Subgroups — #13058553

Order 1344.

596 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	C₂×D₂₁⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₂₁	C₂	1	1	yes	maximal, characteristic
3	C₇×#887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6450	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620966	C₂	1	2	yes	maximal
6	C₄×D₂₁⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₄×D₂₁⋊C₄	C₂	1	2	yes	maximal
8	Dic₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
9	Dic₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620886	C₂	1	2	yes	maximal, complemented
11	#10620885	C₂	1	2	yes	maximal, complemented
12	C₄×Q₈×D₇	—	3	1	no	maximal, Hall
13	C₂×C₄×D₂₁	C₂²	1	2	yes	—
14	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
15	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
17	C₂×#4248	C₂²	1	2	yes	—
18	C₂×D₂₁⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×D₂₁⋊C₄	C₂²	1	2	yes	—
20	C₂×D₂₁⋊C₄	C₂²	1	1	yes	characteristic
21	D₂₁⋊Q₈	C₄	1	8	yes	complemented
22	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
24	#66629	C₂²	1	2	yes	—
25	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₄×C₇×Dic₃	C₂²	1	2	yes	—
27	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
29	C₃×C₄×Dic₇	C₂²	1	2	yes	—
30	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
31	Dic₃⋊Dic₇	C₂²	1	2	yes	—
32	Dic₇⋊Dic₃	C₂²	1	2	yes	—
33	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
34	Dic₃×Dic₇	C₂²	1	2	yes	—
35	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
36	Dic₃×Dic₇	C₂²	1	2	yes	—
37	C₂×Q₈×D₇	—	3	1	no	—
38	C₄×C₇×Q₈	—	3	1	no	—
39	Q₈×Dic₇	—	3	1	no	—
40	D₇×C₄⋊C₄	—	3	1	no	—
41	D₇×C₄⋊C₄	—	3	2	no	—
42	#6450	—	3	2	no	—
43	#6450	D₃	1	1	yes	characteristic, complemented
44	C₄²×D₇	—	3	1	no	—
45	C₄²×D₇	—	3	2	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	C₄×Dic₁₄	—	3	2	no	—
48	C₄×D₃×Q₈	—	7	1	no	maximal, Hall
49	C₂²×D₂₁	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	2	yes	—
53	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
54	C₄×D₂₁	—	2	4	no	—
55	C₄×D₂₁	C₂×C₄	1	4	yes	—
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₇×Dic₃	C₂³	1	2	yes	—
58	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
59	C₇×Dic₆	C₂×C₄	1	4	yes	complemented
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	#4248	C₂×C₄	1	8	yes	complemented
65	D₂₁⋊C₄	—	2	2	no	—
66	D₂₁⋊C₄	C₂×C₄	1	8	yes	—
67	D₂₁⋊C₄	Q₈	1	4	yes	complemented
68	C₂×C₇×Q₈	—	3	1	no	—
69	Q₈×D₇	—	3	8	no	—
70	C₂×C₄×D₇	—	3	1	no	—
71	C₂×C₄×D₇	—	3	1	no	—
72	C₂×C₄×D₇	—	3	2	no	—
73	C₂×C₄×D₇	—	3	2	no	—
74	C₂×C₄×D₇	—	3	1	no	—
75	C₂×Dic₁₄	—	3	2	no	—
76	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
77	C₇×C₄⋊C₄	—	3	2	no	—
78	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
79	C₄²×C₇	—	3	2	no	—
80	C₄²×C₇	—	3	1	no	—
81	C₄⋊Dic₇	—	3	1	no	—
82	C₄⋊Dic₇	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	Dic₇⋊C₄	—	3	2	no	—
85	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
86	C₄×Dic₇	—	3	1	no	—
87	C₄×Dic₇	—	3	2	no	—
88	C₄×Dic₇	—	3	1	no	—
89	C₄×Dic₇	—	3	2	no	—
90	C₄×Dic₇	C₂×D₃	1	2	yes	—
91	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
92	C₂×D₃×Q₈	—	7	1	no	—
93	C₃×C₄×Q₈	—	7	1	no	—
94	Q₈×Dic₃	—	7	1	no	—
95	D₃×C₄⋊C₄	—	7	1	no	—
96	D₃×C₄⋊C₄	—	7	2	no	—
97	#887	—	7	2	no	—
98	#887	D₇	1	1	yes	characteristic, complemented
99	C₄²×D₃	—	7	1	no	—
100	C₄²×D₃	—	7	2	no	—
101	C₄×Dic₆	—	7	1	no	—
102	C₄×Dic₆	—	7	2	no	—
103	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×D₂₁	C₂²×C₄	1	2	yes	—
105	C₂×D₂₁	C₂×Q₈	1	2	yes	—
106	C₂×D₂₁	#41	1	2	yes	—
107	C₃×C₄×C₇	—	2	2	no	—
108	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
109	Dic₂₁	—	2	2	no	—
110	Dic₂₁	C₂²×C₄	1	2	yes	—
111	C₃×Dic₇	—	2	1	no	—
112	C₃×Dic₇	C₂²×C₄	1	4	yes	—
113	C₃×Dic₇	C₂×Q₈	1	2	yes	complemented
114	C₇×Dic₃	—	2	1	no	—
115	C₇×Dic₃	C₂²×C₄	1	4	yes	—
116	C₇×Dic₃	C₂×Q₈	1	2	yes	complemented
117	C₂×C₄×Q₈	—	21	1	no	Sylow
118	C₂²×D₇	—	3	1	no	—
119	C₇×Q₈	—	3	4	no	—
120	C₂×C₄×C₇	—	3	1	no	—
121	C₂×C₄×C₇	—	3	2	no	—
122	C₂×C₄×C₇	—	3	1	no	—
123	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
124	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂×Dic₇	—	3	1	no	—
126	C₂×Dic₇	—	3	2	no	—
127	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
128	C₂×Dic₇	C₂²×D₃	1	2	yes	—
129	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
130	C₄×D₇	—	3	4	no	—
131	C₄×D₇	—	3	8	no	—
132	C₄×D₇	—	3	4	no	—
133	C₄×D₇	—	6	2	no	—
134	C₄×D₇	—	6	4	no	—
135	Dic₁₄	—	3	8	no	—
136	Dic₁₄	C₄×D₃	1	4	yes	complemented
137	C₂×C₃×Q₈	—	7	1	no	—
138	D₃×Q₈	—	7	8	no	—
139	C₂×C₄×D₃	—	7	1	no	—
140	C₂×C₄×D₃	—	7	1	no	—
141	C₂×C₄×D₃	—	7	2	no	—
142	C₂×C₄×D₃	—	7	2	no	—
143	C₂×C₄×D₃	—	7	1	no	—
144	C₂×Dic₆	—	7	2	no	—
145	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	7	2	no	—
147	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
148	C₃×C₄²	—	7	2	no	—
149	C₃×C₄²	—	7	1	no	—
150	C₄⋊Dic₃	—	7	1	no	—
151	C₄⋊Dic₃	—	7	2	no	—
152	Dic₃⋊C₄	—	7	2	no	—
153	Dic₃⋊C₄	—	7	2	no	—
154	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
155	C₄×Dic₃	—	7	1	no	—
156	C₄×Dic₃	—	7	2	no	—
157	C₄×Dic₃	—	7	1	no	—
158	C₄×Dic₃	—	7	2	no	—
159	C₄×Dic₃	C₂×D₇	1	2	yes	—
160	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
161	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
164	D₂₁	C₄×Q₈	1	4	yes	complemented
165	C₂²×Q₈	—	21	1	no	—
166	C₄×Q₈	—	3	1	no	—
167	C₄×Q₈	—	7	1	no	—
168	C₄×Q₈	—	21	2	no	—
169	C₄×Q₈	—	21	1	no	—
170	C₄×Q₈	—	21	2	no	—
171	C₄×Q₈	—	21	1	no	—
172	C₂×C₄⋊C₄	—	21	2	no	—
173	C₂×C₄⋊C₄	—	21	1	no	—
174	C₂×C₄²	—	21	2	no	—
175	C₂×C₄²	—	21	1	no	—
176	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
177	C₂×D₇	—	3	2	no	—
178	C₂×D₇	—	3	2	no	—
179	C₂×D₇	—	3	2	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	3	4	no	—
182	C₄×C₇	—	3	2	no	—
183	C₄×C₇	—	6	1	no	—
184	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	3	2	no	—
187	Dic₇	—	6	2	no	—
188	Dic₇	C₂×C₄×D₃	1	4	yes	—
189	Dic₇	D₃×Q₈	1	2	yes	complemented
190	C₂²×D₃	—	7	1	no	—
191	C₃×Q₈	—	7	4	no	—
192	C₂×C₃×C₄	—	7	1	no	—
193	C₂×C₃×C₄	—	7	2	no	—
194	C₂×C₃×C₄	—	7	1	no	—
195	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
196	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
197	C₂×Dic₃	—	7	1	no	—
198	C₂×Dic₃	—	7	2	no	—
199	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
200	C₂×Dic₃	C₂²×D₇	1	2	yes	—
201	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
202	C₄×D₃	—	7	4	no	—
203	C₄×D₃	—	7	8	no	—
204	C₄×D₃	—	7	4	no	—
205	C₄×D₃	—	14	2	no	—
206	C₄×D₃	—	14	4	no	—
207	Dic₆	—	7	8	no	—
208	Dic₆	C₄×D₇	1	4	yes	complemented
209	C₃×C₇	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Hall
210	C₂×Q₈	—	3	1	no	—
211	C₂×Q₈	—	7	1	no	—
212	C₂×Q₈	—	21	2	no	—
213	C₂×Q₈	—	21	8	no	—
214	C₂²×C₄	—	21	1	no	—
215	C₂²×C₄	—	21	2	no	—
216	C₂²×C₄	—	21	1	no	—
217	C₂²×C₄	—	21	1	no	—
218	C₂²×C₄	—	21	2	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	—	7	2	no	—
221	C₄⋊C₄	—	21	2	no	—
222	C₄⋊C₄	—	21	2	no	—
223	C₄⋊C₄	—	21	2	no	—
224	C₄⋊C₄	—	21	1	no	—
225	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
226	C₄²	—	3	2	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	7	2	no	—
229	C₄²	—	7	1	no	—
230	C₄²	—	21	2	no	—
231	C₄²	—	21	1	no	—
232	C₄²	—	21	2	no	—
233	C₄²	—	21	1	no	—
234	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
235	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
236	C₂×C₇	D₃×#41	1	1	yes	characteristic
237	D₇	—	3	4	no	—
238	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
239	C₂×D₃	—	7	2	no	—
240	C₂×D₃	—	7	2	no	—
241	C₂×D₃	—	7	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	7	4	no	—
244	C₃×C₄	—	7	2	no	—
245	C₃×C₄	—	14	1	no	—
246	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
247	Dic₃	—	2	1	no	—
248	Dic₃	—	7	2	no	—
249	Dic₃	—	14	2	no	—
250	Dic₃	C₂×C₄×D₇	1	4	yes	—
251	Dic₃	Q₈×D₇	1	2	yes	complemented
252	C₂³	—	21	1	no	—
253	Q₈	—	3	4	no	—
254	Q₈	—	7	4	no	—
255	Q₈	—	21	8	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	7	1	no	—
260	C₂×C₄	—	7	2	no	—
261	C₂×C₄	—	7	1	no	—
262	C₂×C₄	—	21	8	no	—
263	C₂×C₄	—	21	4	no	—
264	C₂×C₄	—	21	1	no	—
265	C₂×C₄	—	21	4	no	—
266	C₂×C₄	—	21	2	no	—
267	C₂×C₄	—	42	4	no	—
268	C₂×C₄	—	42	2	no	—
269	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
270	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
271	C₇	C₄×D₃×Q₈	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
273	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
274	C₂×C₃	D₇×#41	1	1	yes	characteristic
275	D₃	—	7	4	no	—
276	C₂²	—	21	2	no	—
277	C₂²	—	21	2	no	—
278	C₂²	—	21	2	no	—
279	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	3	4	no	—
282	C₄	—	3	2	no	—
283	C₄	—	6	1	no	—
284	C₄	—	7	4	no	—
285	C₄	—	7	2	no	—
286	C₄	—	14	1	no	—
287	C₄	—	21	2	no	—
288	C₄	—	42	2	no	—
289	C₄	C₄×D₃×D₇	1	2	yes	—
290	C₃	C₄×Q₈×D₇	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	21	4	no	—
292	C₂	C₂×C₄×D₃×D₇	1	1	yes	characteristic
293	C₂	#10621571	1	1	yes	characteristic
294	C₂	D₃×Q₈×D₇	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	C₂²×D₂₁	C₂³
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×C₄×Q₈
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
3. C₇×#887	C₂	semidirect
4. C₃×#6450	C₂	semidirect
8. Dic₆×Dic₇	C₂	semidirect
9. Dic₃×Dic₁₄	C₂	semidirect
10. #10620886	C₂	semidirect
11. #10620885	C₂	semidirect
21. D₂₁⋊Q₈	C₄	semidirect
43. #6450	D₃	semidirect
59. C₇×Dic₆	C₂×C₄	semidirect
63. C₃×Dic₁₄	C₂×C₄	semidirect
64. #4248	C₂×C₄	semidirect
67. D₂₁⋊C₄	Q₈	semidirect
98. #887	D₇	semidirect
113. C₃×Dic₇	C₂×Q₈	semidirect
116. C₇×Dic₃	C₂×Q₈	semidirect
136. Dic₁₄	C₄×D₃	semidirect
164. D₂₁	C₄×Q₈	semidirect
189. Dic₇	D₃×Q₈	semidirect
208. Dic₆	C₄×D₇	semidirect
209. C₃×C₇	C₂×C₄×Q₈	semidirect
251. Dic₃	Q₈×D₇	semidirect
271. C₇	C₄×D₃×Q₈	semidirect
290. C₃	C₄×Q₈×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₁⋊Q₈	2
C₄⋊C₄×D₂₁	2
C₇×#887	2
C₃×#6450	2
#10620966	2
C₄×D₂₁⋊C₄	2
Dic₆×Dic₇	2
Dic₃×Dic₁₄	2
#10620886	2
#10620885	2
C₄×Q₈×D₇	3
C₄×D₃×Q₈	7