Subgroups — #13058549

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66672	C₂	1	1	yes	maximal, characteristic
2	#10621375	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#893	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6450	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621008	C₂	1	2	yes	maximal, complemented
6	D₁₂×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620967	C₂	1	2	yes	maximal, complemented
8	C₄×D₂₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620962	C₂	1	2	yes	maximal, complemented
10	C₄×D₃×Dic₇	C₂	1	2	yes	maximal, complemented
11	Dic₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#1159	—	3	1	no	maximal, Hall
13	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
14	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
16	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
17	C₂×#4247	C₂²	1	2	yes	—
18	C₂×D₂₁⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×D₂₁⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×D₃×Dic₇	C₂²	1	2	yes	—
21	C₂×D₃×Dic₇	C₂²	1	2	yes	—
22	#66672	C₄	1	8	yes	complemented
23	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	#66631	C₂²	1	2	yes	—
25	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₁	C₂²	1	2	yes	—
27	C₇×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄×Dic₇	C₂²	1	2	yes	—
31	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
32	Dic₇⋊Dic₃	C₂²	1	2	yes	—
33	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
34	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
37	C₂×#1159	—	3	1	no	—
38	C₄×C₇×D₄	—	3	1	no	—
39	D₄×Dic₇	—	3	1	no	—
40	C₄×C₂²⋊D₇	—	3	2	no	—
41	C₂×C₄×Dic₇	—	3	2	no	—
42	#6455	—	3	1	no	—
43	#6450	D₃	1	1	yes	characteristic, complemented
44	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
45	#6440	—	3	2	no	—
46	C₄²×D₇	—	3	1	no	—
47	C₄×Dic₁₄	—	3	1	no	—
48	C₄×#239	—	7	1	no	maximal, Hall
49	C₂²×D₂₁	C₂³	1	1	yes	characteristic
50	C₂²×D₃×C₇	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	2	yes	—
54	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
55	C₄×D₂₁	C₂×C₄	1	4	yes	complemented
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
58	C₇×D₁₂	C₂×C₄	1	4	yes	complemented
59	C₄×D₃×C₇	—	2	4	no	—
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₇	C₂³	1	2	yes	—
62	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
63	C₃×Dic₁₄	C₂×C₄	1	4	yes	complemented
64	#4247	C₂×C₄	1	8	yes	complemented
65	D₂₁⋊C₄	—	2	2	no	—
66	D₂₁⋊C₄	—	2	2	no	—
67	D₃×Dic₇	—	2	4	no	—
68	D₃×Dic₇	C₂×C₄	1	8	yes	complemented
69	C₂×C₇×D₄	—	3	1	no	—
70	C₂²×C₄×C₇	—	3	2	no	—
71	C₂×C₂²⋊D₇	—	3	2	no	—
72	C₂²×Dic₇	—	3	2	no	—
73	C₂²×Dic₇	—	3	2	no	—
74	#1159	—	3	8	no	—
75	C₂×C₄×D₇	—	3	1	no	—
76	C₂×C₄×D₇	—	3	1	no	—
77	C₂×C₄×D₇	—	3	1	no	—
78	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
79	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
80	C₇×C₂²⋊C₄	—	3	2	no	—
81	C₄²×C₇	—	3	1	no	—
82	C₂²⋊Dic₇	—	3	2	no	—
83	(C₂×D₇)⋊C₄	—	3	2	no	—
84	C₄⋊Dic₇	—	3	1	no	—
85	Dic₇⋊C₄	—	3	2	no	—
86	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
87	C₄×Dic₇	—	3	2	no	—
88	C₄×Dic₇	—	3	2	no	—
89	C₄×Dic₇	—	3	1	no	—
90	C₄×Dic₇	—	3	1	no	—
91	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
92	C₄×Dic₇	C₂×D₃	1	2	yes	—
93	C₂×#239	—	7	1	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	Q₈×Dic₃	—	7	1	no	—
96	#893	—	7	2	no	—
97	#893	D₇	1	1	yes	characteristic, complemented
98	#892	—	7	1	no	—
99	#892	—	7	2	no	—
100	C₄×D₁₂	—	7	1	no	—
101	C₄×D₁₂	—	7	2	no	—
102	C₄²×D₃	—	7	2	no	—
103	C₄²×D₃	—	7	1	no	—
104	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
105	C₂×D₂₁	—	2	1	no	—
106	C₂×D₂₁	—	2	1	no	—
107	C₂×D₂₁	C₂²×C₄	1	2	yes	—
108	C₂×D₃×C₇	—	2	2	no	—
109	C₂×D₃×C₇	—	2	2	no	—
110	C₂×D₃×C₇	C₂²×C₄	1	4	yes	—
111	C₃×C₄×C₇	—	2	2	no	—
112	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
113	Dic₂₁	—	2	2	no	—
114	Dic₂₁	C₂²×C₄	1	2	yes	—
115	C₃×Dic₇	—	2	1	no	—
116	C₃×Dic₇	C₂²×C₄	1	4	yes	—
117	C₃×Dic₇	#41	1	2	yes	complemented
118	C₇×Dic₃	—	2	1	no	—
119	C₇×Dic₃	#41	1	2	yes	complemented
120	C₄×#41	—	21	1	no	Sylow
121	C₂³×C₇	—	3	2	no	—
122	C₂²×D₇	—	3	1	no	—
123	C₇×D₄	—	3	4	no	—
124	C₂×C₄×C₇	—	3	1	no	—
125	C₂×C₄×C₇	—	3	1	no	—
126	C₂×C₄×C₇	—	6	4	no	—
127	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
128	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
129	C₂²⋊D₇	—	3	8	no	—
130	C₂×Dic₇	—	3	1	no	—
131	C₂×Dic₇	—	3	8	no	—
132	C₂×Dic₇	—	3	2	no	—
133	C₂×Dic₇	—	6	4	no	—
134	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
135	C₂×Dic₇	C₂²×D₃	1	2	yes	—
136	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
137	C₄×D₇	—	3	4	no	—
138	C₄×D₇	—	6	2	no	—
139	C₄×D₇	—	6	2	no	—
140	Dic₁₄	C₄×D₃	1	4	yes	complemented
141	C₂×C₃×Q₈	—	7	1	no	—
142	#239	—	7	8	no	—
143	C₂×D₁₂	—	7	2	no	—
144	C₂×D₁₂	C₂×D₇	1	1	yes	characteristic
145	C₂×C₄×D₃	—	7	2	no	—
146	C₂×C₄×D₃	—	7	1	no	—
147	C₂×C₄×D₃	—	7	2	no	—
148	C₂×C₄×D₃	—	7	1	no	—
149	C₂×C₄×D₃	—	7	1	no	—
150	C₂×C₄×D₃	C₂×D₇	1	2	yes	—
151	C₃×C₄⋊C₄	—	7	2	no	—
152	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
153	C₃×C₄²	—	7	2	no	—
154	C₃×C₄²	—	7	1	no	—
155	(C₂×D₃)⋊C₄	—	7	2	no	—
156	(C₂×D₃)⋊C₄	—	7	2	no	—
157	(C₂×D₃)⋊C₄	C₂×D₇	1	2	yes	—
158	C₄⋊Dic₃	—	7	1	no	—
159	C₄⋊Dic₃	—	7	2	no	—
160	C₄×Dic₃	—	7	2	no	—
161	C₄×Dic₃	—	7	1	no	—
162	C₄×Dic₃	—	7	1	no	—
163	C₄×Dic₃	—	7	2	no	—
164	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
165	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
168	D₂₁	—	2	2	no	—
169	D₃×C₇	—	2	4	no	—
170	C₂×#41	—	21	1	no	—
171	C₄×Q₈	—	7	1	no	—
172	C₄×Q₈	—	21	1	no	—
173	C₄×D₄	—	3	1	no	—
174	C₄×D₄	—	21	2	no	—
175	C₄×D₄	—	21	1	no	—
176	C₄×D₄	—	21	2	no	—
177	#117	—	21	2	no	—
178	#117	—	21	1	no	—
179	C₂×C₄²	—	21	1	no	—
180	C₂×C₄²	—	21	2	no	—
181	C₂²×C₇	—	3	4	no	—
182	C₂²×C₇	—	6	2	no	—
183	C₂²×C₇	—	6	2	no	—
184	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
185	C₂×D₇	—	3	2	no	—
186	C₂×D₇	—	6	1	no	—
187	C₂×D₇	—	6	1	no	—
188	C₄×C₇	—	2	2	no	—
189	C₄×C₇	—	3	2	no	—
190	C₄×C₇	—	6	1	no	—
191	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
192	Dic₇	—	2	1	no	—
193	Dic₇	—	3	2	no	—
194	Dic₇	—	6	2	no	—
195	Dic₇	C₂×C₄×D₃	1	4	yes	—
196	Dic₇	#239	1	2	yes	complemented
197	C₂²×D₃	—	7	1	no	—
198	C₂²×D₃	C₂²×D₇	1	2	yes	—
199	C₃×Q₈	—	7	4	no	—
200	C₂×C₃×C₄	—	7	2	no	—
201	C₂×C₃×C₄	—	7	1	no	—
202	C₂×C₃×C₄	—	7	1	no	—
203	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
204	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
205	C₂×Dic₃	—	7	1	no	—
206	C₂×Dic₃	—	7	2	no	—
207	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
208	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
209	D₁₂	—	7	8	no	—
210	D₁₂	C₄×D₇	1	4	yes	complemented
211	C₄×D₃	—	2	4	no	—
212	C₄×D₃	—	7	8	no	—
213	C₄×D₃	—	7	4	no	—
214	C₄×D₃	—	14	2	no	—
215	C₄×D₃	—	14	4	no	—
216	C₄×D₃	—	14	2	no	—
217	C₃×C₇	C₄×#41	1	1	yes	characteristic, complemented, Hall
218	#41	—	21	8	no	—
219	C₂×Q₈	—	7	1	no	—
220	C₂×D₄	—	3	1	no	—
221	C₂×D₄	—	21	2	no	—
222	C₂²×C₄	—	3	2	no	—
223	C₂²×C₄	—	21	1	no	—
224	C₂²×C₄	—	21	1	no	—
225	C₂²×C₄	—	21	2	no	—
226	C₂²×C₄	—	21	2	no	—
227	C₂²×C₄	—	21	1	no	—
228	C₄⋊C₄	—	7	2	no	—
229	C₄⋊C₄	—	21	2	no	—
230	C₄⋊C₄	—	21	1	no	—
231	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
232	C₂²⋊C₄	—	3	2	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	—	7	1	no	—
238	C₄²	—	21	1	no	—
239	C₄²	—	21	1	no	—
240	C₄²	—	21	2	no	—
241	C₄²	—	21	2	no	—
242	C₂×C₇	—	6	4	no	—
243	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
244	C₂×C₇	C₂×#239	1	1	yes	characteristic
245	C₂×C₇	D₃×#41	1	1	yes	characteristic
246	D₇	—	6	2	no	—
247	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
248	C₂×D₃	—	2	2	no	—
249	C₂×D₃	—	2	2	no	—
250	C₂×D₃	—	7	2	no	—
251	C₂×D₃	—	14	1	no	—
252	C₂×D₃	—	14	1	no	—
253	C₂×D₃	C₂×C₄×D₇	1	4	yes	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	7	4	no	—
256	C₃×C₄	—	7	2	no	—
257	C₃×C₄	—	14	1	no	—
258	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
259	Dic₃	—	2	1	no	—
260	Dic₃	—	7	2	no	—
261	Dic₃	—	14	2	no	—
262	Dic₃	#1159	1	2	yes	complemented
263	C₂³	—	3	2	no	—
264	C₂³	—	21	1	no	—
265	Q₈	—	7	4	no	—
266	D₄	—	3	4	no	—
267	D₄	—	21	8	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	6	4	no	—
271	C₂×C₄	—	7	1	no	—
272	C₂×C₄	—	7	2	no	—
273	C₂×C₄	—	7	1	no	—
274	C₂×C₄	—	21	8	no	—
275	C₂×C₄	—	21	1	no	—
276	C₂×C₄	—	21	4	no	—
277	C₂×C₄	—	21	2	no	—
278	C₂×C₄	—	42	2	no	—
279	C₂×C₄	—	42	2	no	—
280	C₂×C₄	—	42	4	no	—
281	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
282	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
283	C₇	C₄×#239	1	1	yes	characteristic, complemented, Sylow
284	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
285	C₂×C₃	C₂×#1159	1	1	yes	characteristic
286	C₂×C₃	D₇×#41	1	1	yes	characteristic
287	D₃	—	2	4	no	—
288	D₃	—	14	2	no	—
289	C₂²	—	3	4	no	—
290	C₂²	—	6	2	no	—
291	C₂²	—	6	2	no	—
292	C₂²	—	21	2	no	—
293	C₂²	—	42	1	no	—
294	C₂²	—	42	1	no	—
295	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	3	2	no	—
298	C₄	—	6	1	no	—
299	C₄	—	7	4	no	—
300	C₄	—	7	2	no	—
301	C₄	—	14	1	no	—
302	C₄	—	21	2	no	—
303	C₄	—	42	2	no	—
304	C₄	C₄×D₃×D₇	1	2	yes	—
305	C₃	C₄×#1159	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	6	4	no	—
307	C₂	—	42	2	no	—
308	C₂	C₂×C₄×D₃×D₇	1	1	yes	characteristic
309	C₂	D₃×#1159	1	1	yes	characteristic
310	C₂	D₇×#239	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Fitting	C₃×C₇×C₄⋊C₄	C₂²
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×#66672	C₂
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	C₄×#41
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #10621375	C₂	semidirect
3. C₇×#893	C₂	semidirect
4. C₃×#6450	C₂	semidirect
5. #10621008	C₂	semidirect
6. D₁₂×Dic₇	C₂	semidirect
7. #10620967	C₂	semidirect
8. C₄×D₂₁⋊C₄	C₂	semidirect
9. #10620962	C₂	semidirect
10. C₄×D₃×Dic₇	C₂	semidirect
11. Dic₃×Dic₁₄	C₂	semidirect
22. #66672	C₄	semidirect
43. #6450	D₃	semidirect
55. C₄×D₂₁	C₂×C₄	semidirect
58. C₇×D₁₂	C₂×C₄	semidirect
63. C₃×Dic₁₄	C₂×C₄	semidirect
64. #4247	C₂×C₄	semidirect
68. D₃×Dic₇	C₂×C₄	semidirect
97. #893	D₇	semidirect
117. C₃×Dic₇	#41	semidirect
119. C₇×Dic₃	#41	semidirect
140. Dic₁₄	C₄×D₃	semidirect
196. Dic₇	#239	semidirect
210. D₁₂	C₄×D₇	semidirect
217. C₃×C₇	C₄×#41	semidirect
262. Dic₃	#1159	semidirect
283. C₇	C₄×#239	semidirect
305. C₃	C₄×#1159	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66672	2
#10621375	2
C₇×#893	2
C₃×#6450	2
#10621008	2
D₁₂×Dic₇	2
#10620967	2
C₄×D₂₁⋊C₄	2
#10620962	2
C₄×D₃×Dic₇	2
Dic₃×Dic₁₄	2
C₄×#1159	3
C₄×#239	7