Subgroups — #13057793

Order 1344.

456 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	C₄×D₈₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621030	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620991	C₂	1	2	yes	maximal, complemented
6	#10620980	C₂	1	1	yes	maximal, characteristic
7	#10620980	C₂	1	1	yes	maximal, characteristic
8	#10620973	C₂	1	2	yes	maximal, complemented
9	#10620966	C₂	1	2	yes	maximal, complemented
10	#10620950	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620939	C₂	1	2	yes	maximal, complemented
12	#90381	—	3	1	no	maximal, Hall
13	C₂×D₈₄	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₁	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
16	C₂×#4248	C₂²	1	2	yes	—
17	C₂×D₂₁⋊C₄	C₂²	1	2	yes	—
18	C₂×D₂₁⋊C₄	C₂²	1	2	yes	—
19	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
20	#66631	C₂²	1	2	yes	—
21	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
22	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
31	Dic₃⋊Dic₇	C₂²	1	2	yes	—
32	#66577	C₂²	1	2	yes	—
33	Dic₇⋊Dic₃	C₂²	1	2	yes	—
34	#66575	C₂²	1	2	yes	—
35	#66575	C₂²	1	2	yes	—
36	C₇×#125	—	3	1	no	—
37	#6460	—	3	2	no	—
38	#6459	—	3	2	no	—
39	#6459	—	3	2	no	—
40	#6456	—	3	1	no	—
41	#6456	—	3	1	no	—
42	D₇×C₄⋊C₄	—	3	2	no	—
43	#6453	—	3	1	no	—
44	#6451	—	3	1	no	—
45	C₄×D₂₈	—	3	1	no	—
46	C₄×Dic₁₄	D₃	1	1	yes	characteristic, complemented
47	#5579	—	7	1	no	maximal, Hall
48	C₂²×D₂₁	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	2	yes	—
53	D₈₄	Q₈	1	4	yes	complemented
54	C₄×D₂₁	—	4	2	no	—
55	C₂×C₇×Dic₃	C₂³	1	2	yes	—
56	C₂×C₇×Dic₃	C₂³	1	2	yes	—
57	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
61	C₃×Dic₁₄	—	2	2	no	—
62	#4248	—	4	2	no	—
63	D₂₁⋊C₄	—	2	4	no	—
64	D₂₁⋊C₄	—	4	2	no	—
65	C₂×D₂₈	—	3	1	no	—
66	C₂×C₄×D₇	—	3	2	no	—
67	C₂×C₄×D₇	—	3	2	no	—
68	C₂×C₄×D₇	—	3	2	no	—
69	C₂×Dic₁₄	—	3	2	no	—
70	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	3	1	no	—
72	C₇×C₄⋊C₄	—	3	2	no	—
73	C₇×C₄⋊C₄	—	3	2	no	—
74	C₇×C₄⋊C₄	—	3	1	no	—
75	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
76	(C₂×D₇)⋊C₄	—	3	2	no	—
77	(C₂×D₇)⋊C₄	—	3	2	no	—
78	(C₂×D₇)⋊C₄	—	3	2	no	—
79	C₄⋊Dic₇	—	3	1	no	—
80	C₄⋊Dic₇	—	3	2	no	—
81	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
82	Dic₇⋊C₄	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	Dic₇⋊C₄	C₂×D₃	1	1	yes	characteristic
85	Dic₇⋊C₄	C₂×D₃	1	1	yes	characteristic
86	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
88	C₃×C₄×Q₈	—	7	1	no	—
89	#946	—	7	2	no	—
90	#896	—	7	2	no	—
91	#896	—	7	2	no	—
92	D₃×C₄⋊C₄	—	7	2	no	—
93	C₄×D₁₂	—	7	1	no	—
94	C₄×D₁₂	—	7	1	no	—
95	C₄×D₁₂	—	7	1	no	—
96	#870	—	7	1	no	—
97	#870	D₇	1	1	yes	characteristic, complemented
98	#869	—	7	1	no	—
99	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
100	C₂×D₂₁	—	2	2	no	—
101	C₂×D₂₁	—	2	2	no	—
102	C₂×D₂₁	C₂×Q₈	1	4	yes	—
103	C₃×C₄×C₇	—	2	1	no	—
104	C₃×C₄×C₇	—	2	1	no	—
105	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
106	Dic₂₁	—	2	2	no	—
107	C₃×Dic₇	—	2	1	no	—
108	C₃×Dic₇	—	2	1	no	—
109	C₃×Dic₇	—	2	1	no	—
110	C₃×Dic₇	#41	1	2	yes	—
111	C₇×Dic₃	—	2	2	no	—
112	C₇×Dic₃	—	2	2	no	—
113	#554	—	21	1	no	Sylow
114	C₂²×D₇	—	3	2	no	—
115	C₂×C₄×C₇	—	3	2	no	—
116	C₂×C₄×C₇	—	3	2	no	—
117	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×Dic₇	—	3	2	no	—
121	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
122	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
123	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
125	D₂₈	—	3	4	no	—
126	C₄×D₇	—	6	4	no	—
127	C₄×D₇	—	12	2	no	—
128	C₄×D₇	—	12	2	no	—
129	Dic₁₄	—	2	2	no	—
130	Dic₁₄	—	12	2	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×D₁₂	—	7	1	no	—
133	C₂×C₄×D₃	—	7	2	no	—
134	C₂×C₄×D₃	—	7	2	no	—
135	C₂×C₄×D₃	—	7	2	no	—
136	C₂×Dic₆	—	7	2	no	—
137	C₃×C₄⋊C₄	—	7	1	no	—
138	C₃×C₄⋊C₄	—	7	1	no	—
139	C₃×C₄⋊C₄	—	7	1	no	—
140	C₃×C₄²	—	7	1	no	—
141	C₃×C₄²	—	7	1	no	—
142	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
143	(C₂×D₃)⋊C₄	—	7	2	no	—
144	(C₂×D₃)⋊C₄	—	7	2	no	—
145	(C₂×D₃)⋊C₄	—	7	2	no	—
146	C₄⋊Dic₃	—	7	1	no	—
147	C₄⋊Dic₃	—	7	2	no	—
148	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
149	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
150	Dic₃⋊C₄	—	7	2	no	—
151	Dic₃⋊C₄	—	7	2	no	—
152	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
153	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
154	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
157	D₂₁	—	2	4	no	—
158	C₄⋊Q₈	—	21	1	no	—
159	#125	—	3	1	no	—
160	#125	—	21	1	no	—
161	C₂²⋊Q₈	—	21	2	no	—
162	C₂²⋊Q₈	—	21	2	no	—
163	C₂²⋊Q₈	—	21	2	no	—
164	C₄×Q₈	—	7	1	no	—
165	C₄×D₄	—	21	1	no	—
166	C₄×D₄	—	21	1	no	—
167	C₄×D₄	—	21	1	no	—
168	C₂×C₄⋊C₄	—	21	2	no	—
169	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
170	C₂×D₇	—	3	4	no	—
171	C₂×D₇	—	6	2	no	—
172	C₂×D₇	—	6	2	no	—
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	1	no	—
175	C₄×C₇	—	6	2	no	—
176	C₄×C₇	—	6	2	no	—
177	C₄×C₇	D₃×Q₈	1	2	yes	—
178	Dic₇	—	2	1	no	—
179	Dic₇	—	2	1	no	—
180	Dic₇	—	2	1	no	—
181	Dic₇	—	6	2	no	—
182	Dic₇	#235	1	2	yes	—
183	C₂²×D₃	—	7	2	no	—
184	C₃×Q₈	—	14	2	no	—
185	C₂×C₃×C₄	—	7	1	no	—
186	C₂×C₃×C₄	—	7	1	no	—
187	C₂×C₃×C₄	—	7	1	no	—
188	C₂×C₃×C₄	—	7	1	no	—
189	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
192	C₂×Dic₃	—	7	2	no	—
193	C₂×Dic₃	C₂²×D₇	1	2	yes	—
194	C₂×Dic₃	C₂²×D₇	1	2	yes	—
195	D₁₂	—	7	4	no	—
196	C₄×D₃	—	14	4	no	—
197	C₄×D₃	—	28	2	no	—
198	C₄×D₃	—	28	2	no	—
199	Dic₆	—	28	2	no	—
200	C₃×C₇	#554	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	7	1	no	—
202	C₂×Q₈	—	21	2	no	—
203	C₂×D₄	—	21	1	no	—
204	C₂²×C₄	—	21	2	no	—
205	C₂²×C₄	—	21	2	no	—
206	C₂²×C₄	—	21	2	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	1	no	—
210	C₄⋊C₄	—	3	2	no	—
211	C₄⋊C₄	—	7	1	no	—
212	C₄⋊C₄	—	7	1	no	—
213	C₄⋊C₄	—	7	1	no	—
214	C₄⋊C₄	—	21	2	no	—
215	C₄⋊C₄	—	21	2	no	—
216	C₄⋊C₄	—	21	2	no	—
217	C₄⋊C₄	—	21	1	no	—
218	C₂²⋊C₄	—	21	2	no	—
219	C₂²⋊C₄	—	21	2	no	—
220	C₂²⋊C₄	—	21	2	no	—
221	C₄²	—	7	1	no	—
222	C₄²	—	7	1	no	—
223	C₄²	D₃×D₇	1	1	yes	characteristic
224	C₂×C₇	C₂×#235	1	1	yes	characteristic
225	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
226	C₂×C₇	#1009	1	1	yes	characteristic
227	D₇	—	6	4	no	—
228	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
229	C₂×D₃	—	7	4	no	—
230	C₂×D₃	—	14	2	no	—
231	C₂×D₃	—	14	2	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	7	2	no	—
235	C₃×C₄	—	14	1	no	—
236	C₃×C₄	—	14	1	no	—
237	C₃×C₄	—	14	1	no	—
238	C₃×C₄	Q₈×D₇	1	2	yes	—
239	Dic₃	—	2	2	no	—
240	Dic₃	—	2	2	no	—
241	Dic₃	—	14	2	no	—
242	C₂³	—	21	2	no	—
243	Q₈	—	14	2	no	—
244	Q₈	—	84	2	no	—
245	D₄	—	21	4	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	7	1	no	—
249	C₂×C₄	—	7	1	no	—
250	C₂×C₄	—	7	1	no	—
251	C₂×C₄	—	7	1	no	—
252	C₂×C₄	—	21	2	no	—
253	C₂×C₄	—	42	4	no	—
254	C₂×C₄	—	84	2	no	—
255	C₂×C₄	—	84	2	no	—
256	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
257	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
258	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
259	C₇	#5579	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
261	C₂×C₃	D₇×#41	1	1	yes	characteristic
262	C₂×C₃	#6553	1	1	yes	characteristic
263	D₃	—	14	4	no	—
264	C₂²	—	21	4	no	—
265	C₂²	—	42	2	no	—
266	C₂²	—	42	2	no	—
267	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	6	2	no	—
271	C₄	—	6	2	no	—
272	C₄	—	7	2	no	—
273	C₄	—	14	1	no	—
274	C₄	—	14	1	no	—
275	C₄	—	14	1	no	—
276	C₄	—	42	2	no	—
277	C₄	D₂₁⋊Q₈	1	2	yes	—
278	C₃	#90381	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	42	4	no	—
280	C₂	C₂×D₂₁⋊Q₈	1	1	yes	characteristic
281	C₂	D₇×#235	1	1	yes	characteristic
282	C₂	#10621566	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₄²×C₇	C₂²
prime radical	C₂×D₈₄	C₂²
hypercenter	C₄²	D₃×D₇
hypocenter	C₃×C₇	#554
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₇×#870	C₂	semidirect
3. C₃×C₄×Dic₁₄	C₂	semidirect
4. #10621030	C₂	semidirect
5. #10620991	C₂	semidirect
8. #10620973	C₂	semidirect
9. #10620966	C₂	semidirect
10. #10620950	C₂	semidirect
11. #10620939	C₂	semidirect
46. C₄×Dic₁₄	D₃	semidirect
53. D₈₄	Q₈	semidirect
97. #870	D₇	semidirect
200. C₃×C₇	#554	semidirect
259. C₇	#5579	semidirect
278. C₃	#90381	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₈₄	2
C₇×#870	2
C₃×C₄×Dic₁₄	2
#10621030	2
#10620991	2
#10620980	2
#10620973	2
#10620966	2
#10620950	2
#10620939	2
#90381	3
#5579	7