Subgroups — #13061658

Order 1344.

424 conjugacy classes of subgroups.

The subgroup lattice has 280 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	#10621430	C₂	1	1	yes	maximal, characteristic
3	#10621427	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621381	C₂	1	2	yes	maximal, complemented
5	#10621379	C₂	1	2	yes	maximal, complemented
6	#10621376	C₂	1	1	yes	maximal, characteristic
7	#10621370	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621359	C₂	1	2	yes	maximal, complemented
9	#10621358	C₂	1	1	yes	maximal, characteristic
10	#10621355	C₂	1	2	yes	maximal, complemented
11	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#90259	—	3	1	no	maximal, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂×D₈₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁	C₂²	1	2	yes	—
16	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₇	C₂²	1	2	yes	—
22	#66631	C₂²	1	2	yes	—
23	#66631	C₂²	1	2	yes	—
24	#66631	C₂²	1	2	yes	—
25	#66631	C₂²	1	2	yes	—
26	#66631	C₂²	1	2	yes	—
27	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
28	#66629	C₂²	1	1	yes	characteristic
29	#66629	C₂²	1	1	yes	characteristic
30	#66629	C₂²	1	2	yes	—
31	#66629	C₂²	1	2	yes	—
32	#66629	C₂²	1	2	yes	—
33	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁	C₂²	1	2	yes	—
36	C₄×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
37	#6510	—	3	1	no	—
38	#6507	—	3	1	no	—
39	#6461	—	3	2	no	—
40	#6459	—	3	2	no	—
41	#6456	—	3	1	no	—
42	#6450	—	3	1	no	—
43	#6439	—	3	2	no	—
44	#6438	—	3	1	no	—
45	#6435	—	3	2	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5568	—	7	1	no	maximal, Hall
48	C₂²×D₂₁	C₂³	1	2	yes	—
49	C₃×C₇×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
57	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
58	C₂×Dic₂₁	C₂³	1	2	yes	—
59	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
60	D₈₄	—	2	2	no	—
61	C₄×D₂₁	—	2	4	no	—
62	Dic₄₂	—	2	2	no	—
63	Dic₄₂	—	2	2	no	—
64	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
65	C₂×D₂₈	—	3	1	no	—
66	C₂×C₄×D₇	—	3	2	no	—
67	C₂×Dic₁₄	—	3	1	no	—
68	C₂×Dic₁₄	—	3	1	no	—
69	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
70	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
71	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
72	C₄²×C₇	C₂×D₃	1	2	yes	—
73	(C₂×D₇)⋊C₄	—	3	2	no	—
74	(C₂×D₇)⋊C₄	—	3	2	no	—
75	(C₂×D₇)⋊C₄	—	3	2	no	—
76	(C₂×D₇)⋊C₄	—	3	2	no	—
77	(C₂×D₇)⋊C₄	—	3	2	no	—
78	C₄⋊Dic₇	—	3	1	no	—
79	Dic₇⋊C₄	—	3	1	no	—
80	Dic₇⋊C₄	—	3	1	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	1	no	—
86	C₄×Dic₇	—	3	2	no	—
87	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, complemented
88	#947	—	7	1	no	—
89	#944	—	7	1	no	—
90	#898	—	7	2	no	—
91	#896	—	7	2	no	—
92	#893	—	7	1	no	—
93	#887	—	7	1	no	—
94	#876	—	7	2	no	—
95	#875	—	7	1	no	—
96	#872	—	7	2	no	—
97	C₄×Dic₆	—	7	1	no	—
98	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
99	C₂×D₂₁	—	2	2	no	—
100	C₂×D₂₁	—	2	2	no	—
101	C₂×D₂₁	—	2	2	no	—
102	C₃×C₄×C₇	—	2	1	no	—
103	C₃×C₄×C₇	—	2	2	no	—
104	C₃×C₄×C₇	—	2	2	no	—
105	C₃×C₄×C₇	—	2	1	no	—
106	C₃×C₄×C₇	#41	1	2	yes	—
107	Dic₂₁	—	2	2	no	—
108	Dic₂₁	—	2	1	no	—
109	Dic₂₁	—	2	1	no	—
110	Dic₂₁	—	2	1	no	—
111	Dic₂₁	#41	1	2	yes	—
112	#556	—	21	1	no	Sylow
113	C₂²×D₇	—	3	2	no	—
114	C₇×Q₈	—	2	2	no	—
115	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
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	1	yes	characteristic
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×Dic₇	—	3	1	no	—
121	C₂×Dic₇	—	3	1	no	—
122	C₂×Dic₇	—	3	1	no	—
123	C₂×Dic₇	—	3	2	no	—
124	C₂×Dic₇	—	3	1	no	—
125	D₂₈	—	6	2	no	—
126	C₄×D₇	—	6	4	no	—
127	Dic₁₄	—	6	2	no	—
128	Dic₁₄	—	6	2	no	—
129	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
130	C₂×D₁₂	—	7	1	no	—
131	C₂×C₄×D₃	—	7	2	no	—
132	C₂×Dic₆	—	7	1	no	—
133	C₂×Dic₆	—	7	1	no	—
134	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
135	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
136	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
137	C₃×C₄²	C₂×D₇	1	2	yes	—
138	(C₂×D₃)⋊C₄	—	7	2	no	—
139	(C₂×D₃)⋊C₄	—	7	2	no	—
140	(C₂×D₃)⋊C₄	—	7	2	no	—
141	(C₂×D₃)⋊C₄	—	7	2	no	—
142	(C₂×D₃)⋊C₄	—	7	2	no	—
143	C₄⋊Dic₃	—	7	1	no	—
144	Dic₃⋊C₄	—	7	1	no	—
145	Dic₃⋊C₄	—	7	1	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	Dic₃⋊C₄	—	7	2	no	—
149	C₄×Dic₃	—	7	1	no	—
150	C₄×Dic₃	—	7	1	no	—
151	C₄×Dic₃	—	7	2	no	—
152	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
155	D₂₁	—	4	2	no	—
156	C₄⋊Q₈	—	21	1	no	—
157	#126	—	21	2	no	—
158	#126	—	21	2	no	—
159	#124	—	21	1	no	—
160	#124	—	21	1	no	—
161	C₂²⋊Q₈	—	21	2	no	—
162	C₄×Q₈	—	21	1	no	—
163	C₄×Q₈	—	21	1	no	—
164	C₄×Q₈	D₂₁	1	1	yes	characteristic, complemented
165	C₄×D₄	—	21	1	no	—
166	#117	—	21	2	no	—
167	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
168	C₂×D₇	—	6	2	no	—
169	C₂×D₇	—	6	2	no	—
170	C₂×D₇	—	6	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	#235	1	2	yes	—
176	Dic₇	—	3	2	no	—
177	Dic₇	—	6	2	no	—
178	Dic₇	—	6	1	no	—
179	Dic₇	—	6	1	no	—
180	Dic₇	—	6	1	no	—
181	C₂²×D₃	—	7	2	no	—
182	C₃×Q₈	—	2	2	no	—
183	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
184	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
188	C₂×Dic₃	—	7	1	no	—
189	C₂×Dic₃	—	7	1	no	—
190	C₂×Dic₃	—	7	1	no	—
191	C₂×Dic₃	—	7	2	no	—
192	C₂×Dic₃	—	7	1	no	—
193	D₁₂	—	14	2	no	—
194	C₄×D₃	—	14	4	no	—
195	Dic₆	—	14	2	no	—
196	Dic₆	—	14	2	no	—
197	C₃×C₇	#556	1	1	yes	characteristic, complemented, Hall
198	C₂×Q₈	—	21	1	no	—
199	C₂×Q₈	—	21	1	no	—
200	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
201	C₂×D₄	—	21	1	no	—
202	C₂²×C₄	—	21	2	no	—
203	C₄⋊C₄	—	21	1	no	—
204	C₄⋊C₄	—	21	2	no	—
205	C₄⋊C₄	—	21	2	no	—
206	C₄⋊C₄	—	21	1	no	—
207	C₄⋊C₄	—	21	2	no	—
208	C₄⋊C₄	—	21	1	no	—
209	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
211	C₂²⋊C₄	—	21	2	no	—
212	C₂²⋊C₄	—	21	2	no	—
213	C₂²⋊C₄	—	21	2	no	—
214	C₂²⋊C₄	—	21	2	no	—
215	C₂²⋊C₄	—	21	2	no	—
216	C₄²	—	21	2	no	—
217	C₄²	—	21	1	no	—
218	C₄²	—	21	1	no	—
219	C₄²	C₂×D₂₁	1	2	yes	—
220	C₄²	C₂×D₂₁	1	1	yes	characteristic
221	C₂×C₇	C₂×#235	1	1	yes	characteristic
222	C₂×C₇	#1007	1	1	yes	characteristic
223	C₂×C₇	D₃×#41	1	1	yes	characteristic
224	D₇	—	12	2	no	—
225	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
226	C₂×D₃	—	14	2	no	—
227	C₂×D₃	—	14	2	no	—
228	C₂×D₃	—	14	2	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	#1157	1	2	yes	—
234	Dic₃	—	7	2	no	—
235	Dic₃	—	14	2	no	—
236	Dic₃	—	14	1	no	—
237	Dic₃	—	14	1	no	—
238	Dic₃	—	14	1	no	—
239	C₂³	—	21	2	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	42	2	no	—
242	Q₈	—	42	2	no	—
243	D₄	—	42	2	no	—
244	C₂×C₄	—	21	1	no	—
245	C₂×C₄	—	21	1	no	—
246	C₂×C₄	—	21	1	no	—
247	C₂×C₄	—	21	2	no	—
248	C₂×C₄	—	21	1	no	—
249	C₂×C₄	—	42	4	no	—
250	C₂×C₄	C₂²×D₂₁	1	2	yes	—
251	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₂₁	1	2	yes	—
253	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
255	C₇	#5568	1	1	yes	characteristic, complemented, Sylow
256	C₂×C₃	C₂×#1157	1	1	yes	characteristic
257	C₂×C₃	#6551	1	1	yes	characteristic
258	C₂×C₃	D₇×#41	1	1	yes	characteristic
259	D₃	—	28	2	no	—
260	C₂²	—	42	2	no	—
261	C₂²	—	42	2	no	—
262	C₂²	—	42	2	no	—
263	C₂²	C₂³×D₂₁	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	21	2	no	—
269	C₄	—	42	2	no	—
270	C₄	—	42	1	no	—
271	C₄	—	42	1	no	—
272	C₄	—	42	1	no	—
273	C₄	#66728	1	2	yes	—
274	C₃	#90259	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	84	2	no	—
276	C₂	C₂×#66728	1	1	yes	characteristic
277	C₂	#10621660	1	1	yes	characteristic
278	C₂	#41×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₄×C₇×Q₈	C₂
prime radical	C₂×D₈₄	C₂²
hypercenter	C₄×Q₈	D₂₁
hypocenter	C₃×C₇	#556
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ 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
4. #10621381	C₂	semidirect
5. #10621379	C₂	semidirect
7. #10621370	C₂	semidirect
8. #10621359	C₂	semidirect
10. #10621355	C₂	semidirect
11. C₄×Dic₄₂	C₂	semidirect
36. C₄×C₇×Q₈	D₃	semidirect
87. C₃×C₄×Q₈	D₇	semidirect
164. C₄×Q₈	D₂₁	semidirect
197. C₃×C₇	#556	semidirect
255. C₇	#5568	semidirect
274. C₃	#90259	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
#10621430	2
#10621427	2
#10621381	2
#10621379	2
#10621376	2
#10621370	2
#10621359	2
#10621358	2
#10621355	2
C₄×Dic₄₂	2
#90259	3
#5568	7