Subgroups — #13061639

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×#66730	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#4268	C₂	1	2	yes	maximal, complemented
5	C₂×#66629	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621370	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621364	C₂	1	2	yes	maximal, complemented
9	#10621360	C₂	1	2	yes	maximal, complemented
10	#10621355	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#90237	—	3	1	no	maximal, Hall
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
15	C₂×#4268	C₂²	1	2	yes	—
16	C₂²×Dic₂₁	C₂²	1	2	yes	—
17	C₂²×Dic₂₁	C₂²	1	2	yes	—
18	#66730	C₄	1	8	yes	complemented
19	C₂×C₄×D₂₁	C₂²	1	2	yes	—
20	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
22	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
25	#66636	C₂²	1	2	yes	—
26	#66631	C₂²	1	1	yes	characteristic
27	#66631	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
29	#66629	C₂²	1	2	yes	—
30	#66629	C₂²	1	1	yes	characteristic
31	#66629	C₂²	1	1	yes	characteristic
32	#66629	C₂²	1	2	yes	—
33	#66629	C₂²	1	2	yes	—
34	C₄×Dic₂₁	C₂²	1	2	yes	—
35	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₁	C₂²	1	2	yes	—
37	C₂×#1159	—	3	1	no	—
38	C₄×C₇×D₄	D₃	1	1	yes	characteristic, complemented
39	D₄×Dic₇	—	3	1	no	—
40	C₄×C₂²⋊D₇	—	3	2	no	—
41	C₂×Dic₇⋊C₄	—	3	2	no	—
42	D₇×C₄⋊C₄	—	3	1	no	—
43	#6450	—	3	1	no	—
44	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
45	#6440	—	3	2	no	—
46	#6435	—	3	1	no	—
47	C₄×Dic₁₄	—	3	1	no	—
48	#5546	—	7	1	no	maximal, Hall
49	C₂³×C₃×C₇	C₂³	1	2	yes	—
50	C₂²×D₂₁	C₂³	1	1	yes	characteristic
51	C₃×C₇×D₄	C₂×C₄	1	4	yes	complemented
52	C₂×C₃×C₄×C₇	—	4	2	no	—
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
57	#4268	C₂×C₄	1	8	yes	complemented
58	C₂×Dic₂₁	—	4	2	no	—
59	C₂×Dic₂₁	C₂×C₄	1	8	yes	complemented
60	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₁	C₂³	1	2	yes	—
63	C₂×Dic₂₁	C₂³	1	2	yes	—
64	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
65	C₄×D₂₁	—	4	2	no	—
66	C₄×D₂₁	C₂×C₄	1	4	yes	complemented
67	Dic₄₂	C₂×C₄	1	4	yes	complemented
68	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
70	C₂×C₂²⋊D₇	—	3	2	no	—
71	C₂²×Dic₇	—	3	2	no	—
72	C₂²×Dic₇	—	3	2	no	—
73	#1159	—	3	8	no	—
74	C₂×C₄×D₇	—	3	1	no	—
75	C₂×C₄×D₇	—	3	2	no	—
76	C₂×Dic₁₄	—	3	1	no	—
77	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
78	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
79	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
80	C₂²⋊Dic₇	—	3	2	no	—
81	(C₂×D₇)⋊C₄	—	3	1	no	—
82	(C₂×D₇)⋊C₄	—	3	1	no	—
83	C₄⋊Dic₇	—	3	1	no	—
84	Dic₇⋊C₄	—	3	2	no	—
85	Dic₇⋊C₄	—	3	1	no	—
86	Dic₇⋊C₄	—	3	1	no	—
87	Dic₇⋊C₄	—	3	2	no	—
88	Dic₇⋊C₄	—	3	2	no	—
89	C₄×Dic₇	—	3	2	no	—
90	C₄×Dic₇	—	3	1	no	—
91	C₄×Dic₇	—	3	2	no	—
92	C₂×#237	—	7	1	no	—
93	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
94	D₄×Dic₃	—	7	1	no	—
95	C₄×#67	—	7	2	no	—
96	C₂×Dic₃⋊C₄	—	7	2	no	—
97	D₃×C₄⋊C₄	—	7	1	no	—
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	—	2	2	no	—
104	C₂²×C₃×C₇	—	2	2	no	—
105	C₂²×C₃×C₇	C₂²×C₄	1	4	yes	—
106	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
107	C₂×D₂₁	—	2	1	no	—
108	C₂×D₂₁	—	2	1	no	—
109	C₂×D₂₁	C₂²×C₄	1	2	yes	—
110	C₃×C₄×C₇	—	2	2	no	—
111	C₃×C₄×C₇	—	2	1	no	—
112	C₃×C₄×C₇	—	2	1	no	—
113	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
114	Dic₂₁	—	2	2	no	—
115	Dic₂₁	—	2	1	no	—
116	Dic₂₁	—	2	1	no	—
117	Dic₂₁	C₂²×C₄	1	4	yes	—
118	Dic₂₁	C₂²×C₄	1	2	yes	—
119	#520	—	21	1	no	Sylow
120	C₂³×C₇	C₂²×D₃	1	2	yes	—
121	C₂²×D₇	—	3	1	no	—
122	C₇×D₄	C₄×D₃	1	4	yes	complemented
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
126	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
128	C₂²⋊D₇	—	3	8	no	—
129	C₂×Dic₇	—	3	8	no	—
130	C₂×Dic₇	—	3	2	no	—
131	C₂×Dic₇	—	3	1	no	—
132	C₂×Dic₇	—	3	1	no	—
133	C₂×Dic₇	—	3	1	no	—
134	C₂×Dic₇	—	3	2	no	—
135	C₂×Dic₇	—	12	2	no	—
136	C₄×D₇	—	3	4	no	—
137	C₄×D₇	—	12	2	no	—
138	Dic₁₄	—	3	4	no	—
139	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
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	—	7	1	no	—
146	C₂×C₄×D₃	—	7	2	no	—
147	C₂×Dic₆	—	7	1	no	—
148	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
149	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
150	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
151	#217	—	7	2	no	—
152	(C₂×D₃)⋊C₄	—	7	1	no	—
153	(C₂×D₃)⋊C₄	—	7	1	no	—
154	C₄⋊Dic₃	—	7	1	no	—
155	Dic₃⋊C₄	—	7	2	no	—
156	Dic₃⋊C₄	—	7	1	no	—
157	Dic₃⋊C₄	—	7	1	no	—
158	Dic₃⋊C₄	—	7	2	no	—
159	Dic₃⋊C₄	—	7	2	no	—
160	C₄×Dic₃	—	7	2	no	—
161	C₄×Dic₃	—	7	1	no	—
162	C₄×Dic₃	—	7	2	no	—
163	C₂×C₃×C₇	—	2	4	no	—
164	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
167	D₂₁	—	2	2	no	—
168	C₂×#41	—	21	1	no	—
169	C₄×Q₈	—	21	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	D₂₁	1	1	yes	characteristic, complemented
175	#117	—	21	2	no	—
176	#117	—	21	1	no	—
177	C₂×C₄⋊C₄	—	21	2	no	—
178	C₂×C₄⋊C₄	—	21	1	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	C₂×C₄×D₃	1	4	yes	—
182	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
183	C₂×D₇	—	3	2	no	—
184	C₂×D₇	—	6	1	no	—
185	C₂×D₇	—	6	1	no	—
186	C₄×C₇	—	2	2	no	—
187	C₄×C₇	—	2	1	no	—
188	C₄×C₇	—	2	1	no	—
189	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
190	Dic₇	—	3	4	no	—
191	Dic₇	—	3	2	no	—
192	Dic₇	—	6	2	no	—
193	Dic₇	—	6	1	no	—
194	Dic₇	—	6	1	no	—
195	C₂³×C₃	C₂²×D₇	1	2	yes	—
196	C₂²×D₃	—	7	1	no	—
197	C₃×D₄	C₄×D₇	1	4	yes	complemented
198	C₂×C₃×C₄	—	4	2	no	—
199	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
201	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
203	#67	—	7	8	no	—
204	C₂×Dic₃	—	7	8	no	—
205	C₂×Dic₃	—	7	2	no	—
206	C₂×Dic₃	—	7	1	no	—
207	C₂×Dic₃	—	7	1	no	—
208	C₂×Dic₃	—	7	1	no	—
209	C₂×Dic₃	—	7	2	no	—
210	C₂×Dic₃	—	28	2	no	—
211	C₄×D₃	—	7	4	no	—
212	C₄×D₃	—	28	2	no	—
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	—	21	2	no	—
218	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
219	C₂²×C₄	—	21	2	no	—
220	C₂²×C₄	—	21	1	no	—
221	C₂²×C₄	—	21	2	no	—
222	C₂²×C₄	—	21	2	no	—
223	C₂²×C₄	C₂×D₂₁	1	2	yes	—
224	C₄⋊C₄	—	21	2	no	—
225	C₄⋊C₄	—	21	1	no	—
226	C₄⋊C₄	—	21	2	no	—
227	C₄⋊C₄	—	21	2	no	—
228	C₄⋊C₄	—	21	1	no	—
229	C₄⋊C₄	—	21	1	no	—
230	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
231	C₂²⋊C₄	—	21	2	no	—
232	C₂²⋊C₄	—	21	1	no	—
233	C₂²⋊C₄	—	21	1	no	—
234	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
235	C₄²	—	21	2	no	—
236	C₄²	—	21	2	no	—
237	C₄²	—	21	1	no	—
238	C₄²	C₂×D₂₁	1	1	yes	characteristic
239	C₂×C₇	—	2	4	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	—	6	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	C₂×C₄×D₇	1	4	yes	—
247	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
248	C₂×D₃	—	7	2	no	—
249	C₂×D₃	—	14	1	no	—
250	C₂×D₃	—	14	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
255	Dic₃	—	7	4	no	—
256	Dic₃	—	7	2	no	—
257	Dic₃	—	14	2	no	—
258	Dic₃	—	14	1	no	—
259	Dic₃	—	14	1	no	—
260	C₂³	—	21	1	no	—
261	C₂³	C₂²×D₂₁	1	2	yes	—
262	Q₈	—	21	4	no	—
263	D₄	—	21	8	no	—
264	D₄	C₄×D₂₁	1	4	yes	complemented
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	21	8	no	—
267	C₂×C₄	—	21	2	no	—
268	C₂×C₄	—	21	1	no	—
269	C₂×C₄	—	21	4	no	—
270	C₂×C₄	—	21	1	no	—
271	C₂×C₄	—	21	2	no	—
272	C₂×C₄	—	21	1	no	—
273	C₂×C₄	—	84	2	no	—
274	C₂×C₄	—	84	2	no	—
275	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₂₁	1	2	yes	—
277	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
279	C₇	#5546	1	1	yes	characteristic, complemented, Sylow
280	C₂×C₃	—	2	4	no	—
281	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
282	C₂×C₃	#6548	1	1	yes	characteristic
283	C₂×C₃	#6554	1	1	yes	characteristic
284	D₃	—	14	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	21	2	no	—
288	C₂²	—	42	1	no	—
289	C₂²	—	42	1	no	—
290	C₂²	C₂×C₄×D₂₁	1	4	yes	—
291	C₂²	C₂³×D₂₁	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	21	2	no	—
296	C₄	—	21	4	no	—
297	C₄	—	42	2	no	—
298	C₄	—	42	1	no	—
299	C₄	—	42	1	no	—
300	C₄	C₂×C₄×D₂₁	1	2	yes	—
301	C₃	#90237	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	2	4	no	—
303	C₂	—	42	2	no	—
304	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
305	C₂	#10621657	1	1	yes	characteristic
306	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator	C₂×C₃×C₇	C₂³×C₄
Fitting	C₃×C₄×C₇×D₄	C₂
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×#66730	C₂
hypercenter	C₄×D₄	D₂₁
hypocenter	C₃×C₇	#520
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×D₄	C₂	semidirect
3. D₄×Dic₂₁	C₂	semidirect
4. C₄×#4268	C₂	semidirect
5. C₂×#66629	C₂	semidirect
6. C₄⋊C₄×D₂₁	C₂	semidirect
7. #10621370	C₂	semidirect
8. #10621364	C₂	semidirect
9. #10621360	C₂	semidirect
10. #10621355	C₂	semidirect
11. C₄×Dic₄₂	C₂	semidirect
18. #66730	C₄	semidirect
38. C₄×C₇×D₄	D₃	semidirect
51. C₃×C₇×D₄	C₂×C₄	semidirect
57. #4268	C₂×C₄	semidirect
59. C₂×Dic₂₁	C₂×C₄	semidirect
66. C₄×D₂₁	C₂×C₄	semidirect
67. Dic₄₂	C₂×C₄	semidirect
93. C₃×C₄×D₄	D₇	semidirect
122. C₇×D₄	C₄×D₃	semidirect
174. C₄×D₄	D₂₁	semidirect
197. C₃×D₄	C₄×D₇	semidirect
214. C₃×C₇	#520	semidirect
264. D₄	C₄×D₂₁	semidirect
279. C₇	#5546	semidirect
301. C₃	#90237	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66730	2
C₃×C₄×C₇×D₄	2
D₄×Dic₂₁	2
C₄×#4268	2
C₂×#66629	2
C₄⋊C₄×D₂₁	2
#10621370	2
#10621364	2
#10621360	2
#10621355	2
C₄×Dic₄₂	2
#90237	3
#5546	7