Subgroups — #13061728

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 300 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	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#10621380	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₄×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621372	C₂	1	2	yes	maximal, complemented
9	#10621371	C₂	1	1	yes	maximal, characteristic, complemented
10	#10621364	C₂	1	2	yes	maximal
11	#10621361	C₂	1	2	yes	maximal, complemented
12	#90326	—	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	C₂²×Dic₂₁	C₂²	1	2	yes	—
17	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₂₁	C₂²	1	2	yes	—
19	C₂×Dic₄₂	C₂²	1	2	yes	—
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	#66636	C₂²	1	1	yes	characteristic
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	2	yes	—
29	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
30	#66629	C₂²	1	2	yes	—
31	#66629	C₂²	1	2	yes	—
32	#66629	C₂²	1	2	yes	—
33	#66629	C₂²	1	1	yes	characteristic
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	C₂×C₄⋊Dic₇	—	3	1	no	—
41	#6460	—	3	2	no	—
42	D₇×C₄⋊C₄	—	3	1	no	—
43	#6452	—	3	2	no	—
44	#6451	—	3	1	no	—
45	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
46	#6441	—	3	2	no	—
47	#5635	—	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	2	yes	—
53	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	#4268	Q₈	1	4	yes	complemented
57	C₂×Dic₂₁	—	4	2	no	—
58	C₂×Dic₂₁	C₂³	1	2	yes	—
59	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁	C₂³	1	2	yes	—
62	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
63	C₄×D₂₁	—	2	2	no	—
64	C₄×D₂₁	—	4	2	no	—
65	Dic₄₂	—	4	2	no	—
66	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
68	C₂×C₂²⋊D₇	—	3	1	no	—
69	C₂²×Dic₇	—	3	2	no	—
70	C₂×C₄×D₇	—	3	1	no	—
71	C₂×C₄×D₇	—	3	2	no	—
72	C₂×Dic₁₄	—	3	2	no	—
73	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
74	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
75	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
76	C₂²⋊Dic₇	—	3	1	no	—
77	(C₂×D₇)⋊C₄	—	3	2	no	—
78	(C₂×D₇)⋊C₄	—	3	1	no	—
79	C₄⋊Dic₇	—	3	2	no	—
80	C₄⋊Dic₇	—	3	1	no	—
81	C₄⋊Dic₇	—	3	2	no	—
82	C₄⋊Dic₇	—	3	1	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	Dic₇⋊C₄	—	3	2	no	—
85	Dic₇⋊C₄	—	3	2	no	—
86	Dic₇⋊C₄	—	3	1	no	—
87	C₄×Dic₇	—	3	2	no	—
88	C₄×Dic₇	—	3	1	no	—
89	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
90	#946	—	7	1	no	—
91	Q₈×Dic₃	—	7	1	no	—
92	C₄×#67	—	7	1	no	—
93	C₂×C₄⋊Dic₃	—	7	1	no	—
94	#897	—	7	2	no	—
95	D₃×C₄⋊C₄	—	7	1	no	—
96	#889	—	7	2	no	—
97	#888	—	7	1	no	—
98	#67⋊C₄	—	7	2	no	—
99	#878	—	7	2	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	—	2	1	no	—
102	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	—
103	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×D₂₁	—	2	1	no	—
105	C₂×D₂₁	—	2	1	no	—
106	C₂×D₂₁	C₂×Q₈	1	2	yes	—
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×C₄×C₇	—	2	2	no	—
109	C₃×C₄×C₇	—	2	2	no	—
110	C₃×C₄×C₇	#41	1	2	yes	—
111	Dic₂₁	—	2	1	no	—
112	Dic₂₁	—	2	1	no	—
113	Dic₂₁	—	2	2	no	—
114	Dic₂₁	—	2	2	no	—
115	Dic₂₁	C₂×Q₈	1	2	yes	—
116	#554	—	21	1	no	Sylow
117	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂²×D₇	—	3	1	no	—
119	C₇×Q₈	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
122	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
123	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂²⋊D₇	—	3	4	no	—
126	C₂×Dic₇	—	3	1	no	—
127	C₂×Dic₇	—	3	2	no	—
128	C₂×Dic₇	—	3	1	no	—
129	C₂×Dic₇	—	3	1	no	—
130	C₂×Dic₇	—	3	2	no	—
131	C₂×Dic₇	—	12	2	no	—
132	C₄×D₇	—	6	2	no	—
133	C₄×D₇	—	12	2	no	—
134	Dic₁₄	—	12	2	no	—
135	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
137	C₂×#67	—	7	1	no	—
138	C₂²×Dic₃	—	7	2	no	—
139	C₂×C₄×D₃	—	7	1	no	—
140	C₂×C₄×D₃	—	7	2	no	—
141	C₂×Dic₆	—	7	2	no	—
142	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
143	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
144	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
145	#217	—	7	1	no	—
146	(C₂×D₃)⋊C₄	—	7	2	no	—
147	(C₂×D₃)⋊C₄	—	7	1	no	—
148	C₄⋊Dic₃	—	7	2	no	—
149	C₄⋊Dic₃	—	7	1	no	—
150	C₄⋊Dic₃	—	7	2	no	—
151	C₄⋊Dic₃	—	7	1	no	—
152	Dic₃⋊C₄	—	7	2	no	—
153	Dic₃⋊C₄	—	7	2	no	—
154	Dic₃⋊C₄	—	7	2	no	—
155	Dic₃⋊C₄	—	7	1	no	—
156	C₄×Dic₃	—	7	2	no	—
157	C₄×Dic₃	—	7	1	no	—
158	C₂×C₃×C₇	—	2	2	no	—
159	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
162	D₂₁	—	2	2	no	—
163	C₄⋊Q₈	—	21	1	no	—
164	#125	—	21	2	no	—
165	C₂²⋊Q₈	—	21	1	no	—
166	C₂²⋊Q₈	—	21	2	no	—
167	C₂²⋊Q₈	—	21	2	no	—
168	C₂²⋊Q₈	D₂₁	1	1	yes	characteristic, complemented
169	C₄×Q₈	—	21	1	no	—
170	C₄×D₄	—	21	1	no	—
171	C₄×D₄	—	21	2	no	—
172	C₂×C₄⋊C₄	—	21	1	no	—
173	C₂×C₄⋊C₄	—	21	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	D₃×Q₈	1	2	yes	—
177	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
178	C₂×D₇	—	3	2	no	—
179	C₂×D₇	—	6	1	no	—
180	C₂×D₇	—	6	1	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	#237	1	2	yes	—
185	Dic₇	—	3	2	no	—
186	Dic₇	—	6	1	no	—
187	Dic₇	—	6	1	no	—
188	Dic₇	—	6	2	no	—
189	Dic₇	—	6	2	no	—
190	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
191	C₂²×D₃	—	7	1	no	—
192	C₃×Q₈	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
195	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
196	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
198	#67	—	7	4	no	—
199	C₂×Dic₃	—	7	1	no	—
200	C₂×Dic₃	—	7	2	no	—
201	C₂×Dic₃	—	7	1	no	—
202	C₂×Dic₃	—	7	1	no	—
203	C₂×Dic₃	—	7	2	no	—
204	C₂×Dic₃	—	28	2	no	—
205	C₄×D₃	—	14	2	no	—
206	C₄×D₃	—	28	2	no	—
207	Dic₆	—	28	2	no	—
208	C₃×C₇	#554	1	1	yes	characteristic, complemented, Hall
209	C₂×Q₈	—	21	2	no	—
210	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
211	C₂×D₄	—	21	1	no	—
212	C₂²×C₄	—	21	2	no	—
213	C₂²×C₄	—	21	1	no	—
214	C₂²×C₄	—	21	2	no	—
215	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
216	C₄⋊C₄	—	21	2	no	—
217	C₄⋊C₄	—	21	1	no	—
218	C₄⋊C₄	—	21	2	no	—
219	C₄⋊C₄	—	21	2	no	—
220	C₄⋊C₄	—	21	1	no	—
221	C₄⋊C₄	—	21	2	no	—
222	C₄⋊C₄	—	21	1	no	—
223	C₄⋊C₄	—	21	2	no	—
224	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
225	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
226	C₂²⋊C₄	—	21	1	no	—
227	C₂²⋊C₄	—	21	1	no	—
228	C₂²⋊C₄	—	21	2	no	—
229	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
230	C₄²	—	21	2	no	—
231	C₄²	—	21	1	no	—
232	C₂×C₇	—	2	2	no	—
233	C₂×C₇	C₂×#237	1	1	yes	characteristic
234	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
235	C₂×C₇	#1009	1	1	yes	characteristic
236	D₇	—	6	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	Q₈×D₇	1	2	yes	—
240	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
241	C₂×D₃	—	7	2	no	—
242	C₂×D₃	—	14	1	no	—
243	C₂×D₃	—	14	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	#1159	1	2	yes	—
248	Dic₃	—	7	2	no	—
249	Dic₃	—	14	1	no	—
250	Dic₃	—	14	1	no	—
251	Dic₃	—	14	2	no	—
252	Dic₃	—	14	2	no	—
253	C₂³	—	21	1	no	—
254	C₂³	C₂²×D₂₁	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	84	2	no	—
257	D₄	—	21	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	21	1	no	—
260	C₂×C₄	—	21	1	no	—
261	C₂×C₄	—	21	2	no	—
262	C₂×C₄	—	21	1	no	—
263	C₂×C₄	—	21	2	no	—
264	C₂×C₄	—	42	2	no	—
265	C₂×C₄	—	84	2	no	—
266	C₂×C₄	—	84	2	no	—
267	C₂×C₄	C₂²×D₂₁	1	2	yes	—
268	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₂₁	1	2	yes	—
270	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
271	C₇	#5635	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	C₂×#1159	1	1	yes	characteristic
274	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
275	C₂×C₃	#6553	1	1	yes	characteristic
276	D₃	—	14	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	21	2	no	—
280	C₂²	—	42	1	no	—
281	C₂²	—	42	1	no	—
282	C₂²	Q₈×D₂₁	1	2	yes	—
283	C₂²	C₂³×D₂₁	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	21	2	no	—
288	C₄	—	42	1	no	—
289	C₄	—	42	2	no	—
290	C₄	—	42	2	no	—
291	C₄	—	42	1	no	—
292	C₄	#66730	1	2	yes	—
293	C₃	#90326	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	42	2	no	—
296	C₂	C₂×#66730	1	1	yes	characteristic
297	C₂	C₂×Q₈×D₂₁	1	1	yes	characteristic
298	C₂	#10621662	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	#554
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. C₂×C₄⋊Dic₂₁	C₂	semidirect
6. #10621380	C₂	semidirect
7. C₄⋊C₄×D₂₁	C₂	semidirect
8. #10621372	C₂	semidirect
9. #10621371	C₂	semidirect
11. #10621361	C₂	semidirect
36. C₇×C₂²⋊Q₈	D₃	semidirect
56. #4268	Q₈	semidirect
89. C₃×C₂²⋊Q₈	D₇	semidirect
168. C₂²⋊Q₈	D₂₁	semidirect
208. C₃×C₇	#554	semidirect
271. C₇	#5635	semidirect
293. C₃	#90326	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
C₂×C₄⋊Dic₂₁	2
#10621380	2
C₄⋊C₄×D₂₁	2
#10621372	2
#10621371	2
#10621364	2
#10621361	2
#90326	3
#5635	7