Subgroups — #13058711

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	#10621373	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6456	C₂	1	1	yes	maximal, characteristic
4	C₄×#4248	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620996	C₂	1	1	yes	maximal, characteristic
6	Dic₃×D₂₈	C₂	1	1	yes	maximal, characteristic
7	#10620984	C₂	1	2	yes	maximal, complemented
8	#10620983	C₂	1	2	yes	maximal, complemented
9	D₇×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
10	#10620950	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620921	C₂	1	2	yes	maximal, complemented
12	#90421	—	3	1	no	maximal, Hall
13	C₂×C₇×Dic₆	C₂²	1	2	yes	—
14	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
16	C₂×#4248	C₂²	1	1	yes	characteristic
17	C₂×Dic₃×D₇	C₂²	1	2	yes	—
18	C₂×Dic₃×D₇	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₂₁	C₂²	1	2	yes	—
22	#66629	C₂²	1	2	yes	—
23	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
24	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
27	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
28	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	Dic₃⋊Dic₇	C₂²	1	1	yes	characteristic
31	#66577	C₂²	1	2	yes	—
32	#66577	C₂²	1	1	yes	characteristic
33	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
34	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
35	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
36	C₇×C₄⋊Q₈	—	3	1	no	—
37	#6509	—	3	2	no	—
38	#6509	—	3	2	no	—
39	#6460	—	3	2	no	—
40	#6456	—	3	1	no	—
41	#6456	D₃	1	1	yes	characteristic
42	D₇×C₄⋊C₄	—	3	2	no	—
43	#6453	—	3	1	no	—
44	#6453	—	3	1	no	—
45	C₄×D₂₈	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5539	—	7	1	no	maximal, Hall
48	C₂²×C₃×D₇	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	2	yes	—
53	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
54	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
55	C₂×C₇×Dic₃	C₂³	1	2	yes	—
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₇×Dic₆	—	4	2	no	—
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₃×D₂₈	Q₈	1	4	yes	complemented
61	C₃×C₄×D₇	—	4	2	no	—
62	#4248	—	2	2	no	—
63	Dic₃×D₇	—	2	4	no	—
64	Dic₃×D₇	—	4	2	no	—
65	C₂×C₇×Q₈	—	3	2	no	—
66	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄×D₇	—	3	2	no	—
68	C₂×C₄×D₇	—	3	2	no	—
69	C₂×C₄×D₇	C₂×D₃	1	2	yes	—
70	C₂×Dic₁₄	—	3	1	no	—
71	C₇×C₄⋊C₄	—	3	2	no	—
72	C₇×C₄⋊C₄	—	3	1	no	—
73	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
74	C₄²×C₇	—	3	1	no	—
75	(C₂×D₇)⋊C₄	—	3	2	no	—
76	(C₂×D₇)⋊C₄	—	3	2	no	—
77	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	—
78	C₄⋊Dic₇	—	3	1	no	—
79	C₄⋊Dic₇	—	3	2	no	—
80	C₄⋊Dic₇	—	3	1	no	—
81	C₄⋊Dic₇	—	3	2	no	—
82	Dic₇⋊C₄	—	3	2	no	—
83	Dic₇⋊C₄	—	3	1	no	—
84	Dic₇⋊C₄	—	3	1	no	—
85	Dic₇⋊C₄	—	3	2	no	—
86	C₄×Dic₇	—	3	1	no	—
87	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
88	C₃×C₄×D₄	—	7	1	no	—
89	D₄×Dic₃	—	7	1	no	—
90	D₄×Dic₃	—	7	1	no	—
91	#924	—	7	2	no	—
92	C₂×Dic₃⋊C₄	—	7	2	no	—
93	#890	—	7	1	no	—
94	#888	D₇	1	1	yes	characteristic, complemented
95	#878	—	7	2	no	—
96	#878	—	7	2	no	—
97	#870	—	7	1	no	—
98	C₄×Dic₆	—	7	1	no	—
99	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×D₇	—	2	2	no	—
101	C₂×C₃×D₇	—	2	2	no	—
102	C₂×C₃×D₇	C₂×Q₈	1	4	yes	—
103	C₃×C₄×C₇	—	2	2	no	—
104	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
105	Dic₂₁	—	2	2	no	—
106	Dic₂₁	—	2	1	no	—
107	Dic₂₁	—	2	1	no	—
108	C₃×Dic₇	—	2	1	no	—
109	C₃×Dic₇	—	2	1	no	—
110	C₇×Dic₃	—	2	2	no	—
111	C₇×Dic₃	—	2	1	no	—
112	C₇×Dic₃	#41	1	2	yes	—
113	#554	—	21	1	no	Sylow
114	C₂²×D₇	C₂²×D₃	1	2	yes	—
115	C₇×Q₈	—	12	2	no	—
116	C₂×C₄×C₇	—	3	1	no	—
117	C₂×C₄×C₇	—	3	1	no	—
118	C₂×C₄×C₇	—	3	2	no	—
119	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
120	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
121	C₂×Dic₇	—	3	1	no	—
122	C₂×Dic₇	—	3	1	no	—
123	C₂×Dic₇	—	3	2	no	—
124	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
125	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
126	D₂₈	Dic₆	1	4	yes	complemented
127	C₄×D₇	—	4	2	no	—
128	C₄×D₇	—	6	4	no	—
129	C₄×D₇	—	12	2	no	—
130	Dic₁₄	—	6	2	no	—
131	C₂×C₃×D₄	—	7	1	no	—
132	C₂²×C₃×C₄	—	7	2	no	—
133	C₂²×Dic₃	—	7	2	no	—
134	C₂²×Dic₃	—	7	2	no	—
135	C₂×Dic₆	—	7	1	no	—
136	C₂×Dic₆	C₂×D₇	1	2	yes	—
137	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
138	C₃×C₂²⋊C₄	—	7	2	no	—
139	C₃×C₄²	—	7	1	no	—
140	#217	—	7	2	no	—
141	#217	—	7	2	no	—
142	C₄⋊Dic₃	—	7	1	no	—
143	C₄⋊Dic₃	—	7	2	no	—
144	C₄⋊Dic₃	—	7	1	no	—
145	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	—	7	1	no	—
148	Dic₃⋊C₄	—	7	2	no	—
149	Dic₃⋊C₄	—	7	1	no	—
150	Dic₃⋊C₄	—	7	2	no	—
151	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
152	C₄×Dic₃	—	7	1	no	—
153	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
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	C₃×D₇	—	2	4	no	—
158	C₄⋊Q₈	—	3	1	no	—
159	#125	—	21	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	—	21	1	no	—
165	C₄×D₄	—	7	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	—	2	2	no	—
171	C₂×D₇	—	2	2	no	—
172	C₂×D₇	C₂×Dic₆	1	4	yes	—
173	C₄×C₇	—	2	2	no	—
174	C₄×C₇	—	3	2	no	—
175	C₄×C₇	—	6	2	no	—
176	C₄×C₇	—	6	1	no	—
177	C₄×C₇	C₂×Dic₆	1	2	yes	—
178	Dic₇	—	2	1	no	—
179	Dic₇	—	2	1	no	—
180	Dic₇	—	6	1	no	—
181	Dic₇	—	6	1	no	—
182	Dic₇	—	6	2	no	—
183	C₂³×C₃	—	7	2	no	—
184	C₃×D₄	—	7	4	no	—
185	C₂×C₃×C₄	—	7	1	no	—
186	C₂×C₃×C₄	—	7	1	no	—
187	C₂×C₃×C₄	—	28	2	no	—
188	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
190	C₂×Dic₃	—	7	1	no	—
191	C₂×Dic₃	—	7	1	no	—
192	C₂×Dic₃	—	7	2	no	—
193	C₂×Dic₃	—	14	4	no	—
194	C₂×Dic₃	—	28	2	no	—
195	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
196	C₂×Dic₃	C₂²×D₇	1	2	yes	—
197	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
198	Dic₆	—	4	2	no	—
199	Dic₆	—	14	2	no	—
200	C₃×C₇	#554	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	3	2	no	—
202	C₂×Q₈	—	21	1	no	—
203	C₂×D₄	—	7	1	no	—
204	C₂²×C₄	—	7	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	—	21	2	no	—
210	C₄⋊C₄	—	21	1	no	—
211	C₄⋊C₄	—	21	1	no	—
212	C₄⋊C₄	—	21	1	no	—
213	C₄⋊C₄	—	21	2	no	—
214	C₄⋊C₄	—	21	2	no	—
215	C₄⋊C₄	—	21	1	no	—
216	C₄⋊C₄	—	21	2	no	—
217	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	21	2	no	—
220	C₂²⋊C₄	—	21	2	no	—
221	C₄²	—	3	1	no	—
222	C₄²	—	7	1	no	—
223	C₄²	—	21	1	no	—
224	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
225	C₂×C₇	#1004	1	1	yes	characteristic
226	C₂×C₇	D₃×#41	1	1	yes	characteristic
227	D₇	—	2	4	no	—
228	C₂²×C₃	—	7	4	no	—
229	C₂²×C₃	—	14	2	no	—
230	C₂²×C₃	—	14	2	no	—
231	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	14	1	no	—
234	C₃×C₄	—	14	1	no	—
235	C₃×C₄	Q₈×D₇	1	2	yes	—
236	Dic₃	—	2	1	no	—
237	Dic₃	—	2	2	no	—
238	Dic₃	—	14	1	no	—
239	Dic₃	—	14	1	no	—
240	Dic₃	—	14	2	no	—
241	Dic₃	#1161	1	2	yes	—
242	C₂³	—	7	2	no	—
243	Q₈	—	12	2	no	—
244	Q₈	—	42	2	no	—
245	D₄	—	7	4	no	—
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	7	1	no	—
250	C₂×C₄	—	7	1	no	—
251	C₂×C₄	—	21	1	no	—
252	C₂×C₄	—	21	1	no	—
253	C₂×C₄	—	21	2	no	—
254	C₂×C₄	—	28	2	no	—
255	C₂×C₄	—	42	4	no	—
256	C₂×C₄	—	84	2	no	—
257	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
258	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
259	C₇	#5539	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	—	14	4	no	—
261	C₂×C₃	#6548	1	1	yes	characteristic
262	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
263	C₂×C₃	C₂×#1161	1	1	yes	characteristic
264	C₂²	—	7	4	no	—
265	C₂²	—	14	2	no	—
266	C₂²	—	14	2	no	—
267	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	3	2	no	—
270	C₄	—	6	2	no	—
271	C₄	—	6	1	no	—
272	C₄	—	14	1	no	—
273	C₄	—	14	1	no	—
274	C₄	—	42	1	no	—
275	C₄	—	42	2	no	—
276	C₄	—	42	1	no	—
277	C₄	D₇×Dic₆	1	2	yes	—
278	C₃	#90421	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	14	4	no	—
280	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
281	C₂	#10621572	1	1	yes	characteristic
282	C₂	D₃×#1161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×C₄⋊C₄	C₂²
prime radical	C₂×C₃×D₂₈	C₂²
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	#554
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #10621373	C₂	semidirect
2. C₇×#888	C₂	semidirect
4. C₄×#4248	C₂	semidirect
7. #10620984	C₂	semidirect
8. #10620983	C₂	semidirect
9. D₇×Dic₃⋊C₄	C₂	semidirect
10. #10620950	C₂	semidirect
11. #10620921	C₂	semidirect
60. C₃×D₂₈	Q₈	semidirect
94. #888	D₇	semidirect
126. D₂₈	Dic₆	semidirect
200. C₃×C₇	#554	semidirect
259. C₇	#5539	semidirect
278. C₃	#90421	semidirect

Maximal subgroups

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

subgroup	index
#10621373	2
C₇×#888	2
C₃×#6456	2
C₄×#4248	2
#10620996	2
Dic₃×D₂₈	2
#10620984	2
#10620983	2
D₇×Dic₃⋊C₄	2
#10620950	2
#10620921	2
#90421	3
#5539	7