Subgroups — #13061719

Order 1344.

384 conjugacy classes of subgroups.

The subgroup lattice has 272 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
2	Q₈×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621427	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621409	C₂	1	1	yes	maximal, characteristic
5	#10621407	C₂	1	1	yes	maximal, characteristic
6	#10621373	C₂	1	2	yes	maximal, complemented
7	#10621373	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621372	C₂	1	2	yes	maximal, complemented
9	#10621370	C₂	1	1	yes	maximal, characteristic, complemented
10	#10621362	C₂	1	2	yes	maximal
11	#10621362	C₂	1	2	yes	maximal
12	#90316	—	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×Dic₄₂	C₂²	1	1	yes	characteristic
16	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	#66636	C₂²	1	2	yes	—
20	#66636	C₂²	1	1	yes	characteristic
21	#66636	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₂₁	C₂²	1	2	yes	—
23	C₄⋊Dic₂₁	C₂²	1	2	yes	—
24	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
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	2	yes	—
33	C₄×Dic₂₁	C₂²	1	2	yes	—
34	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
36	C₇×C₂²⋊Q₈	D₃	1	1	yes	characteristic
37	Q₈×Dic₇	—	3	1	no	—
38	#6507	—	3	1	no	—
39	#6489	—	3	1	no	—
40	#6487	—	3	1	no	—
41	#6453	—	3	2	no	—
42	#6453	—	3	1	no	—
43	#6452	—	3	2	no	—
44	#6450	—	3	1	no	—
45	#6442	—	3	2	no	—
46	#6442	—	3	2	no	—
47	#5625	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₃×C₇×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₇	—	2	2	no	—
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	1	yes	characteristic
57	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
58	C₂×Dic₂₁	C₂³	1	2	yes	—
59	C₂×Dic₂₁	C₂³	1	2	yes	—
60	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
61	Dic₄₂	—	2	2	no	—
62	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
64	C₂×Dic₁₄	—	3	1	no	—
65	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
67	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
68	C₂²⋊Dic₇	—	3	2	no	—
69	C₂²⋊Dic₇	—	3	1	no	—
70	C₂²⋊Dic₇	—	3	1	no	—
71	C₄⋊Dic₇	—	3	2	no	—
72	C₄⋊Dic₇	—	3	2	no	—
73	C₄⋊Dic₇	—	3	1	no	—
74	C₄⋊Dic₇	—	3	1	no	—
75	C₄⋊Dic₇	—	3	1	no	—
76	Dic₇⋊C₄	—	3	2	no	—
77	Dic₇⋊C₄	—	3	2	no	—
78	Dic₇⋊C₄	—	3	2	no	—
79	Dic₇⋊C₄	—	3	2	no	—
80	Dic₇⋊C₄	—	3	2	no	—
81	C₄×Dic₇	—	3	2	no	—
82	C₄×Dic₇	—	3	2	no	—
83	C₄×Dic₇	—	3	1	no	—
84	C₄×Dic₇	—	3	1	no	—
85	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic
86	Q₈×Dic₃	—	7	1	no	—
87	#944	—	7	1	no	—
88	#926	—	7	1	no	—
89	#924	—	7	1	no	—
90	#890	—	7	2	no	—
91	#890	—	7	1	no	—
92	#889	—	7	2	no	—
93	#887	—	7	1	no	—
94	#879	—	7	2	no	—
95	#879	—	7	2	no	—
96	C₂²×C₃×C₇	—	2	1	no	—
97	C₂²×C₃×C₇	—	2	1	no	—
98	C₂²×C₃×C₇	—	2	1	no	—
99	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
100	C₃×C₄×C₇	—	2	1	no	—
101	C₃×C₄×C₇	—	2	2	no	—
102	C₃×C₄×C₇	—	2	2	no	—
103	C₃×C₄×C₇	#41	1	2	yes	—
104	Dic₂₁	—	2	2	no	—
105	Dic₂₁	—	2	1	no	—
106	Dic₂₁	—	2	2	no	—
107	Dic₂₁	—	2	1	no	—
108	Dic₂₁	—	2	1	no	—
109	Dic₂₁	—	2	1	no	—
110	#541	—	21	1	no	Sylow
111	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
112	C₇×Q₈	—	2	2	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
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	1	yes	characteristic
118	C₂×Dic₇	—	3	1	no	—
119	C₂×Dic₇	—	3	1	no	—
120	C₂×Dic₇	—	3	1	no	—
121	C₂×Dic₇	—	3	2	no	—
122	C₂×Dic₇	—	3	2	no	—
123	C₂×Dic₇	—	3	1	no	—
124	Dic₁₄	—	6	2	no	—
125	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
127	C₂×Dic₆	—	7	1	no	—
128	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
130	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
131	#217	—	7	2	no	—
132	#217	—	7	1	no	—
133	#217	—	7	1	no	—
134	C₄⋊Dic₃	—	7	2	no	—
135	C₄⋊Dic₃	—	7	2	no	—
136	C₄⋊Dic₃	—	7	1	no	—
137	C₄⋊Dic₃	—	7	1	no	—
138	C₄⋊Dic₃	—	7	1	no	—
139	Dic₃⋊C₄	—	7	2	no	—
140	Dic₃⋊C₄	—	7	2	no	—
141	Dic₃⋊C₄	—	7	2	no	—
142	Dic₃⋊C₄	—	7	2	no	—
143	Dic₃⋊C₄	—	7	2	no	—
144	C₄×Dic₃	—	7	2	no	—
145	C₄×Dic₃	—	7	2	no	—
146	C₄×Dic₃	—	7	1	no	—
147	C₄×Dic₃	—	7	1	no	—
148	C₂×C₃×C₇	—	4	1	no	—
149	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
152	C₄⋊Q₈	—	21	1	no	—
153	#126	—	21	2	no	—
154	#126	—	21	2	no	—
155	#125	—	21	2	no	—
156	#125	—	21	1	no	—
157	#125	—	21	2	no	—
158	C₂²⋊Q₈	—	21	1	no	—
159	C₂²⋊Q₈	D₂₁	1	1	yes	characteristic
160	C₄×Q₈	—	21	1	no	—
161	C₄×Q₈	—	21	1	no	—
162	#117	—	21	1	no	—
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	#237	1	2	yes	—
171	Dic₇	—	6	2	no	—
172	Dic₇	—	6	1	no	—
173	Dic₇	—	6	2	no	—
174	Dic₇	—	6	1	no	—
175	Dic₇	—	6	1	no	—
176	Dic₇	—	6	1	no	—
177	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
178	C₃×Q₈	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
182	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
184	C₂×Dic₃	—	7	1	no	—
185	C₂×Dic₃	—	7	1	no	—
186	C₂×Dic₃	—	7	1	no	—
187	C₂×Dic₃	—	7	2	no	—
188	C₂×Dic₃	—	7	2	no	—
189	C₂×Dic₃	—	7	1	no	—
190	Dic₆	—	14	2	no	—
191	C₃×C₇	#541	1	1	yes	characteristic, complemented, Hall
192	C₂×Q₈	—	21	1	no	—
193	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
195	C₄⋊C₄	—	21	2	no	—
196	C₄⋊C₄	—	21	2	no	—
197	C₄⋊C₄	—	21	2	no	—
198	C₄⋊C₄	—	21	2	no	—
199	C₄⋊C₄	—	21	2	no	—
200	C₄⋊C₄	—	21	1	no	—
201	C₄⋊C₄	—	21	1	no	—
202	C₄⋊C₄	—	21	1	no	—
203	C₄⋊C₄	—	21	2	no	—
204	C₄⋊C₄	—	21	2	no	—
205	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
206	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
207	C₂²⋊C₄	—	21	1	no	—
208	C₂²⋊C₄	—	21	1	no	—
209	C₂²⋊C₄	—	21	2	no	—
210	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
211	C₄²	—	21	1	no	—
212	C₄²	—	21	1	no	—
213	C₄²	—	21	2	no	—
214	C₄²	—	21	2	no	—
215	C₂×C₇	—	4	1	no	—
216	C₂×C₇	C₂×#237	1	1	yes	characteristic
217	C₂×C₇	#1007	1	1	yes	characteristic
218	C₂×C₇	#1010	1	1	yes	characteristic
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	#1159	1	2	yes	—
227	Dic₃	—	14	2	no	—
228	Dic₃	—	14	1	no	—
229	Dic₃	—	14	2	no	—
230	Dic₃	—	14	1	no	—
231	Dic₃	—	14	1	no	—
232	Dic₃	—	14	1	no	—
233	C₂³	C₂²×D₂₁	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	42	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	21	1	no	—
238	C₂×C₄	—	21	1	no	—
239	C₂×C₄	—	21	1	no	—
240	C₂×C₄	—	21	2	no	—
241	C₂×C₄	—	21	1	no	—
242	C₂×C₄	—	21	2	no	—
243	C₂×C₄	C₂²×D₂₁	1	2	yes	—
244	C₂×C₄	C₂²×D₂₁	1	2	yes	—
245	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
247	C₇	#5625	1	1	yes	characteristic, complemented, Sylow
248	C₂×C₃	—	4	1	no	—
249	C₂×C₃	C₂×#1159	1	1	yes	characteristic
250	C₂×C₃	#6551	1	1	yes	characteristic
251	C₂×C₃	#6554	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂³×D₂₁	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	42	2	no	—
260	C₄	—	42	1	no	—
261	C₄	—	42	1	no	—
262	C₄	—	42	2	no	—
263	C₄	—	42	1	no	—
264	C₄	—	42	1	no	—
265	C₄	#66730	1	2	yes	—
266	C₃	#90316	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	4	1	no	—
268	C₂	C₂×#66730	1	1	yes	characteristic
269	C₂	#10621660	1	1	yes	characteristic
270	C₂	#10621663	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	C₂³×C₃×C₇	C₂³
hypercenter	C₂²⋊Q₈	D₂₁
hypocenter	C₃×C₇	#541
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
2. Q₈×Dic₂₁	C₂	semidirect
3. #10621427	C₂	semidirect
6. #10621373	C₂	semidirect
7. #10621373	C₂	semidirect
8. #10621372	C₂	semidirect
9. #10621370	C₂	semidirect
191. C₃×C₇	#541	semidirect
247. C₇	#5625	semidirect
266. C₃	#90316	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
Q₈×Dic₂₁	2
#10621427	2
#10621409	2
#10621407	2
#10621373	2
#10621372	2
#10621370	2
#10621362	2
#90316	3
#5625	7