Subgroups — #13059614

Order 1344.

472 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621409	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₇×#67	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6493	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621111	C₂	1	2	yes	maximal, complemented
5	#10621096	C₂	1	2	yes	maximal, complemented
6	#10621031	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621026	C₂	1	2	yes	maximal, complemented
8	#10621018	C₂	1	1	yes	maximal, characteristic, complemented
9	Dic₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620956	C₂	1	2	yes	maximal, complemented
11	#10620900	C₂	1	1	yes	maximal, characteristic, complemented
12	#90254	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₇×#67	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
18	C₂×#4248	C₂²	1	2	yes	—
19	C₂×#4245	C₂²	1	2	yes	—
20	C₂×Dic₃×D₇	C₂²	1	2	yes	—
21	#66636	C₂²	1	2	yes	—
22	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₁	C₂²	1	2	yes	—
25	C₇×#217	C₂²	1	1	yes	characteristic
26	C₇×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
27	C₇×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
29	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
31	Dic₃⋊Dic₇	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	C₄×C₇×D₄	—	3	1	no	—
37	#6501	—	3	2	no	—
38	#6493	—	3	1	no	—
39	#6493	D₃	1	1	yes	characteristic, complemented
40	#6489	—	3	1	no	—
41	#6489	—	3	1	no	—
42	#6455	—	3	2	no	—
43	#6448	—	3	2	no	—
44	#6448	—	3	2	no	—
45	C₄×D₂₈	—	3	1	no	—
46	#6432	—	3	1	no	—
47	#5616	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
50	C₂²×C₃×D₇	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	—	2	2	no	—
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₁	C₂³	1	2	yes	—
55	C₂×Dic₂₁	C₂³	1	2	yes	—
56	C₇×#67	—	2	2	no	—
57	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
60	C₄×D₃×C₇	—	2	2	no	—
61	C₃×C₂²⋊D₇	—	4	2	no	—
62	C₂×C₃×Dic₇	C₂³	1	2	yes	—
63	C₃×D₂₈	—	2	2	no	—
64	#4248	—	4	2	no	—
65	#4245	—	4	2	no	—
66	Dic₃×D₇	—	2	4	no	—
67	C₂×C₇×D₄	—	3	1	no	—
68	C₂²×C₄×C₇	—	3	1	no	—
69	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
70	C₂×C₂²⋊D₇	—	3	2	no	—
71	C₂×C₂²⋊D₇	C₂×D₃	1	2	yes	—
72	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₄×D₇	—	3	2	no	—
74	C₂×Dic₁₄	—	3	2	no	—
75	C₇×C₄⋊C₄	—	3	1	no	—
76	C₇×C₂²⋊C₄	—	3	1	no	—
77	C₇×C₂²⋊C₄	—	3	1	no	—
78	C₄²×C₇	—	3	1	no	—
79	C₂²⋊Dic₇	—	3	2	no	—
80	C₂²⋊Dic₇	—	3	2	no	—
81	(C₂×D₇)⋊C₄	—	3	2	no	—
82	(C₂×D₇)⋊C₄	—	3	2	no	—
83	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	—
84	C₄⋊Dic₇	—	3	1	no	—
85	C₄⋊Dic₇	—	3	1	no	—
86	C₄⋊Dic₇	—	3	2	no	—
87	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
88	C₄×Dic₇	—	3	2	no	—
89	C₄×Dic₇	—	3	2	no	—
90	C₃×#121	—	7	1	no	—
91	#939	—	7	2	no	—
92	#938	—	7	1	no	—
93	#936	—	7	2	no	—
94	D₄×Dic₃	—	7	1	no	—
95	C₄×#67	D₇	1	1	yes	characteristic, complemented
96	#926	—	7	1	no	—
97	#892	—	7	1	no	—
98	#888	—	7	1	no	—
99	#885	—	7	2	no	—
100	#877	—	7	2	no	—
101	C₂²×C₃×C₇	—	2	1	no	—
102	C₂²×C₃×C₇	—	2	1	no	—
103	C₂²×C₃×C₇	—	2	1	no	—
104	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
105	C₂×D₃×C₇	—	2	1	no	—
106	C₂×D₃×C₇	—	2	1	no	—
107	C₂×D₃×C₇	—	2	1	no	—
108	C₂×C₃×D₇	—	2	2	no	—
109	C₂×C₃×D₇	—	2	2	no	—
110	C₂×C₃×D₇	—	2	2	no	—
111	C₃×C₄×C₇	—	2	1	no	—
112	C₃×C₄×C₇	#41	1	2	yes	—
113	Dic₂₁	—	2	2	no	—
114	Dic₂₁	—	2	2	no	—
115	C₃×Dic₇	—	2	2	no	—
116	C₇×Dic₃	—	2	1	no	—
117	C₇×Dic₃	—	2	1	no	—
118	C₇×Dic₃	#41	1	2	yes	—
119	#555	—	21	1	no	Sylow
120	C₂³×C₇	—	3	1	no	—
121	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
122	C₂²×D₇	C₂²×D₃	1	2	yes	—
123	C₇×D₄	—	6	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	3	1	no	—
126	C₂×C₄×C₇	—	3	1	no	—
127	C₂×C₄×C₇	—	3	1	no	—
128	C₂×C₄×C₇	—	6	2	no	—
129	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
131	C₂²⋊D₇	—	4	2	no	—
132	C₂²⋊D₇	—	12	2	no	—
133	C₂×Dic₇	—	3	2	no	—
134	C₂×Dic₇	—	3	2	no	—
135	C₂×Dic₇	C₂²×D₃	1	2	yes	—
136	D₂₈	—	2	2	no	—
137	C₄×D₇	—	6	4	no	—
138	Dic₁₄	—	12	2	no	—
139	C₂×C₃×D₄	—	7	2	no	—
140	C₂×C₃×D₄	—	7	1	no	—
141	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
142	C₂×#67	—	7	2	no	—
143	C₂×#67	C₂×D₇	1	1	yes	characteristic
144	C₂²×Dic₃	—	7	2	no	—
145	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
146	C₂×Dic₆	—	7	2	no	—
147	C₃×C₄⋊C₄	—	7	1	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	#217	—	7	2	no	—
150	#217	—	7	2	no	—
151	#217	—	7	2	no	—
152	#217	C₂×D₇	1	1	yes	characteristic
153	(C₂×D₃)⋊C₄	—	7	2	no	—
154	(C₂×D₃)⋊C₄	C₂×D₇	1	1	yes	characteristic
155	C₄⋊Dic₃	—	7	1	no	—
156	C₄⋊Dic₃	—	7	1	no	—
157	Dic₃⋊C₄	—	7	2	no	—
158	Dic₃⋊C₄	C₂×D₇	1	1	yes	characteristic
159	C₄×Dic₃	—	7	2	no	—
160	C₄×Dic₃	—	7	2	no	—
161	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
162	C₂×C₃×C₇	—	4	1	no	—
163	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₃×D₇	—	4	2	no	—
167	D₃×C₇	—	4	1	no	—
168	C₄⋊Q₈	—	21	1	no	—
169	#124	—	21	2	no	—
170	#124	—	21	2	no	—
171	#121	—	7	1	no	—
172	#121	—	21	1	no	—
173	#121	—	21	2	no	—
174	C₄×D₄	—	3	1	no	—
175	C₄×D₄	—	21	1	no	—
176	#117	—	21	2	no	—
177	#117	—	21	1	no	—
178	#117	—	21	1	no	—
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	—	2	1	no	—
181	C₂²×C₇	—	2	1	no	—
182	C₂²×C₇	—	6	1	no	—
183	C₂²×C₇	—	6	1	no	—
184	C₂²×C₇	—	6	1	no	—
185	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
186	C₂×D₇	—	2	2	no	—
187	C₂×D₇	—	2	2	no	—
188	C₂×D₇	—	2	2	no	—
189	C₄×C₇	—	2	1	no	—
190	C₄×C₇	—	3	2	no	—
191	C₄×C₇	—	6	1	no	—
192	C₄×C₇	—	6	1	no	—
193	C₄×C₇	#237	1	2	yes	—
194	Dic₇	—	2	2	no	—
195	Dic₇	—	6	2	no	—
196	Dic₇	—	6	2	no	—
197	C₂³×C₃	—	7	2	no	—
198	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
199	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
200	C₃×D₄	—	14	2	no	—
201	C₃×D₄	—	28	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	7	2	no	—
204	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
206	#67	—	2	2	no	—
207	#67	—	28	2	no	—
208	C₂×Dic₃	—	7	2	no	—
209	C₂×Dic₃	—	7	2	no	—
210	C₂×Dic₃	—	14	4	no	—
211	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
212	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
213	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
214	C₄×D₃	—	2	2	no	—
215	Dic₆	—	28	2	no	—
216	C₃×C₇	#555	1	1	yes	characteristic, complemented, Hall
217	C₂×Q₈	—	21	2	no	—
218	C₂×D₄	—	3	1	no	—
219	C₂×D₄	—	7	2	no	—
220	C₂×D₄	—	7	1	no	—
221	C₂×D₄	—	21	2	no	—
222	C₂²×C₄	—	3	1	no	—
223	C₂²×C₄	—	21	2	no	—
224	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	7	1	no	—
227	C₄⋊C₄	—	21	2	no	—
228	C₄⋊C₄	—	21	1	no	—
229	C₄⋊C₄	—	21	1	no	—
230	C₂²⋊C₄	—	3	1	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	7	2	no	—
233	C₂²⋊C₄	—	21	2	no	—
234	C₂²⋊C₄	—	21	2	no	—
235	C₂²⋊C₄	—	21	2	no	—
236	C₂²⋊C₄	—	21	2	no	—
237	C₄²	—	3	1	no	—
238	C₄²	—	21	2	no	—
239	C₄²	—	21	2	no	—
240	C₂×C₇	—	4	1	no	—
241	C₂×C₇	—	12	1	no	—
242	C₂×C₇	C₂×#237	1	1	yes	characteristic
243	C₂×C₇	#1004	1	1	yes	characteristic
244	C₂×C₇	D₃×#41	1	1	yes	characteristic
245	D₇	—	4	2	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	14	2	no	—
250	C₂²×C₃	—	14	2	no	—
251	C₂²×C₃	—	14	2	no	—
252	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	14	2	no	—
258	C₃×C₄	#1159	1	2	yes	—
259	Dic₃	—	2	1	no	—
260	Dic₃	—	2	1	no	—
261	Dic₃	—	14	2	no	—
262	Dic₃	—	14	2	no	—
263	Dic₃	#1157	1	2	yes	—
264	C₂³	—	3	1	no	—
265	C₂³	—	7	2	no	—
266	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
267	Q₈	—	84	2	no	—
268	D₄	—	6	2	no	—
269	D₄	—	14	2	no	—
270	D₄	—	28	2	no	—
271	D₄	—	84	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	6	2	no	—
277	C₂×C₄	—	7	2	no	—
278	C₂×C₄	—	21	2	no	—
279	C₂×C₄	—	21	2	no	—
280	C₂×C₄	—	42	4	no	—
281	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
282	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
283	C₇	#5616	1	1	yes	characteristic, complemented, Sylow
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	—	28	2	no	—
286	C₂×C₃	C₂×#1157	1	1	yes	characteristic
287	C₂×C₃	C₂×#1159	1	1	yes	characteristic
288	C₂×C₃	#6553	1	1	yes	characteristic
289	D₃	—	4	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	6	1	no	—
294	C₂²	—	6	1	no	—
295	C₂²	—	6	1	no	—
296	C₂²	—	14	2	no	—
297	C₂²	—	14	2	no	—
298	C₂²	—	14	2	no	—
299	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	3	2	no	—
302	C₄	—	6	1	no	—
303	C₄	—	6	1	no	—
304	C₄	—	14	2	no	—
305	C₄	—	42	2	no	—
306	C₄	—	42	2	no	—
307	C₄	#66684	1	2	yes	—
308	C₃	#90254	1	1	yes	characteristic, complemented, Sylow
309	C₂	—	4	1	no	—
310	C₂	—	12	1	no	—
311	C₂	—	28	2	no	—
312	C₂	D₃×#1157	1	1	yes	characteristic
313	C₂	#10621564	1	1	yes	characteristic
314	C₂	C₂×#66684	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
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#555
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
1. #10621409	C₂	semidirect
2. C₄×C₇×#67	C₂	semidirect
3. C₃×#6493	C₂	semidirect
4. #10621111	C₂	semidirect
5. #10621096	C₂	semidirect
6. #10621031	C₂	semidirect
7. #10621026	C₂	semidirect
8. #10621018	C₂	semidirect
9. Dic₃×D₂₈	C₂	semidirect
10. #10620956	C₂	semidirect
11. #10620900	C₂	semidirect
39. #6493	D₃	semidirect
95. C₄×#67	D₇	semidirect
216. C₃×C₇	#555	semidirect
283. C₇	#5616	semidirect
308. C₃	#90254	semidirect

Maximal subgroups

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

subgroup	index
#10621409	2
C₄×C₇×#67	2
C₃×#6493	2
#10621111	2
#10621096	2
#10621031	2
#10621026	2
#10621018	2
Dic₃×D₂₈	2
#10620956	2
#10620900	2
#90254	3
#5616	7