Subgroups — #13061718

Order 1344.

424 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621429	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#4268	C₂	1	1	yes	maximal, characteristic
5	#10621409	C₂	1	1	yes	maximal, characteristic, complemented
6	#10621381	C₂	1	2	yes	maximal, complemented
7	#10621375	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621371	C₂	1	1	yes	maximal, characteristic, complemented
9	#10621370	C₂	1	2	yes	maximal, complemented
10	#10621368	C₂	1	2	yes	maximal
11	#10621362	C₂	1	2	yes	maximal, complemented
12	#90332	—	3	1	no	maximal, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×#4268	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₂	C₂²	1	2	yes	—
18	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	#66636	C₂²	1	1	yes	characteristic
22	#66636	C₂²	1	2	yes	—
23	#66631	C₂²	1	2	yes	—
24	#66631	C₂²	1	2	yes	—
25	#66631	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₁	C₂²	1	2	yes	—
27	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
29	#66629	C₂²	1	2	yes	—
30	#66629	C₂²	1	2	yes	—
31	#66629	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₁	C₂²	1	2	yes	—
33	C₄×Dic₂₁	C₂²	1	2	yes	—
34	C₄×Dic₂₁	C₂²	1	2	yes	—
35	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
36	C₇×C₂²⋊Q₈	D₃	1	1	yes	characteristic, complemented
37	#6509	—	3	1	no	—
38	Q₈×Dic₇	—	3	1	no	—
39	C₄×C₂²⋊D₇	—	3	1	no	—
40	#6489	—	3	1	no	—
41	#6461	—	3	2	no	—
42	#6455	—	3	1	no	—
43	#6451	—	3	1	no	—
44	#6450	—	3	2	no	—
45	#6448	—	3	2	no	—
46	#6442	—	3	2	no	—
47	#5641	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×D₂₁	C₂³	1	1	yes	characteristic
50	C₃×C₇×Q₈	—	2	2	no	—
51	C₂×C₃×C₄×C₇	—	2	2	no	—
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	#4268	—	2	2	no	—
57	C₂×Dic₂₁	C₂³	1	2	yes	—
58	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
59	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁	C₂³	1	2	yes	—
61	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
62	C₄×D₂₁	—	2	2	no	—
63	Dic₄₂	—	2	4	no	—
64	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	—	3	1	no	—
67	C₂×C₄×D₇	—	3	1	no	—
68	C₂×Dic₁₄	—	3	2	no	—
69	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
70	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
72	C₂²⋊Dic₇	—	3	1	no	—
73	C₂²⋊Dic₇	—	3	2	no	—
74	(C₂×D₇)⋊C₄	—	3	2	no	—
75	(C₂×D₇)⋊C₄	—	3	2	no	—
76	(C₂×D₇)⋊C₄	—	3	1	no	—
77	C₄⋊Dic₇	—	3	2	no	—
78	C₄⋊Dic₇	—	3	1	no	—
79	C₄⋊Dic₇	—	3	1	no	—
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	Dic₇⋊C₄	—	3	1	no	—
83	C₄×Dic₇	—	3	2	no	—
84	C₄×Dic₇	—	3	2	no	—
85	C₄×Dic₇	—	3	2	no	—
86	C₄×Dic₇	—	3	1	no	—
87	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
88	#946	—	7	1	no	—
89	Q₈×Dic₃	—	7	1	no	—
90	C₄×#67	—	7	1	no	—
91	#926	—	7	1	no	—
92	#898	—	7	2	no	—
93	#892	—	7	1	no	—
94	#888	—	7	1	no	—
95	#887	—	7	2	no	—
96	#885	—	7	2	no	—
97	#879	—	7	2	no	—
98	C₂²×C₃×C₇	—	2	1	no	—
99	C₂²×C₃×C₇	—	2	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
102	C₂×D₂₁	—	2	1	no	—
103	C₂×D₂₁	—	2	1	no	—
104	C₂×D₂₁	—	2	1	no	—
105	C₃×C₄×C₇	—	2	2	no	—
106	C₃×C₄×C₇	—	2	1	no	—
107	C₃×C₄×C₇	—	2	2	no	—
108	C₃×C₄×C₇	#41	1	2	yes	—
109	Dic₂₁	—	2	2	no	—
110	Dic₂₁	—	2	1	no	—
111	Dic₂₁	—	2	2	no	—
112	Dic₂₁	—	2	1	no	—
113	Dic₂₁	#41	1	2	yes	—
114	#556	—	21	1	no	Sylow
115	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
116	C₂²×D₇	—	3	1	no	—
117	C₇×Q₈	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
121	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
122	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
123	C₂²⋊D₇	—	6	2	no	—
124	C₂×Dic₇	—	3	1	no	—
125	C₂×Dic₇	—	3	2	no	—
126	C₂×Dic₇	—	3	1	no	—
127	C₂×Dic₇	—	3	1	no	—
128	C₂×Dic₇	—	3	2	no	—
129	C₄×D₇	—	6	2	no	—
130	Dic₁₄	—	6	4	no	—
131	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
133	C₂×#67	—	7	1	no	—
134	C₂×C₄×D₃	—	7	1	no	—
135	C₂×Dic₆	—	7	2	no	—
136	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
137	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
138	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
139	#217	—	7	1	no	—
140	#217	—	7	2	no	—
141	(C₂×D₃)⋊C₄	—	7	2	no	—
142	(C₂×D₃)⋊C₄	—	7	2	no	—
143	(C₂×D₃)⋊C₄	—	7	1	no	—
144	C₄⋊Dic₃	—	7	2	no	—
145	C₄⋊Dic₃	—	7	1	no	—
146	C₄⋊Dic₃	—	7	1	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	Dic₃⋊C₄	—	7	2	no	—
149	Dic₃⋊C₄	—	7	1	no	—
150	C₄×Dic₃	—	7	2	no	—
151	C₄×Dic₃	—	7	2	no	—
152	C₄×Dic₃	—	7	2	no	—
153	C₄×Dic₃	—	7	1	no	—
154	C₂×C₃×C₇	—	4	1	no	—
155	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
158	D₂₁	—	4	1	no	—
159	C₄⋊Q₈	—	21	1	no	—
160	#126	—	21	2	no	—
161	#126	—	21	2	no	—
162	#124	—	21	2	no	—
163	C₂²⋊Q₈	—	21	1	no	—
164	C₂²⋊Q₈	D₂₁	1	1	yes	characteristic, complemented
165	C₄×Q₈	—	21	1	no	—
166	C₄×Q₈	—	21	2	no	—
167	C₄×D₄	—	21	1	no	—
168	#117	—	21	1	no	—
169	#117	—	21	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
174	C₂×D₇	—	6	1	no	—
175	C₂×D₇	—	6	1	no	—
176	C₂×D₇	—	6	1	no	—
177	C₄×C₇	—	2	1	no	—
178	C₄×C₇	—	2	2	no	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	#237	1	2	yes	—
181	Dic₇	—	3	2	no	—
182	Dic₇	—	6	1	no	—
183	Dic₇	—	6	2	no	—
184	Dic₇	—	6	1	no	—
185	Dic₇	—	6	2	no	—
186	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
187	C₂²×D₃	—	7	1	no	—
188	C₃×Q₈	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
192	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
193	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
194	#67	—	14	2	no	—
195	C₂×Dic₃	—	7	1	no	—
196	C₂×Dic₃	—	7	2	no	—
197	C₂×Dic₃	—	7	1	no	—
198	C₂×Dic₃	—	7	1	no	—
199	C₂×Dic₃	—	7	2	no	—
200	C₄×D₃	—	14	2	no	—
201	Dic₆	—	14	4	no	—
202	C₃×C₇	#556	1	1	yes	characteristic, complemented, Hall
203	C₂×Q₈	—	21	2	no	—
204	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
205	C₂×D₄	—	21	1	no	—
206	C₂²×C₄	—	21	1	no	—
207	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
208	C₄⋊C₄	—	21	1	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	2	no	—
211	C₄⋊C₄	—	21	1	no	—
212	C₄⋊C₄	—	21	1	no	—
213	C₄⋊C₄	—	21	2	no	—
214	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
215	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
216	C₂²⋊C₄	—	21	2	no	—
217	C₂²⋊C₄	—	21	1	no	—
218	C₂²⋊C₄	—	21	2	no	—
219	C₂²⋊C₄	—	21	1	no	—
220	C₂²⋊C₄	—	21	2	no	—
221	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
222	C₄²	—	21	2	no	—
223	C₄²	—	21	2	no	—
224	C₄²	—	21	1	no	—
225	C₄²	—	21	2	no	—
226	C₂×C₇	—	4	1	no	—
227	C₂×C₇	C₂×#237	1	1	yes	characteristic
228	C₂×C₇	#1007	1	1	yes	characteristic
229	C₂×C₇	D₃×#41	1	1	yes	characteristic
230	D₇	—	12	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
235	C₂×D₃	—	14	1	no	—
236	C₂×D₃	—	14	1	no	—
237	C₂×D₃	—	14	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	#1159	1	2	yes	—
242	Dic₃	—	7	2	no	—
243	Dic₃	—	14	1	no	—
244	Dic₃	—	14	2	no	—
245	Dic₃	—	14	1	no	—
246	Dic₃	—	14	2	no	—
247	C₂³	—	21	1	no	—
248	C₂³	C₂²×D₂₁	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	42	4	no	—
251	D₄	—	42	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	21	1	no	—
254	C₂×C₄	—	21	2	no	—
255	C₂×C₄	—	21	1	no	—
256	C₂×C₄	—	21	1	no	—
257	C₂×C₄	—	21	2	no	—
258	C₂×C₄	—	42	2	no	—
259	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₂₁	1	2	yes	—
261	C₂×C₄	C₂²×D₂₁	1	2	yes	—
262	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
263	C₇	#5641	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₃	—	4	1	no	—
265	C₂×C₃	C₂×#1159	1	1	yes	characteristic
266	C₂×C₃	#6551	1	1	yes	characteristic
267	C₂×C₃	D₇×#41	1	1	yes	characteristic
268	D₃	—	28	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	42	1	no	—
273	C₂²	—	42	1	no	—
274	C₂²	—	42	1	no	—
275	C₂²	C₂³×D₂₁	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	21	2	no	—
280	C₄	—	42	1	no	—
281	C₄	—	42	2	no	—
282	C₄	—	42	1	no	—
283	C₄	—	42	2	no	—
284	C₄	#66730	1	2	yes	—
285	C₃	#90332	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	4	1	no	—
287	C₂	—	84	1	no	—
288	C₂	C₂×#66730	1	1	yes	characteristic
289	C₂	#10621660	1	1	yes	characteristic
290	C₂	#41×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×C₂²⋊Q₈	C₂
prime radical	C₂×#4268	C₂²
hypercenter	C₂²⋊Q₈	D₂₁
hypocenter	C₃×C₇	#556
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
2. #10621429	C₂	semidirect
3. Q₈×Dic₂₁	C₂	semidirect
5. #10621409	C₂	semidirect
6. #10621381	C₂	semidirect
7. #10621375	C₂	semidirect
8. #10621371	C₂	semidirect
9. #10621370	C₂	semidirect
11. #10621362	C₂	semidirect
36. C₇×C₂²⋊Q₈	D₃	semidirect
87. C₃×C₂²⋊Q₈	D₇	semidirect
164. C₂²⋊Q₈	D₂₁	semidirect
202. C₃×C₇	#556	semidirect
263. C₇	#5641	semidirect
285. C₃	#90332	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×C₂²⋊Q₈	2
#10621429	2
Q₈×Dic₂₁	2
C₄×#4268	2
#10621409	2
#10621381	2
#10621375	2
#10621371	2
#10621370	2
#10621368	2
#10621362	2
#90332	3
#5641	7