Subgroups — #13061668

Order 1344.

444 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621429	C₂	1	1	yes	maximal, characteristic
3	#10621427	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621380	C₂	1	1	yes	maximal, characteristic
5	#10621379	C₂	1	2	yes	maximal
6	#10621372	C₂	1	2	yes	maximal, complemented
7	#10621371	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621355	C₂	1	2	yes	maximal
9	C₄²×D₂₁	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₄₂	C₂	1	2	yes	maximal, complemented
11	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#90269	—	3	1	no	maximal, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₂	C₂²	1	2	yes	—
18	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₃×C₄²×C₇	C₂²	1	2	yes	—
23	#66631	C₂²	1	2	yes	—
24	#66631	C₂²	1	2	yes	—
25	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₁	C₂²	1	2	yes	—
27	#66629	C₂²	1	2	yes	—
28	#66629	C₂²	1	2	yes	—
29	#66629	C₂²	1	2	yes	—
30	#66629	C₂²	1	2	yes	—
31	#66629	C₂²	1	2	yes	—
32	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁	C₂²	1	2	yes	—
34	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
36	C₄×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
37	#6509	—	3	1	no	—
38	#6507	—	3	1	no	—
39	#6460	—	3	1	no	—
40	#6459	—	3	2	no	—
41	#6452	—	3	2	no	—
42	#6451	—	3	1	no	—
43	#6435	—	3	2	no	—
44	C₄²×D₇	—	3	1	no	—
45	C₄×Dic₁₄	—	3	2	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5578	—	7	1	no	maximal, Hall
48	C₂²×D₂₁	C₂³	1	1	yes	characteristic
49	C₃×C₇×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
56	C₂×Dic₂₁	C₂³	1	2	yes	—
57	C₂×Dic₂₁	C₂³	1	2	yes	—
58	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
59	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
60	C₄×D₂₁	—	2	2	no	—
61	C₄×D₂₁	—	2	2	no	—
62	C₄×D₂₁	Q₈	1	4	yes	complemented
63	Dic₄₂	—	2	2	no	—
64	Dic₄₂	—	2	4	no	—
65	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄×D₇	—	3	1	no	—
67	C₂×C₄×D₇	—	3	1	no	—
68	C₂×C₄×D₇	—	3	1	no	—
69	C₂×Dic₁₄	—	3	2	no	—
70	C₂×Dic₁₄	—	3	1	no	—
71	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
73	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
74	C₄²×C₇	C₂×D₃	1	2	yes	—
75	(C₂×D₇)⋊C₄	—	3	2	no	—
76	(C₂×D₇)⋊C₄	—	3	2	no	—
77	C₄⋊Dic₇	—	3	1	no	—
78	C₄⋊Dic₇	—	3	2	no	—
79	Dic₇⋊C₄	—	3	2	no	—
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	Dic₇⋊C₄	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	—	3	2	no	—
86	C₄×Dic₇	—	3	1	no	—
87	C₄×Dic₇	—	3	1	no	—
88	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, complemented
89	#946	—	7	1	no	—
90	#944	—	7	1	no	—
91	#897	—	7	1	no	—
92	#896	—	7	2	no	—
93	#889	—	7	2	no	—
94	#888	—	7	1	no	—
95	#872	—	7	2	no	—
96	C₄²×D₃	—	7	1	no	—
97	C₄×Dic₆	—	7	2	no	—
98	C₄×Dic₆	—	7	1	no	—
99	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
100	C₂×D₂₁	—	2	1	no	—
101	C₂×D₂₁	—	2	1	no	—
102	C₂×D₂₁	C₂×Q₈	1	2	yes	—
103	C₃×C₄×C₇	—	2	2	no	—
104	C₃×C₄×C₇	—	2	1	no	—
105	C₃×C₄×C₇	—	2	2	no	—
106	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
107	C₃×C₄×C₇	#41	1	2	yes	—
108	Dic₂₁	—	2	2	no	—
109	Dic₂₁	—	2	2	no	—
110	Dic₂₁	—	2	1	no	—
111	Dic₂₁	C₂×Q₈	1	2	yes	—
112	Dic₂₁	#41	1	2	yes	—
113	#533	—	21	1	no	Sylow
114	C₂²×D₇	—	3	1	no	—
115	C₇×Q₈	—	2	2	no	—
116	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
118	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
121	C₂×Dic₇	—	3	1	no	—
122	C₂×Dic₇	—	3	1	no	—
123	C₂×Dic₇	—	3	1	no	—
124	C₂×Dic₇	—	3	2	no	—
125	C₂×Dic₇	—	3	2	no	—
126	C₄×D₇	—	3	4	no	—
127	C₄×D₇	—	6	2	no	—
128	C₄×D₇	—	6	2	no	—
129	Dic₁₄	—	6	4	no	—
130	Dic₁₄	—	6	2	no	—
131	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
132	C₂×C₄×D₃	—	7	1	no	—
133	C₂×C₄×D₃	—	7	1	no	—
134	C₂×C₄×D₃	—	7	1	no	—
135	C₂×Dic₆	—	7	2	no	—
136	C₂×Dic₆	—	7	1	no	—
137	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
138	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
139	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
140	C₃×C₄²	C₂×D₇	1	2	yes	—
141	(C₂×D₃)⋊C₄	—	7	2	no	—
142	(C₂×D₃)⋊C₄	—	7	2	no	—
143	C₄⋊Dic₃	—	7	1	no	—
144	C₄⋊Dic₃	—	7	2	no	—
145	Dic₃⋊C₄	—	7	2	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	Dic₃⋊C₄	—	7	2	no	—
149	Dic₃⋊C₄	—	7	2	no	—
150	C₄×Dic₃	—	7	1	no	—
151	C₄×Dic₃	—	7	2	no	—
152	C₄×Dic₃	—	7	1	no	—
153	C₄×Dic₃	—	7	1	no	—
154	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
157	D₂₁	—	2	2	no	—
158	C₄⋊Q₈	—	21	1	no	—
159	C₄⋊Q₈	—	21	1	no	—
160	#125	—	21	2	no	—
161	C₂²⋊Q₈	—	21	1	no	—
162	C₂²⋊Q₈	—	21	2	no	—
163	C₂²⋊Q₈	—	21	1	no	—
164	C₄×Q₈	—	21	2	no	—
165	C₄×Q₈	—	21	1	no	—
166	C₄×Q₈	D₂₁	1	1	yes	characteristic, complemented
167	#117	—	21	2	no	—
168	C₂×C₄²	—	21	1	no	—
169	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
170	C₂×D₇	—	3	2	no	—
171	C₂×D₇	—	6	1	no	—
172	C₂×D₇	—	6	1	no	—
173	C₄×C₇	—	2	2	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	#235	1	2	yes	—
177	C₄×C₇	D₃×Q₈	1	2	yes	—
178	Dic₇	—	3	2	no	—
179	Dic₇	—	3	2	no	—
180	Dic₇	—	6	2	no	—
181	Dic₇	—	6	1	no	—
182	Dic₇	—	6	2	no	—
183	C₂²×D₃	—	7	1	no	—
184	C₃×Q₈	—	2	2	no	—
185	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
187	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
188	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
190	C₂×Dic₃	—	7	1	no	—
191	C₂×Dic₃	—	7	1	no	—
192	C₂×Dic₃	—	7	1	no	—
193	C₂×Dic₃	—	7	2	no	—
194	C₂×Dic₃	—	7	2	no	—
195	C₄×D₃	—	7	4	no	—
196	C₄×D₃	—	14	2	no	—
197	C₄×D₃	—	14	2	no	—
198	Dic₆	—	14	4	no	—
199	Dic₆	—	14	2	no	—
200	C₃×C₇	#533	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	21	2	no	—
202	C₂×Q₈	—	21	1	no	—
203	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
204	C₂²×C₄	—	21	1	no	—
205	C₂²×C₄	—	21	1	no	—
206	C₂²×C₄	—	21	1	no	—
207	C₄⋊C₄	—	21	2	no	—
208	C₄⋊C₄	—	21	2	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	2	no	—
211	C₄⋊C₄	—	21	2	no	—
212	C₄⋊C₄	—	21	2	no	—
213	C₄⋊C₄	—	21	1	no	—
214	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
215	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
216	C₂²⋊C₄	—	21	2	no	—
217	C₂²⋊C₄	—	21	2	no	—
218	C₄²	—	21	1	no	—
219	C₄²	—	21	1	no	—
220	C₄²	—	21	1	no	—
221	C₄²	—	21	2	no	—
222	C₄²	C₂×D₂₁	1	1	yes	characteristic
223	C₄²	C₂×D₂₁	1	2	yes	—
224	C₂×C₇	C₂×#235	1	1	yes	characteristic
225	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
226	C₂×C₇	D₃×#41	1	1	yes	characteristic
227	D₇	—	6	2	no	—
228	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
229	C₂×D₃	—	7	2	no	—
230	C₂×D₃	—	14	1	no	—
231	C₂×D₃	—	14	1	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	#1157	1	2	yes	—
236	C₃×C₄	Q₈×D₇	1	2	yes	—
237	Dic₃	—	7	2	no	—
238	Dic₃	—	7	2	no	—
239	Dic₃	—	14	2	no	—
240	Dic₃	—	14	1	no	—
241	Dic₃	—	14	2	no	—
242	C₂³	—	21	1	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	42	4	no	—
245	Q₈	—	42	2	no	—
246	C₂×C₄	—	21	1	no	—
247	C₂×C₄	—	21	1	no	—
248	C₂×C₄	—	21	4	no	—
249	C₂×C₄	—	21	1	no	—
250	C₂×C₄	—	21	2	no	—
251	C₂×C₄	—	21	2	no	—
252	C₂×C₄	—	42	2	no	—
253	C₂×C₄	—	42	2	no	—
254	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₂₁	1	2	yes	—
256	C₂×C₄	C₂²×D₂₁	1	2	yes	—
257	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
259	C₇	#5578	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	C₂×#1157	1	1	yes	characteristic
261	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
262	C₂×C₃	D₇×#41	1	1	yes	characteristic
263	D₃	—	14	2	no	—
264	C₂²	—	21	2	no	—
265	C₂²	—	42	1	no	—
266	C₂²	—	42	1	no	—
267	C₂²	C₂³×D₂₁	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	21	2	no	—
272	C₄	—	21	2	no	—
273	C₄	—	42	1	no	—
274	C₄	—	42	2	no	—
275	C₄	—	42	2	no	—
276	C₄	#66728	1	2	yes	—
277	C₄	Q₈×D₂₁	1	2	yes	—
278	C₃	#90269	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	42	2	no	—
280	C₂	C₂×#66728	1	1	yes	characteristic
281	C₂	C₂×Q₈×D₂₁	1	1	yes	characteristic
282	C₂	#41×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Frattini	C₂²	C₂³×D₂₁
Fitting	C₃×C₄×C₇×Q₈	C₂
prime radical	C₂²×D₂₁	C₂³
hypercenter	C₄×Q₈	D₂₁
hypocenter	C₃×C₇	#533
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₇×Q₈	C₂	semidirect
3. #10621427	C₂	semidirect
6. #10621372	C₂	semidirect
7. #10621371	C₂	semidirect
10. C₄×Dic₄₂	C₂	semidirect
11. C₄×Dic₄₂	C₂	semidirect
36. C₄×C₇×Q₈	D₃	semidirect
62. C₄×D₂₁	Q₈	semidirect
88. C₃×C₄×Q₈	D₇	semidirect
166. C₄×Q₈	D₂₁	semidirect
200. C₃×C₇	#533	semidirect
259. C₇	#5578	semidirect
278. C₃	#90269	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇×Q₈	2
#10621429	2
#10621427	2
#10621380	2
#10621379	2
#10621372	2
#10621371	2
#10621355	2
C₄²×D₂₁	2
C₄×Dic₄₂	2
#90269	3
#5578	7