Subgroups — #13061624

Order 1344.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621413	C₂	1	1	yes	maximal, characteristic
3	#10621407	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621381	C₂	1	2	yes	maximal, complemented
5	#10621379	C₂	1	2	yes	maximal, complemented
6	#10621368	C₂	1	2	yes	maximal, complemented
7	#10621366	C₂	1	2	yes	maximal, complemented
8	#10621358	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#10621352	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#90225	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×#4268	C₂²	1	2	yes	—
15	C₂×D₈₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₁	C₂²	1	2	yes	—
17	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₂	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
25	#66636	C₂²	1	2	yes	—
26	#66631	C₂²	1	2	yes	—
27	#66631	C₂²	1	2	yes	—
28	#66631	C₂²	1	2	yes	—
29	#66631	C₂²	1	2	yes	—
30	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
31	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
32	C₄⋊Dic₂₁	C₂²	1	2	yes	—
33	#66629	C₂²	1	2	yes	—
34	#66629	C₂²	1	2	yes	—
35	C₄×Dic₂₁	C₂²	1	2	yes	—
36	C₇×#117	D₃	1	1	yes	characteristic, complemented
37	#6493	—	3	1	no	—
38	#6487	—	3	1	no	—
39	#6461	—	3	2	no	—
40	#6459	—	3	2	no	—
41	#6448	—	3	2	no	—
42	#6446	—	3	2	no	—
43	#6438	—	3	1	no	—
44	C₄×D₂₈	—	3	1	no	—
45	#6432	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5534	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×D₂₁	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	—	2	2	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
57	#4268	—	4	2	no	—
58	C₂×Dic₂₁	C₂³	1	2	yes	—
59	C₂×Dic₂₁	C₂³	1	2	yes	—
60	C₂×Dic₂₁	C₂³	1	2	yes	—
61	D₈₄	—	2	2	no	—
62	C₄×D₂₁	—	4	2	no	—
63	Dic₄₂	—	2	2	no	—
64	Dic₄₂	—	4	2	no	—
65	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	—	3	2	no	—
67	C₂×D₂₈	—	3	1	no	—
68	C₂×C₄×D₇	—	3	2	no	—
69	C₂×Dic₁₄	—	3	1	no	—
70	C₂×Dic₁₄	—	3	2	no	—
71	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	C₇×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
74	C₇×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
75	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
77	C₂²⋊Dic₇	—	3	2	no	—
78	(C₂×D₇)⋊C₄	—	3	2	no	—
79	(C₂×D₇)⋊C₄	—	3	2	no	—
80	(C₂×D₇)⋊C₄	—	3	2	no	—
81	(C₂×D₇)⋊C₄	—	3	2	no	—
82	C₄⋊Dic₇	—	3	1	no	—
83	C₄⋊Dic₇	—	3	1	no	—
84	C₄⋊Dic₇	—	3	2	no	—
85	Dic₇⋊C₄	—	3	2	no	—
86	Dic₇⋊C₄	—	3	2	no	—
87	C₄×Dic₇	—	3	2	no	—
88	C₃×#117	D₇	1	1	yes	characteristic, complemented
89	#930	—	7	1	no	—
90	#924	—	7	1	no	—
91	#898	—	7	2	no	—
92	#896	—	7	2	no	—
93	#885	—	7	2	no	—
94	#883	—	7	2	no	—
95	#875	—	7	1	no	—
96	C₄×D₁₂	—	7	1	no	—
97	#869	—	7	1	no	—
98	C₄×Dic₆	—	7	1	no	—
99	C₂²×C₃×C₇	—	2	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	—	2	1	no	—
102	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
103	C₂×D₂₁	—	2	2	no	—
104	C₂×D₂₁	—	2	2	no	—
105	C₂×D₂₁	—	2	2	no	—
106	C₃×C₄×C₇	—	2	1	no	—
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×C₄×C₇	—	2	1	no	—
109	C₃×C₄×C₇	—	2	1	no	—
110	C₃×C₄×C₇	—	2	1	no	—
111	C₃×C₄×C₇	#41	1	2	yes	—
112	Dic₂₁	—	2	2	no	—
113	Dic₂₁	—	2	2	no	—
114	Dic₂₁	—	2	2	no	—
115	#542	—	21	1	no	Sylow
116	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
117	C₂²×D₇	—	3	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂²⋊D₇	—	12	2	no	—
126	C₂×Dic₇	—	3	2	no	—
127	C₂×Dic₇	—	3	2	no	—
128	C₂×Dic₇	—	3	2	no	—
129	D₂₈	—	6	2	no	—
130	C₄×D₇	—	12	2	no	—
131	Dic₁₄	—	6	2	no	—
132	Dic₁₄	—	12	2	no	—
133	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
134	C₂×#67	—	7	2	no	—
135	C₂×D₁₂	—	7	1	no	—
136	C₂×C₄×D₃	—	7	2	no	—
137	C₂×Dic₆	—	7	1	no	—
138	C₂×Dic₆	—	7	2	no	—
139	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
141	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
142	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
143	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
144	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
145	#217	—	7	2	no	—
146	(C₂×D₃)⋊C₄	—	7	2	no	—
147	(C₂×D₃)⋊C₄	—	7	2	no	—
148	(C₂×D₃)⋊C₄	—	7	2	no	—
149	(C₂×D₃)⋊C₄	—	7	2	no	—
150	C₄⋊Dic₃	—	7	1	no	—
151	C₄⋊Dic₃	—	7	1	no	—
152	C₄⋊Dic₃	—	7	2	no	—
153	Dic₃⋊C₄	—	7	2	no	—
154	Dic₃⋊C₄	—	7	2	no	—
155	C₄×Dic₃	—	7	2	no	—
156	C₂×C₃×C₇	—	4	1	no	—
157	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
158	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
160	D₂₁	—	4	2	no	—
161	C₄⋊Q₈	—	21	1	no	—
162	#126	—	21	2	no	—
163	#124	—	21	2	no	—
164	#124	—	21	1	no	—
165	#123	—	21	2	no	—
166	C₂²⋊Q₈	—	21	1	no	—
167	C₂²⋊Q₈	—	21	2	no	—
168	#121	—	21	1	no	—
169	C₄×Q₈	—	21	1	no	—
170	C₄×D₄	—	21	1	no	—
171	#117	D₂₁	1	1	yes	characteristic, complemented
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
176	C₂×D₇	—	6	2	no	—
177	C₂×D₇	—	6	2	no	—
178	C₂×D₇	—	6	2	no	—
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	#235	1	2	yes	—
185	Dic₇	—	6	2	no	—
186	Dic₇	—	6	2	no	—
187	Dic₇	—	6	2	no	—
188	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
189	C₂²×D₃	—	7	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
197	#67	—	28	2	no	—
198	C₂×Dic₃	—	7	2	no	—
199	C₂×Dic₃	—	7	2	no	—
200	C₂×Dic₃	—	7	2	no	—
201	D₁₂	—	14	2	no	—
202	C₄×D₃	—	28	2	no	—
203	Dic₆	—	14	2	no	—
204	Dic₆	—	28	2	no	—
205	C₃×C₇	#542	1	1	yes	characteristic, complemented, Hall
206	C₂×Q₈	—	21	1	no	—
207	C₂×Q₈	—	21	2	no	—
208	C₂×D₄	—	21	2	no	—
209	C₂×D₄	—	21	1	no	—
210	C₂²×C₄	—	21	2	no	—
211	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
212	C₄⋊C₄	—	21	2	no	—
213	C₄⋊C₄	—	21	2	no	—
214	C₄⋊C₄	—	21	1	no	—
215	C₄⋊C₄	—	21	1	no	—
216	C₄⋊C₄	—	21	2	no	—
217	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
218	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
219	C₂²⋊C₄	—	21	2	no	—
220	C₂²⋊C₄	—	21	2	no	—
221	C₂²⋊C₄	—	21	2	no	—
222	C₂²⋊C₄	—	21	2	no	—
223	C₂²⋊C₄	—	21	2	no	—
224	C₂²⋊C₄	C₂×D₂₁	1	1	yes	characteristic
225	C₂²⋊C₄	C₂×D₂₁	1	1	yes	characteristic
226	C₄²	—	21	2	no	—
227	C₄²	C₂×D₂₁	1	1	yes	characteristic
228	C₄²	C₂×D₂₁	1	1	yes	characteristic
229	C₂×C₇	—	4	1	no	—
230	C₂×C₇	C₂×#235	1	1	yes	characteristic
231	C₂×C₇	#1009	1	1	yes	characteristic
232	C₂×C₇	#1010	1	1	yes	characteristic
233	D₇	—	12	2	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
238	C₂×D₃	—	14	2	no	—
239	C₂×D₃	—	14	2	no	—
240	C₂×D₃	—	14	2	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	#1157	1	2	yes	—
247	Dic₃	—	14	2	no	—
248	Dic₃	—	14	2	no	—
249	Dic₃	—	14	2	no	—
250	C₂³	—	21	2	no	—
251	C₂³	C₂²×D₂₁	1	1	yes	characteristic
252	Q₈	—	42	2	no	—
253	Q₈	—	84	2	no	—
254	D₄	—	42	2	no	—
255	D₄	—	84	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	21	2	no	—
258	C₂×C₄	—	21	2	no	—
259	C₂×C₄	—	21	2	no	—
260	C₂×C₄	—	84	2	no	—
261	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
267	C₇	#5534	1	1	yes	characteristic, complemented, Sylow
268	C₂×C₃	—	4	1	no	—
269	C₂×C₃	C₂×#1157	1	1	yes	characteristic
270	C₂×C₃	#6553	1	1	yes	characteristic
271	C₂×C₃	#6554	1	1	yes	characteristic
272	D₃	—	28	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	42	2	no	—
277	C₂²	—	42	2	no	—
278	C₂²	—	42	2	no	—
279	C₂²	C₂³×D₂₁	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	42	2	no	—
286	C₄	—	42	2	no	—
287	C₄	—	42	2	no	—
288	C₄	#66728	1	2	yes	—
289	C₃	#90225	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	4	1	no	—
291	C₂	—	84	2	no	—
292	C₂	C₂×#66728	1	1	yes	characteristic
293	C₂	#10621662	1	1	yes	characteristic
294	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×#117	C₂
prime radical	#10621413	C₂
hypercenter	#117	D₂₁
hypocenter	C₃×C₇	#542
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#117	C₂	semidirect
3. #10621407	C₂	semidirect
4. #10621381	C₂	semidirect
5. #10621379	C₂	semidirect
6. #10621368	C₂	semidirect
7. #10621366	C₂	semidirect
8. #10621358	C₂	semidirect
9. C₄×D₈₄	C₂	semidirect
10. #10621352	C₂	semidirect
11. C₄×Dic₄₂	C₂	semidirect
36. C₇×#117	D₃	semidirect
88. C₃×#117	D₇	semidirect
171. #117	D₂₁	semidirect
205. C₃×C₇	#542	semidirect
267. C₇	#5534	semidirect
289. C₃	#90225	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#117	2
#10621413	2
#10621407	2
#10621381	2
#10621379	2
#10621368	2
#10621366	2
#10621358	2
C₄×D₈₄	2
#10621352	2
C₄×Dic₄₂	2
#90225	3
#5534	7