Subgroups — #13060062

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	D₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂²⋊D₇)⋊Dic₃	C₂	1	2	yes	maximal, complemented
5	Dic₃×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
6	C₂×Dic₇⋊Dic₃	C₂	1	2	yes	maximal, complemented
7	#10621090	C₂	1	2	yes	maximal, complemented
8	D₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620919	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620893	C₂	1	1	yes	maximal, characteristic, complemented
11	Dic₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#90237	—	3	1	no	maximal, Hall
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₁	C₂²	1	2	yes	—
15	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₇	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	1	yes	characteristic
20	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₃×D₇	C₂²	1	2	yes	—
22	#66636	C₂²	1	2	yes	—
23	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
25	C₇×#217	C₂²	1	2	yes	—
26	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
28	Dic₇⋊Dic₃	C₂²	1	2	yes	—
29	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
30	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
31	Dic₇⋊Dic₃	C₂²	1	2	yes	—
32	Dic₇⋊Dic₃	C₂²	1	2	yes	—
33	(C₂×D₇)⋊Dic₃	C₂²	1	1	yes	characteristic
34	(C₂×D₇)⋊Dic₃	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	Dic₃×Dic₇	C₂²	1	2	yes	—
37	C₂×#1159	D₃	1	1	yes	characteristic
38	C₄×C₇×D₄	—	3	1	no	—
39	D₄×Dic₇	—	3	1	no	—
40	C₄×C₂²⋊D₇	—	3	2	no	—
41	C₂×Dic₇⋊C₄	—	3	2	no	—
42	D₇×C₄⋊C₄	—	3	1	no	—
43	#6450	—	3	1	no	—
44	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
45	#6440	—	3	2	no	—
46	#6435	—	3	1	no	—
47	C₄×Dic₁₄	—	3	1	no	—
48	#5827	—	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	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	—	4	2	no	—
54	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₁	C₂³	1	2	yes	—
56	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
57	C₂×C₇×Dic₃	—	4	2	no	—
58	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₇×Dic₃	C₂³	1	2	yes	—
60	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
61	C₃×C₂²⋊D₇	C₂×C₄	1	8	yes	complemented
62	C₂×C₃×Dic₇	C₂×C₄	1	8	yes	complemented
63	C₂×C₃×Dic₇	C₂³	1	2	yes	—
64	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
65	C₃×C₄×D₇	C₂×C₄	1	4	yes	complemented
66	C₃×Dic₁₄	C₂×C₄	1	4	yes	complemented
67	Dic₃×D₇	—	4	2	no	—
68	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₄×C₇	—	3	2	no	—
70	C₂×C₂²⋊D₇	C₂×D₃	1	2	yes	—
71	C₂²×Dic₇	—	3	2	no	—
72	C₂²×Dic₇	C₂×D₃	1	2	yes	—
73	#1159	Dic₃	1	8	yes	complemented
74	C₂×C₄×D₇	—	3	2	no	—
75	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
76	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
77	C₇×C₄⋊C₄	—	3	1	no	—
78	C₇×C₂²⋊C₄	—	3	2	no	—
79	C₄²×C₇	—	3	1	no	—
80	C₂²⋊Dic₇	—	3	2	no	—
81	(C₂×D₇)⋊C₄	—	3	1	no	—
82	(C₂×D₇)⋊C₄	—	3	1	no	—
83	C₄⋊Dic₇	—	3	1	no	—
84	Dic₇⋊C₄	—	3	2	no	—
85	Dic₇⋊C₄	—	3	1	no	—
86	Dic₇⋊C₄	—	3	1	no	—
87	Dic₇⋊C₄	—	3	2	no	—
88	Dic₇⋊C₄	—	3	2	no	—
89	C₄×Dic₇	—	3	1	no	—
90	C₄×Dic₇	—	3	2	no	—
91	C₄×Dic₇	—	3	2	no	—
92	C₂×C₃×#41	—	7	1	no	—
93	Q₈×Dic₃	—	7	1	no	—
94	Q₈×Dic₃	—	7	1	no	—
95	D₄×Dic₃	—	7	1	no	—
96	D₄×Dic₃	—	7	2	no	—
97	D₄×Dic₃	—	7	2	no	—
98	D₄×Dic₃	D₇	1	1	yes	characteristic, complemented
99	#926	—	7	2	no	—
100	#926	—	7	1	no	—
101	C₂×C₄⋊Dic₃	—	7	2	no	—
102	C₂×C₄⋊Dic₃	—	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	C₂²×C₄	1	2	yes	—
111	Dic₂₁	—	2	2	no	—
112	Dic₂₁	—	2	1	no	—
113	Dic₂₁	—	2	1	no	—
114	C₃×Dic₇	C₂²×C₄	1	4	yes	—
115	C₃×Dic₇	C₂²×C₄	1	2	yes	—
116	C₇×Dic₃	—	2	2	no	—
117	C₇×Dic₃	—	2	1	no	—
118	C₇×Dic₃	—	2	1	no	—
119	#520	—	21	1	no	Sylow
120	C₂³×C₇	C₂²×D₃	1	2	yes	—
121	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
122	C₇×D₄	C₂×Dic₃	1	4	yes	complemented
123	C₂×C₄×C₇	—	3	1	no	—
124	C₂×C₄×C₇	—	3	2	no	—
125	C₂×C₄×C₇	—	3	1	no	—
126	C₂×C₄×C₇	—	12	2	no	—
127	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
128	C₂²⋊D₇	C₂×Dic₃	1	8	yes	complemented
129	C₂×Dic₇	—	3	1	no	—
130	C₂×Dic₇	—	3	1	no	—
131	C₂×Dic₇	—	3	2	no	—
132	C₂×Dic₇	—	12	2	no	—
133	C₂×Dic₇	C₂×Dic₃	1	8	yes	complemented
134	C₂×Dic₇	C₂²×D₃	1	2	yes	—
135	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
136	C₄×D₇	—	12	2	no	—
137	C₄×D₇	C₂×Dic₃	1	4	yes	complemented
138	Dic₁₄	C₂×Dic₃	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×Dic₃	—	7	2	no	—
146	C₂²×Dic₃	—	7	2	no	—
147	C₂²×Dic₃	C₂×D₇	1	2	yes	—
148	#217	—	7	2	no	—
149	#217	—	7	1	no	—
150	#217	—	7	1	no	—
151	#217	C₂×D₇	1	2	yes	—
152	C₄⋊Dic₃	—	7	1	no	—
153	C₄⋊Dic₃	—	7	2	no	—
154	C₄⋊Dic₃	—	7	1	no	—
155	C₄⋊Dic₃	—	7	2	no	—
156	C₄⋊Dic₃	—	7	2	no	—
157	C₄⋊Dic₃	—	7	1	no	—
158	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
159	C₄×Dic₃	—	7	1	no	—
160	C₄×Dic₃	—	7	2	no	—
161	C₄×Dic₃	—	7	2	no	—
162	C₄×Dic₃	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	—	21	1	no	—
170	C₄×Q₈	—	21	1	no	—
171	C₄×D₄	—	3	1	no	—
172	C₄×D₄	—	21	2	no	—
173	C₄×D₄	—	21	2	no	—
174	C₄×D₄	—	21	1	no	—
175	#117	—	21	1	no	—
176	#117	—	21	2	no	—
177	C₂×C₄⋊C₄	—	21	1	no	—
178	C₂×C₄⋊C₄	—	21	2	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	C₂²×Dic₃	1	4	yes	—
182	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
183	C₂×D₇	—	2	1	no	—
184	C₂×D₇	—	2	1	no	—
185	C₂×D₇	C₂²×Dic₃	1	2	yes	—
186	C₄×C₇	—	6	2	no	—
187	C₄×C₇	—	6	1	no	—
188	C₄×C₇	—	6	1	no	—
189	C₄×C₇	C₂²×Dic₃	1	2	yes	—
190	Dic₇	—	6	2	no	—
191	Dic₇	—	6	1	no	—
192	Dic₇	—	6	1	no	—
193	Dic₇	C₂²×Dic₃	1	4	yes	—
194	Dic₇	C₂²×Dic₃	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	—	7	8	no	—
201	C₂×C₃×C₄	—	7	2	no	—
202	C₂×C₃×C₄	—	7	1	no	—
203	C₂×C₃×C₄	—	7	4	no	—
204	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
205	C₂×Dic₃	—	4	2	no	—
206	C₂×Dic₃	—	7	1	no	—
207	C₂×Dic₃	—	7	2	no	—
208	C₂×Dic₃	—	7	1	no	—
209	C₂×Dic₃	—	28	2	no	—
210	C₂×Dic₃	—	28	2	no	—
211	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
212	C₂×Dic₃	C₂²×D₇	1	2	yes	—
213	C₂×Dic₃	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	D₃×D₇	1	1	yes	characteristic
219	C₂²×C₄	—	3	2	no	—
220	C₂²×C₄	—	7	2	no	—
221	C₂²×C₄	—	7	1	no	—
222	C₂²×C₄	—	21	2	no	—
223	C₂²×C₄	—	21	2	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	21	1	no	—
226	C₄⋊C₄	—	21	2	no	—
227	C₄⋊C₄	—	21	2	no	—
228	C₄⋊C₄	—	21	2	no	—
229	C₄⋊C₄	—	21	1	no	—
230	C₄⋊C₄	—	21	1	no	—
231	C₂²⋊C₄	—	3	2	no	—
232	C₂²⋊C₄	—	21	1	no	—
233	C₂²⋊C₄	—	21	1	no	—
234	C₂²⋊C₄	—	21	2	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	21	2	no	—
237	C₄²	—	21	2	no	—
238	C₄²	—	21	1	no	—
239	C₂×C₇	—	2	4	no	—
240	C₂×C₇	#1009	1	1	yes	characteristic
241	C₂×C₇	#1010	1	1	yes	characteristic
242	C₂×C₇	C₂³×Dic₃	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	—	7	2	no	—
252	C₃×C₄	—	7	4	no	—
253	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
254	Dic₃	—	2	1	no	—
255	Dic₃	—	2	2	no	—
256	Dic₃	—	2	1	no	—
257	Dic₃	—	14	2	no	—
258	Dic₃	—	14	1	no	—
259	Dic₃	—	14	1	no	—
260	C₂³	—	7	1	no	—
261	C₂³	C₂×D₃×D₇	1	2	yes	—
262	Q₈	—	7	4	no	—
263	D₄	—	7	8	no	—
264	D₄	Dic₃×D₇	1	4	yes	complemented
265	C₂×C₄	—	3	1	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	7	8	no	—
269	C₂×C₄	—	7	2	no	—
270	C₂×C₄	—	7	1	no	—
271	C₂×C₄	—	7	4	no	—
272	C₂×C₄	—	12	2	no	—
273	C₂×C₄	—	21	1	no	—
274	C₂×C₄	—	21	2	no	—
275	C₂×C₄	—	21	1	no	—
276	C₂×C₄	—	84	2	no	—
277	C₂×C₄	—	84	2	no	—
278	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
279	C₇	#5827	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×Dic₃×D₇	1	4	yes	—
291	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
292	C₄	—	6	1	no	—
293	C₄	—	6	1	no	—
294	C₄	—	6	2	no	—
295	C₄	—	7	2	no	—
296	C₄	—	7	4	no	—
297	C₄	—	42	1	no	—
298	C₄	—	42	2	no	—
299	C₄	—	42	1	no	—
300	C₄	C₂×Dic₃×D₇	1	2	yes	—
301	C₃	#90237	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	2	4	no	—
303	C₂	—	14	2	no	—
304	C₂	#10621567	1	1	yes	characteristic
305	C₂	#10621574	1	1	yes	characteristic
306	C₂	C₂²×Dic₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Fitting	C₂×C₃×C₇×D₄	C₂²
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×#1159	C₂
hypercenter	C₂×D₄	D₃×D₇
hypocenter	C₃×C₇	#520
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. D₄×Dic₂₁	C₂	semidirect
3. C₇×D₄×Dic₃	C₂	semidirect
4. (C₂²⋊D₇)⋊Dic₃	C₂	semidirect
5. Dic₃×C₂²⋊D₇	C₂	semidirect
6. C₂×Dic₇⋊Dic₃	C₂	semidirect
7. #10621090	C₂	semidirect
8. D₇×C₄⋊Dic₃	C₂	semidirect
9. #10620919	C₂	semidirect
10. #10620893	C₂	semidirect
11. Dic₃×Dic₁₄	C₂	semidirect
18. C₃×#1159	C₄	semidirect
51. C₃×C₇×D₄	C₂×C₄	semidirect
61. C₃×C₂²⋊D₇	C₂×C₄	semidirect
62. C₂×C₃×Dic₇	C₂×C₄	semidirect
65. C₃×C₄×D₇	C₂×C₄	semidirect
66. C₃×Dic₁₄	C₂×C₄	semidirect
73. #1159	Dic₃	semidirect
98. D₄×Dic₃	D₇	semidirect
122. C₇×D₄	C₂×Dic₃	semidirect
128. C₂²⋊D₇	C₂×Dic₃	semidirect
133. C₂×Dic₇	C₂×Dic₃	semidirect
137. C₄×D₇	C₂×Dic₃	semidirect
138. Dic₁₄	C₂×Dic₃	semidirect
199. C₃×D₄	C₄×D₇	semidirect
214. C₃×C₇	#520	semidirect
264. D₄	Dic₃×D₇	semidirect
279. C₇	#5827	semidirect
301. C₃	#90237	semidirect

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
D₄×Dic₂₁	2
C₇×D₄×Dic₃	2
(C₂²⋊D₇)⋊Dic₃	2
Dic₃×C₂²⋊D₇	2
C₂×Dic₇⋊Dic₃	2
#10621090	2
D₇×C₄⋊Dic₃	2
#10620919	2
#10620893	2
Dic₃×Dic₁₄	2
#90237	3
#5827	7