C₂³	—	3	1	no	—
240	Q₈	—	42	4	no	—
241	Q₈	—	42	4	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	4	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	7	1	no	—
246	C₂×C₄	—	7	1	no	—
247	C₂×C₄	—	7	2	no	—
248	C₂×C₄	—	21	1	no	—
249	C₂×C₄	—	21	1	no	—
250	C₂×C₄	—	21	2	no	—
251	C₂×C₄	—	42	2	no	—
252	C₂×C₄	—	42	2	no	—
253	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
254	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
255	C₇	#5688	1	1	yes	characteristic, complemented, Sylow
256	C₂×C₃	C₂²×Dic₁₄	1	1	yes	characteristic
257	C₂×C₃	D₇×#41	1	1	yes	characteristic
258	C₂×C₃	D₇×#41	1	1	yes	characteristic
259	D₃	—	2	2	no	—
260	C₂²	—	3	2	no	—
261	C₂²	—	6	1	no	—
262	C₂²	—	6	1	no	—
263	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	3	2	no	—
266	C₄	—	6	2	no	—
267	C₄	—	14	1	no	—
268	C₄	—	14	1	no	—
269	C₄	—	14	2	no	—
270	C₄	—	21	2	no	—
271	C₄	—	21	2	no	—
272	C₄	—	42	2	no	—
273	C₄	D₃×Dic₁₄	1	2	yes	—
274	C₃	#90208	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	6	2	no	—
276	C₂	C₂×D₃×Dic₁₄	1	1	yes	characteristic
277	C₂	#10621571	1	1	yes	characteristic
278	C₂	#10621581	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #10621371	C₂	semidirect
3. C₃×#6453	C₂	semidirect
4. #10621032	C₂	semidirect
8. #10620950	C₂	semidirect
9. #10620902	C₂	semidirect
11. #10620886	C₂	semidirect
40. #6453	D₃	semidirect
57. C₄×D₃×C₇	Q₈	semidirect
194. C₄×D₃	Dic₁₄	semidirect
197. C₃×C₇	#533	semidirect
255. C₇	#5688	semidirect
274. C₃	#90208	semidirect

Maximal subgroups

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

subgroup	index
#10621371	2
C₇×#892	2
C₃×#6453	2
#10621032	2
#10620998	2
#10620994	2
C₄×D₃×Dic₇	2
#10620950	2
#10620902	2
#10620900	2
#10620886	2
#90208	3
#5688	7