Subgroups — #13057724

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×#66673	C₂	1	1	yes	maximal, characteristic
2	C₄×D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#872	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620990	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620968	C₂	1	2	yes	maximal, complemented
8	#10620967	C₂	1	2	yes	maximal, complemented
9	#10620966	C₂	1	2	yes	maximal, complemented
10	#10620964	C₂	1	2	yes	maximal, complemented
11	Dic₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#90214	—	3	1	no	maximal, Hall
13	C₂×D₈₄	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₁	C₂²	1	2	yes	—
15	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
17	C₂×#4247	C₂²	1	2	yes	—
18	C₂×D₂₁⋊C₄	C₂²	1	2	yes	—
19	C₂×D₂₁⋊C₄	C₂²	1	2	yes	—
20	C₂×D₃×Dic₇	C₂²	1	2	yes	—
21	#66673	C₄	1	8	yes	complemented
22	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
23	#66631	C₂²	1	2	yes	—
24	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
25	C₇×(C₂×D₃)⋊C₄	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	C₇×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
31	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
32	C₃×C₄×Dic₇	C₂²	1	2	yes	—
33	Dic₃⋊Dic₇	C₂²	1	2	yes	—
34	Dic₇⋊Dic₃	C₂²	1	2	yes	—
35	#66575	C₂²	1	2	yes	—
36	Dic₃×Dic₇	C₂²	1	2	yes	—
37	C₂×#1157	—	3	1	no	—
38	C₇×#117	—	3	1	no	—
39	C₂×C₄⋊Dic₇	—	3	1	no	—
40	#6455	—	3	2	no	—
41	D₇×C₄⋊C₄	—	3	2	no	—
42	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
43	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
44	C₄×D₂₈	—	3	1	no	—
45	C₄×D₂₈	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	C₄×Dic₁₄	D₃	1	1	yes	characteristic, complemented
48	#5574	—	7	1	no	maximal, Hall
49	C₂²×D₂₁	C₂³	1	2	yes	—
50	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
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×Dic₂₁	C₂³	1	2	yes	—
55	D₈₄	C₂×C₄	1	4	yes	complemented
56	C₄×D₂₁	—	4	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	C₂×C₄	1	4	yes	complemented
61	C₂×C₃×Dic₇	C₂³	1	2	yes	—
62	C₂×C₃×Dic₇	C₂³	1	2	yes	—
63	C₃×Dic₁₄	C₂×C₄	1	4	yes	complemented
64	#4247	C₂×C₄	1	8	yes	complemented
65	D₂₁⋊C₄	—	4	2	no	—
66	D₂₁⋊C₄	C₂×C₄	1	8	yes	complemented
67	D₃×Dic₇	—	4	2	no	—
68	C₂²×C₄×C₇	—	3	1	no	—
69	C₂×C₂²⋊D₇	—	3	2	no	—
70	C₂²×Dic₇	—	3	2	no	—
71	#1157	—	3	8	no	—
72	C₂×D₂₈	—	3	1	no	—
73	C₂×C₄×D₇	—	3	2	no	—
74	C₂×C₄×D₇	—	3	2	no	—
75	C₂×C₄×D₇	—	3	2	no	—
76	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
77	C₇×C₄⋊C₄	—	3	1	no	—
78	C₇×C₄⋊C₄	—	3	1	no	—
79	C₇×C₂²⋊C₄	—	3	1	no	—
80	C₇×C₂²⋊C₄	—	3	1	no	—
81	C₄²×C₇	—	3	1	no	—
82	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
83	(C₂×D₇)⋊C₄	—	3	2	no	—
84	(C₂×D₇)⋊C₄	—	3	2	no	—
85	C₄⋊Dic₇	—	3	1	no	—
86	C₄⋊Dic₇	—	3	2	no	—
87	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
88	Dic₇⋊C₄	—	3	2	no	—
89	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
90	C₄×Dic₇	—	3	2	no	—
91	C₄×Dic₇	C₂×D₃	1	2	yes	—
92	C₂×#239	—	7	1	no	—
93	C₃×C₄×Q₈	—	7	1	no	—
94	Q₈×Dic₃	—	7	1	no	—
95	#893	—	7	2	no	—
96	#893	—	7	1	no	—
97	D₃×C₄⋊C₄	—	7	1	no	—
98	D₃×C₄⋊C₄	—	7	2	no	—
99	C₄×D₁₂	—	7	1	no	—
100	C₄×D₁₂	—	7	2	no	—
101	#872	—	7	2	no	—
102	#872	D₇	1	1	yes	characteristic, complemented
103	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×D₂₁	—	2	2	no	—
105	C₂×D₂₁	—	2	2	no	—
106	C₂×D₂₁	C₂²×C₄	1	4	yes	—
107	C₂×D₃×C₇	—	2	1	no	—
108	C₂×D₃×C₇	—	2	1	no	—
109	C₂×D₃×C₇	C₂²×C₄	1	2	yes	—
110	C₃×C₄×C₇	—	2	1	no	—
111	C₃×C₄×C₇	—	2	1	no	—
112	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
113	Dic₂₁	—	2	2	no	—
114	C₃×Dic₇	—	2	2	no	—
115	C₃×Dic₇	C₂²×C₄	1	4	yes	—
116	C₇×Dic₃	—	2	1	no	—
117	C₇×Dic₃	—	2	1	no	—
118	C₇×Dic₃	C₂²×C₄	1	2	yes	—
119	#520	—	21	1	no	Sylow
120	C₂³×C₇	—	3	1	no	—
121	C₂²×D₇	—	3	2	no	—
122	C₂×C₄×C₇	—	3	1	no	—
123	C₂×C₄×C₇	—	3	4	no	—
124	C₂×C₄×C₇	—	3	1	no	—
125	C₂×C₄×C₇	—	3	1	no	—
126	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
129	C₂²⋊D₇	—	3	8	no	—
130	C₂×Dic₇	—	3	2	no	—
131	C₂×Dic₇	—	12	2	no	—
132	C₂×Dic₇	C₂²×D₃	1	2	yes	—
133	C₂×Dic₇	C₂²×D₃	1	2	yes	—
134	D₂₈	—	3	4	no	—
135	C₄×D₇	—	3	8	no	—
136	C₄×D₇	—	12	2	no	—
137	C₄×D₇	—	12	2	no	—
138	Dic₁₄	C₄×D₃	1	4	yes	complemented
139	C₂×C₃×Q₈	—	7	1	no	—
140	#239	—	7	8	no	—
141	C₂×D₁₂	—	7	2	no	—
142	C₂×D₁₂	—	7	1	no	—
143	C₂×C₄×D₃	—	7	2	no	—
144	C₂×C₄×D₃	—	7	2	no	—
145	C₂×C₄×D₃	—	7	2	no	—
146	C₂×C₄×D₃	—	7	2	no	—
147	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
148	C₃×C₄⋊C₄	—	7	2	no	—
149	C₃×C₄⋊C₄	—	7	1	no	—
150	C₃×C₄²	—	7	2	no	—
151	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
152	(C₂×D₃)⋊C₄	—	7	2	no	—
153	(C₂×D₃)⋊C₄	—	7	2	no	—
154	(C₂×D₃)⋊C₄	C₂×D₇	1	1	yes	characteristic
155	(C₂×D₃)⋊C₄	C₂×D₇	1	1	yes	characteristic
156	C₄⋊Dic₃	—	7	1	no	—
157	C₄⋊Dic₃	—	7	2	no	—
158	Dic₃⋊C₄	—	7	2	no	—
159	Dic₃⋊C₄	C₂×D₇	1	1	yes	characteristic
160	Dic₃⋊C₄	C₂×D₇	1	1	yes	characteristic
161	C₄×Dic₃	—	7	2	no	—
162	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
163	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
166	D₂₁	—	2	4	no	—
167	D₃×C₇	—	2	2	no	—
168	C₂×#41	—	21	1	no	—
169	C₄×Q₈	—	7	1	no	—
170	C₄×Q₈	—	21	1	no	—
171	C₄×D₄	—	21	1	no	—
172	C₄×D₄	—	21	2	no	—
173	C₄×D₄	—	21	2	no	—
174	C₄×D₄	—	21	1	no	—
175	#117	—	3	1	no	—
176	#117	—	21	2	no	—
177	C₂×C₄⋊C₄	—	21	2	no	—
178	C₂×C₄⋊C₄	—	21	1	no	—
179	C₂²×C₇	—	3	2	no	—
180	C₂²×C₇	—	6	1	no	—
181	C₂²×C₇	—	6	1	no	—
182	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
183	C₂×D₇	—	3	4	no	—
184	C₂×D₇	—	6	2	no	—
185	C₂×D₇	—	6	2	no	—
186	C₄×C₇	—	2	1	no	—
187	C₄×C₇	—	2	1	no	—
188	C₄×C₇	—	3	2	no	—
189	C₄×C₇	—	6	1	no	—
190	C₄×C₇	—	6	1	no	—
191	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
192	Dic₇	—	2	2	no	—
193	Dic₇	—	6	2	no	—
194	Dic₇	C₂×C₄×D₃	1	4	yes	—
195	C₂²×D₃	—	7	2	no	—
196	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
197	C₃×Q₈	—	7	4	no	—
198	C₂×C₃×C₄	—	7	2	no	—
199	C₂×C₃×C₄	—	7	2	no	—
200	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
203	C₂×Dic₃	—	7	2	no	—
204	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
205	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
206	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
207	D₁₂	—	7	8	no	—
208	D₁₂	—	7	4	no	—
209	C₄×D₃	—	7	8	no	—
210	C₄×D₃	—	28	2	no	—
211	C₄×D₃	—	28	2	no	—
212	C₄×D₃	—	28	2	no	—
213	C₄×D₃	C₄×D₇	1	4	yes	complemented
214	C₃×C₇	#520	1	1	yes	characteristic, complemented, Hall
215	#41	—	21	8	no	—
216	C₂×Q₈	—	7	1	no	—
217	C₂×D₄	—	21	2	no	—
218	C₂×D₄	—	21	1	no	—
219	C₂²×C₄	—	3	1	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	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	7	2	no	—
227	C₄⋊C₄	—	7	1	no	—
228	C₄⋊C₄	—	21	2	no	—
229	C₄⋊C₄	—	21	2	no	—
230	C₄⋊C₄	—	21	1	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	1	no	—
233	C₂²⋊C₄	—	21	2	no	—
234	C₂²⋊C₄	—	21	2	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	7	2	no	—
237	C₄²	—	21	2	no	—
238	C₄²	D₃×D₇	1	1	yes	characteristic
239	C₂×C₇	—	6	2	no	—
240	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
241	C₂×C₇	#1007	1	1	yes	characteristic
242	C₂×C₇	#1009	1	1	yes	characteristic
243	D₇	—	6	4	no	—
244	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
245	C₂×D₃	—	2	1	no	—
246	C₂×D₃	—	2	1	no	—
247	C₂×D₃	—	7	4	no	—
248	C₂×D₃	—	14	2	no	—
249	C₂×D₃	—	14	2	no	—
250	C₂×D₃	C₂×C₄×D₇	1	2	yes	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	7	4	no	—
254	C₃×C₄	—	14	2	no	—
255	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
256	Dic₃	—	2	1	no	—
257	Dic₃	—	2	1	no	—
258	Dic₃	—	14	2	no	—
259	Dic₃	C₂×C₄×D₇	1	2	yes	—
260	C₂³	—	3	1	no	—
261	C₂³	—	21	2	no	—
262	Q₈	—	7	4	no	—
263	D₄	—	21	8	no	—
264	D₄	—	21	4	no	—
265	C₂×C₄	—	3	1	no	—
266	C₂×C₄	—	3	4	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	7	2	no	—
270	C₂×C₄	—	7	2	no	—
271	C₂×C₄	—	21	8	no	—
272	C₂×C₄	—	21	2	no	—
273	C₂×C₄	—	84	2	no	—
274	C₂×C₄	—	84	2	no	—
275	C₂×C₄	—	84	2	no	—
276	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
277	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
278	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
279	C₇	#5574	1	1	yes	characteristic, complemented, Sylow
280	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
281	C₂×C₃	#6553	1	1	yes	characteristic
282	C₂×C₃	#6554	1	1	yes	characteristic
283	D₃	—	2	2	no	—
284	D₃	—	14	4	no	—
285	C₂²	—	3	2	no	—
286	C₂²	—	6	1	no	—
287	C₂²	—	6	1	no	—
288	C₂²	—	21	4	no	—
289	C₂²	—	42	2	no	—
290	C₂²	—	42	2	no	—
291	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	3	2	no	—
295	C₄	—	6	1	no	—
296	C₄	—	6	1	no	—
297	C₄	—	7	4	no	—
298	C₄	—	14	2	no	—
299	C₄	—	42	2	no	—
300	C₄	C₄×D₃×D₇	1	2	yes	—
301	C₃	#90214	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	6	2	no	—
303	C₂	—	42	4	no	—
304	C₂	#10621551	1	1	yes	characteristic
305	C₂	C₂×C₄×D₃×D₇	1	1	yes	characteristic
306	C₂	#10621566	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Fitting	C₃×C₄²×C₇	C₂²
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×#66673	C₂
hypercenter	C₄²	D₃×D₇
hypocenter	C₃×C₇	#520
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×D₈₄	C₂	semidirect
3. C₇×#872	C₂	semidirect
4. C₃×C₄×Dic₁₄	C₂	semidirect
5. #10620990	C₂	semidirect
6. D₃×C₄⋊Dic₇	C₂	semidirect
7. #10620968	C₂	semidirect
8. #10620967	C₂	semidirect
9. #10620966	C₂	semidirect
10. #10620964	C₂	semidirect
11. Dic₃×Dic₁₄	C₂	semidirect
21. #66673	C₄	semidirect
47. C₄×Dic₁₄	D₃	semidirect
55. D₈₄	C₂×C₄	semidirect
60. C₄×D₃×C₇	C₂×C₄	semidirect
63. C₃×Dic₁₄	C₂×C₄	semidirect
64. #4247	C₂×C₄	semidirect
66. D₂₁⋊C₄	C₂×C₄	semidirect
102. #872	D₇	semidirect
138. Dic₁₄	C₄×D₃	semidirect
213. C₄×D₃	C₄×D₇	semidirect
214. C₃×C₇	#520	semidirect
279. C₇	#5574	semidirect
301. C₃	#90214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66673	2
C₄×D₈₄	2
C₇×#872	2
C₃×C₄×Dic₁₄	2
#10620990	2
D₃×C₄⋊Dic₇	2
#10620968	2
#10620967	2
#10620966	2
#10620964	2
Dic₃×Dic₁₄	2
#90214	3
#5574	7