Subgroups — #13061761

Order 1344.

424 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	C₃×C₇×#124	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621427	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621419	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic
6	#10621362	C₂	1	2	yes	maximal, complemented
7	#10621362	C₂	1	2	yes	maximal, complemented
8	#10621361	C₂	1	2	yes	maximal, complemented
9	#10621360	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#90362	—	3	1	no	maximal, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₁	C₂²	1	2	yes	—
16	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	#66636	C₂²	1	2	yes	—
22	#66636	C₂²	1	2	yes	—
23	#66636	C₂²	1	2	yes	—
24	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₁	C₂²	1	2	yes	—
27	#66629	C₂²	1	1	yes	characteristic
28	#66629	C₂²	1	2	yes	—
29	#66629	C₂²	1	1	yes	characteristic
30	#66629	C₂²	1	2	yes	—
31	#66629	C₂²	1	2	yes	—
32	C₄×Dic₂₁	C₂²	1	2	yes	—
33	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁	C₂²	1	2	yes	—
36	C₇×#124	D₃	1	1	yes	characteristic
37	Q₈×Dic₇	—	3	1	no	—
38	#6507	—	3	1	no	—
39	#6499	—	3	1	no	—
40	D₄×Dic₇	—	3	1	no	—
41	#6442	—	3	2	no	—
42	#6442	—	3	2	no	—
43	#6441	—	3	2	no	—
44	#6440	—	3	2	no	—
45	C₄×Dic₁₄	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5671	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	2	yes	—
49	C₃×C₇×Q₈	—	2	2	no	—
50	C₃×C₇×D₄	—	2	2	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₁	—	2	4	no	—
56	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
57	C₂×Dic₂₁	C₂³	1	2	yes	—
58	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
59	C₂×Dic₂₁	C₂³	1	2	yes	—
60	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
62	Dic₄₂	—	2	2	no	—
63	Dic₄₂	—	2	2	no	—
64	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
65	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
66	C₂²×Dic₇	—	3	2	no	—
67	C₂×Dic₁₄	—	3	1	no	—
68	C₂×Dic₁₄	—	3	1	no	—
69	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
70	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
71	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
72	C₂²⋊Dic₇	—	3	2	no	—
73	C₂²⋊Dic₇	—	3	2	no	—
74	C₂²⋊Dic₇	—	3	2	no	—
75	C₄⋊Dic₇	—	3	1	no	—
76	C₄⋊Dic₇	—	3	1	no	—
77	C₄⋊Dic₇	—	3	2	no	—
78	Dic₇⋊C₄	—	3	1	no	—
79	Dic₇⋊C₄	—	3	2	no	—
80	Dic₇⋊C₄	—	3	1	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	Dic₇⋊C₄	—	3	2	no	—
83	C₄×Dic₇	—	3	2	no	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	—	3	1	no	—
86	C₄×Dic₇	—	3	2	no	—
87	C₃×#124	D₇	1	1	yes	characteristic
88	Q₈×Dic₃	—	7	1	no	—
89	#944	—	7	1	no	—
90	#936	—	7	1	no	—
91	D₄×Dic₃	—	7	1	no	—
92	#879	—	7	2	no	—
93	#879	—	7	2	no	—
94	#878	—	7	2	no	—
95	#877	—	7	2	no	—
96	C₄×Dic₆	—	7	1	no	—
97	C₄×Dic₆	—	7	1	no	—
98	C₂²×C₃×C₇	—	2	2	no	—
99	C₂²×C₃×C₇	—	2	2	no	—
100	C₂²×C₃×C₇	—	2	2	no	—
101	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
102	C₃×C₄×C₇	—	2	1	no	—
103	C₃×C₄×C₇	—	2	2	no	—
104	C₃×C₄×C₇	—	2	1	no	—
105	C₃×C₄×C₇	#41	1	2	yes	—
106	Dic₂₁	—	2	2	no	—
107	Dic₂₁	—	2	1	no	—
108	Dic₂₁	—	2	2	no	—
109	Dic₂₁	—	2	1	no	—
110	Dic₂₁	—	2	1	no	—
111	Dic₂₁	#41	1	2	yes	—
112	#556	—	21	1	no	Sylow
113	C₂³×C₇	C₂²×D₃	1	2	yes	—
114	C₇×Q₈	—	2	2	no	—
115	C₇×D₄	—	2	2	no	—
116	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
118	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×Dic₇	—	3	1	no	—
121	C₂×Dic₇	—	3	1	no	—
122	C₂×Dic₇	—	3	2	no	—
123	C₂×Dic₇	—	3	1	no	—
124	C₂×Dic₇	—	3	2	no	—
125	C₂×Dic₇	—	3	1	no	—
126	C₂×Dic₇	—	6	4	no	—
127	Dic₁₄	—	6	2	no	—
128	Dic₁₄	—	6	2	no	—
129	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
130	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
131	C₂²×Dic₃	—	7	2	no	—
132	C₂×Dic₆	—	7	1	no	—
133	C₂×Dic₆	—	7	1	no	—
134	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
135	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
136	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
137	#217	—	7	2	no	—
138	#217	—	7	2	no	—
139	#217	—	7	2	no	—
140	C₄⋊Dic₃	—	7	1	no	—
141	C₄⋊Dic₃	—	7	1	no	—
142	C₄⋊Dic₃	—	7	2	no	—
143	Dic₃⋊C₄	—	7	1	no	—
144	Dic₃⋊C₄	—	7	2	no	—
145	Dic₃⋊C₄	—	7	1	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	C₄×Dic₃	—	7	2	no	—
149	C₄×Dic₃	—	7	1	no	—
150	C₄×Dic₃	—	7	1	no	—
151	C₄×Dic₃	—	7	2	no	—
152	C₂×C₃×C₇	—	4	2	no	—
153	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
156	C₄⋊Q₈	—	21	1	no	—
157	#126	—	21	2	no	—
158	#126	—	21	2	no	—
159	#124	—	21	1	no	—
160	#124	D₂₁	1	1	yes	characteristic
161	C₂²⋊Q₈	—	21	2	no	—
162	C₄×Q₈	—	21	1	no	—
163	C₄×Q₈	—	21	1	no	—
164	C₄×Q₈	—	21	1	no	—
165	C₄×D₄	—	21	1	no	—
166	#117	—	21	2	no	—
167	C₂²×C₇	—	2	2	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	#237	1	2	yes	—
175	Dic₇	—	3	2	no	—
176	Dic₇	—	6	2	no	—
177	Dic₇	—	6	1	no	—
178	Dic₇	—	6	2	no	—
179	Dic₇	—	6	1	no	—
180	Dic₇	—	6	1	no	—
181	C₂³×C₃	C₂²×D₇	1	2	yes	—
182	C₃×Q₈	—	2	2	no	—
183	C₃×D₄	—	2	2	no	—
184	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
188	C₂×Dic₃	—	7	1	no	—
189	C₂×Dic₃	—	7	1	no	—
190	C₂×Dic₃	—	7	2	no	—
191	C₂×Dic₃	—	7	1	no	—
192	C₂×Dic₃	—	7	2	no	—
193	C₂×Dic₃	—	7	1	no	—
194	C₂×Dic₃	—	14	4	no	—
195	Dic₆	—	14	2	no	—
196	Dic₆	—	14	2	no	—
197	C₃×C₇	#556	1	1	yes	characteristic, complemented, Hall
198	C₂×Q₈	—	21	1	no	—
199	C₂×Q₈	—	21	1	no	—
200	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
201	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
202	C₂²×C₄	—	21	2	no	—
203	C₄⋊C₄	—	21	1	no	—
204	C₄⋊C₄	—	21	2	no	—
205	C₄⋊C₄	—	21	2	no	—
206	C₄⋊C₄	—	21	1	no	—
207	C₄⋊C₄	—	21	2	no	—
208	C₄⋊C₄	—	21	1	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	1	no	—
211	C₂²⋊C₄	—	21	2	no	—
212	C₂²⋊C₄	—	21	2	no	—
213	C₂²⋊C₄	—	21	2	no	—
214	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
215	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
216	C₄²	—	21	2	no	—
217	C₄²	—	21	2	no	—
218	C₄²	—	21	1	no	—
219	C₄²	—	21	1	no	—
220	C₄²	C₂×D₂₁	1	1	yes	characteristic
221	C₂×C₇	—	4	2	no	—
222	C₂×C₇	C₂×#237	1	1	yes	characteristic
223	C₂×C₇	D₃×#41	1	1	yes	characteristic
224	C₂×C₇	#1010	1	1	yes	characteristic
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	#1159	1	2	yes	—
233	Dic₃	—	7	2	no	—
234	Dic₃	—	14	2	no	—
235	Dic₃	—	14	1	no	—
236	Dic₃	—	14	2	no	—
237	Dic₃	—	14	1	no	—
238	Dic₃	—	14	1	no	—
239	C₂³	C₂²×D₂₁	1	2	yes	—
240	Q₈	—	2	2	no	—
241	Q₈	—	42	2	no	—
242	Q₈	—	42	2	no	—
243	D₄	—	2	2	no	—
244	C₂×C₄	—	21	1	no	—
245	C₂×C₄	—	21	1	no	—
246	C₂×C₄	—	21	2	no	—
247	C₂×C₄	—	21	1	no	—
248	C₂×C₄	—	21	2	no	—
249	C₂×C₄	—	21	1	no	—
250	C₂×C₄	—	42	4	no	—
251	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₂₁	1	2	yes	—
253	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
255	C₇	#5671	1	1	yes	characteristic, complemented, Sylow
256	C₂×C₃	—	4	2	no	—
257	C₂×C₃	C₂×#1159	1	1	yes	characteristic
258	C₂×C₃	D₇×#41	1	1	yes	characteristic
259	C₂×C₃	#6554	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂³×D₂₁	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	21	2	no	—
268	C₄	—	42	2	no	—
269	C₄	—	42	2	no	—
270	C₄	—	42	1	no	—
271	C₄	—	42	1	no	—
272	C₄	—	42	1	no	—
273	C₄	#66730	1	2	yes	—
274	C₃	#90362	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	4	2	no	—
276	C₂	C₂×#66730	1	1	yes	characteristic
277	C₂	#41×D₂₁	1	1	yes	characteristic
278	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×#124	C₂
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	#124	D₂₁
hypocenter	C₃×C₇	#556
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₂₁	C₂	semidirect
3. #10621427	C₂	semidirect
6. #10621362	C₂	semidirect
7. #10621362	C₂	semidirect
8. #10621361	C₂	semidirect
9. #10621360	C₂	semidirect
10. C₄×Dic₄₂	C₂	semidirect
11. C₄×Dic₄₂	C₂	semidirect
197. C₃×C₇	#556	semidirect
255. C₇	#5671	semidirect
274. C₃	#90362	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#124	2
Q₈×Dic₂₁	2
#10621427	2
#10621419	2
D₄×Dic₂₁	2
#10621362	2
#10621361	2
#10621360	2
C₄×Dic₄₂	2
#90362	3
#5671	7