Subgroups — #13061714

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×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	#10621407	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic
6	#10621372	C₂	1	2	yes	maximal, complemented
7	#10621371	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621370	C₂	1	1	yes	maximal, characteristic, complemented
9	#10621370	C₂	1	2	yes	maximal, complemented
10	#10621361	C₂	1	2	yes	maximal
11	#10621360	C₂	1	2	yes	maximal
12	#90313	—	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	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₁	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×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	#66636	C₂²	1	2	yes	—
23	C₄⋊Dic₂₁	C₂²	1	2	yes	—
24	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
25	#66629	C₂²	1	2	yes	—
26	#66629	C₂²	1	2	yes	—
27	#66629	C₂²	1	2	yes	—
28	#66629	C₂²	1	2	yes	—
29	#66629	C₂²	1	2	yes	—
30	C₄×Dic₂₁	C₂²	1	2	yes	—
31	C₄×Dic₂₁	C₂²	1	2	yes	—
32	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
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	#6487	—	3	1	no	—
40	C₂×C₄×Dic₇	—	3	1	no	—
41	#6452	—	3	2	no	—
42	#6451	—	3	1	no	—
43	#6450	—	3	1	no	—
44	#6450	—	3	2	no	—
45	#6441	—	3	2	no	—
46	#6440	—	3	2	no	—
47	#5622	—	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	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₁	—	2	2	no	—
56	C₂×Dic₂₁	Q₈	1	4	yes	complemented
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	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
63	Dic₄₂	—	2	2	no	—
64	Dic₄₂	—	2	4	no	—
65	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
67	C₂²×Dic₇	—	3	1	no	—
68	C₂²×Dic₇	—	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	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
74	C₂²⋊Dic₇	—	3	2	no	—
75	C₄⋊Dic₇	—	3	2	no	—
76	C₄⋊Dic₇	—	3	1	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	Dic₇⋊C₄	—	3	2	no	—
82	C₄×Dic₇	—	3	2	no	—
83	C₄×Dic₇	—	3	2	no	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	—	3	1	no	—
86	C₄×Dic₇	—	3	1	no	—
87	C₄×Dic₇	—	3	1	no	—
88	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic
89	Q₈×Dic₃	—	7	1	no	—
90	#944	—	7	1	no	—
91	#924	—	7	1	no	—
92	C₂×C₄×Dic₃	—	7	1	no	—
93	#889	—	7	2	no	—
94	#888	—	7	1	no	—
95	#887	—	7	1	no	—
96	#887	—	7	2	no	—
97	#878	—	7	2	no	—
98	#877	—	7	2	no	—
99	C₂²×C₃×C₇	—	2	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	—
102	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
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	#41	1	2	yes	—
107	Dic₂₁	—	2	2	no	—
108	Dic₂₁	—	2	2	no	—
109	Dic₂₁	—	2	1	no	—
110	Dic₂₁	C₂×Q₈	1	2	yes	—
111	Dic₂₁	C₂×Q₈	1	2	yes	—
112	Dic₂₁	#41	1	2	yes	—
113	#533	—	21	1	no	Sylow
114	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
115	C₇×Q₈	—	2	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
119	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
120	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
121	C₂×Dic₇	—	3	4	no	—
122	C₂×Dic₇	—	3	1	no	—
123	C₂×Dic₇	—	3	1	no	—
124	C₂×Dic₇	—	3	1	no	—
125	C₂×Dic₇	—	3	1	no	—
126	C₂×Dic₇	—	3	2	no	—
127	C₂×Dic₇	—	3	2	no	—
128	C₂×Dic₇	—	6	2	no	—
129	Dic₁₄	—	6	4	no	—
130	Dic₁₄	—	6	2	no	—
131	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
133	C₂²×Dic₃	—	7	1	no	—
134	C₂²×Dic₃	—	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×C₂²⋊C₄	C₂×D₇	1	2	yes	—
140	#217	—	7	2	no	—
141	C₄⋊Dic₃	—	7	2	no	—
142	C₄⋊Dic₃	—	7	1	no	—
143	Dic₃⋊C₄	—	7	2	no	—
144	Dic₃⋊C₄	—	7	2	no	—
145	Dic₃⋊C₄	—	7	2	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	C₄×Dic₃	—	7	2	no	—
149	C₄×Dic₃	—	7	2	no	—
150	C₄×Dic₃	—	7	1	no	—
151	C₄×Dic₃	—	7	1	no	—
152	C₄×Dic₃	—	7	1	no	—
153	C₄×Dic₃	—	7	1	no	—
154	C₂×C₃×C₇	—	2	2	no	—
155	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
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	D₂₁	1	1	yes	characteristic
164	C₄×Q₈	—	21	1	no	—
165	C₄×Q₈	—	21	1	no	—
166	C₄×Q₈	—	21	2	no	—
167	#117	—	21	2	no	—
168	C₂×C₄²	—	21	1	no	—
169	C₂²×C₇	—	2	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	D₃×Q₈	1	2	yes	—
172	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	2	2	no	—
176	C₄×C₇	#237	1	2	yes	—
177	Dic₇	—	3	2	no	—
178	Dic₇	—	3	2	no	—
179	Dic₇	—	3	2	no	—
180	Dic₇	—	6	2	no	—
181	Dic₇	—	6	2	no	—
182	Dic₇	—	6	1	no	—
183	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
184	C₃×Q₈	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
188	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
190	C₂×Dic₃	—	7	4	no	—
191	C₂×Dic₃	—	7	1	no	—
192	C₂×Dic₃	—	7	1	no	—
193	C₂×Dic₃	—	7	1	no	—
194	C₂×Dic₃	—	7	1	no	—
195	C₂×Dic₃	—	7	2	no	—
196	C₂×Dic₃	—	7	2	no	—
197	C₂×Dic₃	—	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	C₂×D₂₁	1	1	yes	characteristic
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	C₂×D₂₁	1	2	yes	—
218	C₄²	—	21	1	no	—
219	C₄²	—	21	1	no	—
220	C₄²	—	21	1	no	—
221	C₄²	—	21	1	no	—
222	C₄²	—	21	2	no	—
223	C₄²	—	21	2	no	—
224	C₂×C₇	—	2	2	no	—
225	C₂×C₇	C₂×#237	1	1	yes	characteristic
226	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
227	C₂×C₇	D₃×#41	1	1	yes	characteristic
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	Q₈×D₇	1	2	yes	—
231	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	#1159	1	2	yes	—
236	Dic₃	—	7	2	no	—
237	Dic₃	—	7	2	no	—
238	Dic₃	—	7	2	no	—
239	Dic₃	—	14	2	no	—
240	Dic₃	—	14	2	no	—
241	Dic₃	—	14	1	no	—
242	C₂³	C₂²×D₂₁	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	42	4	no	—
245	Q₈	—	42	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	21	4	no	—
248	C₂×C₄	—	21	1	no	—
249	C₂×C₄	—	21	1	no	—
250	C₂×C₄	—	21	1	no	—
251	C₂×C₄	—	21	1	no	—
252	C₂×C₄	—	21	2	no	—
253	C₂×C₄	—	21	2	no	—
254	C₂×C₄	—	42	2	no	—
255	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₂₁	1	2	yes	—
257	C₂×C₄	C₂²×D₂₁	1	2	yes	—
258	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
259	C₇	#5622	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	C₂×#1159	1	1	yes	characteristic
262	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
263	C₂×C₃	D₇×#41	1	1	yes	characteristic
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	Q₈×D₂₁	1	2	yes	—
267	C₂²	C₂³×D₂₁	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	21	2	no	—
272	C₄	—	21	2	no	—
273	C₄	—	21	2	no	—
274	C₄	—	42	2	no	—
275	C₄	—	42	1	no	—
276	C₄	—	42	2	no	—
277	C₄	#66730	1	2	yes	—
278	C₃	#90313	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	2	2	no	—
280	C₂	C₂×#66730	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, 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	#533
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. #10621372	C₂	semidirect
7. #10621371	C₂	semidirect
8. #10621370	C₂	semidirect
9. #10621370	C₂	semidirect
56. C₂×Dic₂₁	Q₈	semidirect
200. C₃×C₇	#533	semidirect
259. C₇	#5622	semidirect
278. C₃	#90313	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
#10621407	2
C₂×C₄×Dic₂₁	2
#10621372	2
#10621371	2
#10621370	2
#10621361	2
#10621360	2
#90313	3
#5622	7