Subgroups — #13058544

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×#66669	C₂	1	1	yes	maximal, characteristic
2	#10621370	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6456	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621003	C₂	1	2	yes	maximal, complemented
6	C₄×#4245	C₂	1	2	yes	maximal, complemented
7	#10620996	C₂	1	1	yes	maximal, characteristic, complemented
8	#10620956	C₂	1	2	yes	maximal, complemented
9	D₇×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
10	#10620933	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620895	C₂	1	1	yes	maximal, characteristic, complemented
12	#90195	—	3	1	no	maximal, Hall
13	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
15	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
18	C₂×#4245	C₂²	1	2	yes	—
19	C₂×Dic₃×D₇	C₂²	1	2	yes	—
20	C₂×Dic₃×D₇	C₂²	1	2	yes	—
21	#66669	C₄	1	8	yes	complemented
22	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	#66629	C₂²	1	2	yes	—
24	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₁	C₂²	1	2	yes	—
26	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	Dic₃⋊Dic₇	C₂²	1	1	yes	characteristic
31	Dic₃⋊Dic₇	C₂²	1	1	yes	characteristic
32	#66577	C₂²	1	2	yes	—
33	(C₂×D₃)⋊Dic₇	C₂²	1	1	yes	characteristic
34	(C₂×D₃)⋊Dic₇	C₂²	1	1	yes	characteristic
35	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
36	Dic₃×Dic₇	C₂²	1	2	yes	—
37	C₂×#1157	—	3	1	no	—
38	C₂×C₇×C₄⋊C₄	—	3	1	no	—
39	C₄×C₂²⋊D₇	—	3	2	no	—
40	C₄×C₂²⋊D₇	—	3	2	no	—
41	#6489	—	3	1	no	—
42	#6456	—	3	1	no	—
43	#6456	D₃	1	1	yes	characteristic, complemented
44	#6455	—	3	2	no	—
45	D₇×C₄⋊C₄	—	3	2	no	—
46	#6450	—	3	1	no	—
47	#6450	—	3	1	no	—
48	#5546	—	7	1	no	maximal, Hall
49	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
50	C₂²×C₃×D₇	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	1	yes	characteristic
54	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₁	C₂³	1	2	yes	—
56	Dic₄₂	C₂×C₄	1	4	yes	complemented
57	C₂×C₇×Dic₃	C₂³	1	2	yes	—
58	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
59	C₄×D₃×C₇	—	4	2	no	—
60	C₄×D₃×C₇	C₂×C₄	1	4	yes	complemented
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
63	C₃×D₂₈	C₂×C₄	1	4	yes	complemented
64	C₃×C₄×D₇	—	4	2	no	—
65	#4245	C₂×C₄	1	8	yes	complemented
66	Dic₃×D₇	—	4	2	no	—
67	Dic₃×D₇	C₂×C₄	1	8	yes	complemented
68	C₂²×C₄×C₇	—	3	1	no	—
69	C₂²×C₄×C₇	—	3	2	no	—
70	C₂×C₂²⋊D₇	—	3	2	no	—
71	#1157	—	3	8	no	—
72	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₄×D₇	—	3	2	no	—
74	C₂×C₄×D₇	—	3	2	no	—
75	C₂×C₄×D₇	C₂×D₃	1	2	yes	—
76	C₂×Dic₁₄	—	3	1	no	—
77	C₇×C₄⋊C₄	—	3	1	no	—
78	C₇×C₄⋊C₄	—	3	2	no	—
79	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
80	C₂²⋊Dic₇	—	3	1	no	—
81	C₂²⋊Dic₇	—	3	1	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	Dic₇⋊C₄	—	3	2	no	—
87	Dic₇⋊C₄	—	3	2	no	—
88	C₄×Dic₇	—	3	2	no	—
89	C₄×Dic₇	—	3	1	no	—
90	C₄×Dic₇	—	3	2	no	—
91	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
92	C₂×#237	—	7	1	no	—
93	C₃×C₄×D₄	—	7	1	no	—
94	D₄×Dic₃	—	7	1	no	—
95	C₄×#67	—	7	2	no	—
96	C₂×Dic₃⋊C₄	—	7	2	no	—
97	D₃×C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
98	#887	—	7	1	no	—
99	#67⋊C₄	—	7	2	no	—
100	#877	—	7	2	no	—
101	#872	—	7	1	no	—
102	C₄×Dic₆	—	7	1	no	—
103	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×D₃×C₇	—	2	1	no	—
105	C₂×D₃×C₇	—	2	1	no	—
106	C₂×D₃×C₇	C₂²×C₄	1	2	yes	—
107	C₂×C₃×D₇	—	2	2	no	—
108	C₂×C₃×D₇	—	2	2	no	—
109	C₂×C₃×D₇	C₂²×C₄	1	4	yes	—
110	C₃×C₄×C₇	—	2	2	no	—
111	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
112	Dic₂₁	—	2	1	no	—
113	Dic₂₁	—	2	1	no	—
114	Dic₂₁	C₂²×C₄	1	4	yes	—
115	C₃×Dic₇	—	2	1	no	—
116	C₃×Dic₇	—	2	1	no	—
117	C₇×Dic₃	—	2	2	no	—
118	C₇×Dic₃	C₂²×C₄	1	2	yes	—
119	#520	—	21	1	no	Sylow
120	C₂³×C₇	—	3	1	no	—
121	C₂²×D₇	C₂²×D₃	1	2	yes	—
122	C₂×C₄×C₇	—	3	1	no	—
123	C₂×C₄×C₇	—	3	4	no	—
124	C₂×C₄×C₇	—	3	2	no	—
125	C₂×C₄×C₇	—	12	2	no	—
126	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
127	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
128	C₂²⋊D₇	—	3	8	no	—
129	C₂×Dic₇	—	3	2	no	—
130	C₂×Dic₇	—	3	1	no	—
131	C₂×Dic₇	—	3	1	no	—
132	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
133	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
134	D₂₈	C₄×D₃	1	4	yes	complemented
135	C₄×D₇	—	3	8	no	—
136	C₄×D₇	—	4	2	no	—
137	C₄×D₇	—	12	2	no	—
138	Dic₁₄	—	3	4	no	—
139	C₂×C₃×D₄	—	7	1	no	—
140	C₂²×C₃×C₄	—	7	2	no	—
141	C₂×#67	—	7	2	no	—
142	C₂²×Dic₃	—	7	2	no	—
143	C₂²×Dic₃	—	7	2	no	—
144	#237	—	7	8	no	—
145	C₂×C₄×D₃	C₂×D₇	1	2	yes	—
146	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
147	C₂×Dic₆	—	7	1	no	—
148	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₄²	—	7	1	no	—
151	#217	—	7	2	no	—
152	(C₂×D₃)⋊C₄	—	7	1	no	—
153	(C₂×D₃)⋊C₄	—	7	1	no	—
154	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
155	Dic₃⋊C₄	—	7	2	no	—
156	Dic₃⋊C₄	—	7	2	no	—
157	Dic₃⋊C₄	—	7	1	no	—
158	Dic₃⋊C₄	—	7	1	no	—
159	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
160	C₄×Dic₃	—	7	1	no	—
161	C₄×Dic₃	—	7	2	no	—
162	C₄×Dic₃	—	7	2	no	—
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	C₃×D₇	—	2	4	no	—
167	D₃×C₇	—	2	2	no	—
168	C₂×#41	—	21	1	no	—
169	C₄×Q₈	—	21	1	no	—
170	C₄×Q₈	—	21	1	no	—
171	C₄×D₄	—	7	1	no	—
172	C₄×D₄	—	21	2	no	—
173	C₄×D₄	—	21	1	no	—
174	C₄×D₄	—	21	2	no	—
175	#117	—	21	1	no	—
176	#117	—	21	2	no	—
177	C₂×C₄⋊C₄	—	3	1	no	—
178	C₂×C₄⋊C₄	—	21	2	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	—	2	2	no	—
184	C₂×D₇	—	2	2	no	—
185	C₂×D₇	C₂×C₄×D₃	1	4	yes	—
186	C₄×C₇	—	2	2	no	—
187	C₄×C₇	—	3	2	no	—
188	C₄×C₇	—	6	2	no	—
189	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
190	Dic₇	—	2	1	no	—
191	Dic₇	—	2	1	no	—
192	Dic₇	—	3	4	no	—
193	Dic₇	—	6	1	no	—
194	Dic₇	—	6	1	no	—
195	C₂³×C₃	—	7	2	no	—
196	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
197	C₃×D₄	—	7	4	no	—
198	C₂×C₃×C₄	—	7	1	no	—
199	C₂×C₃×C₄	—	7	1	no	—
200	C₂×C₃×C₄	—	28	2	no	—
201	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
202	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
203	#67	—	7	8	no	—
204	C₂×Dic₃	—	7	2	no	—
205	C₂×Dic₃	—	7	8	no	—
206	C₂×Dic₃	—	7	1	no	—
207	C₂×Dic₃	—	7	1	no	—
208	C₂×Dic₃	—	28	2	no	—
209	C₂×Dic₃	C₂²×D₇	1	2	yes	—
210	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
211	C₄×D₃	—	4	2	no	—
212	C₄×D₃	C₄×D₇	1	4	yes	complemented
213	Dic₆	—	7	4	no	—
214	C₃×C₇	#520	1	1	yes	characteristic, complemented, Hall
215	#41	—	21	8	no	—
216	C₂×Q₈	—	21	1	no	—
217	C₂×D₄	—	7	1	no	—
218	C₂×D₄	—	21	2	no	—
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	3	2	no	—
221	C₂²×C₄	—	7	2	no	—
222	C₂²×C₄	—	21	2	no	—
223	C₂²×C₄	—	21	2	no	—
224	C₄⋊C₄	—	3	2	no	—
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	21	2	no	—
227	C₄⋊C₄	—	21	1	no	—
228	C₄⋊C₄	—	21	1	no	—
229	C₄⋊C₄	—	21	2	no	—
230	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
231	C₂²⋊C₄	—	7	2	no	—
232	C₂²⋊C₄	—	21	2	no	—
233	C₂²⋊C₄	—	21	1	no	—
234	C₂²⋊C₄	—	21	1	no	—
235	C₄²	—	7	1	no	—
236	C₄²	—	21	2	no	—
237	C₄²	—	21	2	no	—
238	C₄²	—	21	1	no	—
239	C₂×C₇	—	6	2	no	—
240	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
241	C₂×C₇	#1004	1	1	yes	characteristic
242	C₂×C₇	#1010	1	1	yes	characteristic
243	D₇	—	2	4	no	—
244	C₂²×C₃	—	7	4	no	—
245	C₂²×C₃	—	14	2	no	—
246	C₂²×C₃	—	14	2	no	—
247	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	—	2	1	no	—
250	C₂×D₃	C₂×C₄×D₇	1	2	yes	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	14	1	no	—
253	C₃×C₄	—	14	1	no	—
254	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
255	Dic₃	—	2	2	no	—
256	Dic₃	—	7	4	no	—
257	Dic₃	—	14	1	no	—
258	Dic₃	—	14	1	no	—
259	Dic₃	C₂×C₄×D₇	1	2	yes	—
260	C₂³	—	3	1	no	—
261	C₂³	—	7	2	no	—
262	Q₈	—	21	4	no	—
263	D₄	—	7	4	no	—
264	D₄	—	21	8	no	—
265	C₂×C₄	—	3	1	no	—
266	C₂×C₄	—	3	4	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	7	1	no	—
269	C₂×C₄	—	7	1	no	—
270	C₂×C₄	—	12	2	no	—
271	C₂×C₄	—	21	8	no	—
272	C₂×C₄	—	21	2	no	—
273	C₂×C₄	—	21	1	no	—
274	C₂×C₄	—	21	1	no	—
275	C₂×C₄	—	28	2	no	—
276	C₂×C₄	—	84	2	no	—
277	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
278	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
279	C₇	#5546	1	1	yes	characteristic, complemented, Sylow
280	C₂×C₃	—	14	4	no	—
281	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
282	C₂×C₃	#6548	1	1	yes	characteristic
283	C₂×C₃	#6551	1	1	yes	characteristic
284	D₃	—	2	2	no	—
285	C₂²	—	3	2	no	—
286	C₂²	—	6	1	no	—
287	C₂²	—	6	1	no	—
288	C₂²	—	7	4	no	—
289	C₂²	—	14	2	no	—
290	C₂²	—	14	2	no	—
291	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	3	2	no	—
294	C₄	—	6	2	no	—
295	C₄	—	14	1	no	—
296	C₄	—	14	1	no	—
297	C₄	—	21	4	no	—
298	C₄	—	42	1	no	—
299	C₄	—	42	1	no	—
300	C₄	C₄×D₃×D₇	1	2	yes	—
301	C₃	#90195	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	6	2	no	—
303	C₂	—	14	4	no	—
304	C₂	C₂×C₄×D₃×D₇	1	1	yes	characteristic
305	C₂	#10621572	1	1	yes	characteristic
306	C₂	#10621575	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×#66669	C₂
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	#520
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. #10621370	C₂	semidirect
3. D₃×C₇×C₄⋊C₄	C₂	semidirect
4. C₃×#6456	C₂	semidirect
5. #10621003	C₂	semidirect
6. C₄×#4245	C₂	semidirect
7. #10620996	C₂	semidirect
8. #10620956	C₂	semidirect
9. D₇×Dic₃⋊C₄	C₂	semidirect
10. #10620933	C₂	semidirect
11. #10620895	C₂	semidirect
21. #66669	C₄	semidirect
43. #6456	D₃	semidirect
56. Dic₄₂	C₂×C₄	semidirect
60. C₄×D₃×C₇	C₂×C₄	semidirect
63. C₃×D₂₈	C₂×C₄	semidirect
65. #4245	C₂×C₄	semidirect
67. Dic₃×D₇	C₂×C₄	semidirect
97. D₃×C₄⋊C₄	D₇	semidirect
134. D₂₈	C₄×D₃	semidirect
212. C₄×D₃	C₄×D₇	semidirect
214. C₃×C₇	#520	semidirect
279. C₇	#5546	semidirect
301. C₃	#90195	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66669	2
#10621370	2
D₃×C₇×C₄⋊C₄	2
C₃×#6456	2
#10621003	2
C₄×#4245	2
#10620996	2
#10620956	2
D₇×Dic₃⋊C₄	2
#10620933	2
#10620895	2
#90195	3
#5546	7