Subgroups — #13058697

Order 1344.

444 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#6455	C₂	1	1	yes	maximal, characteristic
4	#10621031	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620984	C₂	1	2	yes	maximal
6	#10620983	C₂	1	2	yes	maximal
7	C₄×Dic₃×D₇	C₂	1	1	yes	maximal, characteristic
8	#10620946	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620920	C₂	1	1	yes	maximal, characteristic
10	Dic₆×Dic₇	C₂	1	2	yes	maximal, complemented
11	#10620885	C₂	1	2	yes	maximal, complemented
12	#90419	—	3	1	no	maximal, Hall
13	C₂×C₇×Dic₆	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
15	C₂×#4248	C₂²	1	2	yes	—
16	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₂₁	C₂²	1	2	yes	—
20	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
21	#66629	C₂²	1	2	yes	—
22	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
23	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₇	C₂²	1	2	yes	—
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	Dic₃⋊Dic₇	C₂²	1	2	yes	—
31	Dic₇⋊Dic₃	C₂²	1	2	yes	—
32	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
33	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
34	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	C₇×C₄⋊Q₈	—	3	1	no	—
37	#6509	—	3	2	no	—
38	Q₈×Dic₇	—	3	2	no	—
39	#6460	—	3	2	no	—
40	#6455	—	3	1	no	—
41	#6455	D₃	1	1	yes	characteristic
42	#6453	—	3	1	no	—
43	#6453	—	3	1	no	—
44	#6450	—	3	2	no	—
45	C₄²×D₇	—	3	1	no	—
46	#6432	—	3	1	no	—
47	#5517	—	7	1	no	maximal, Hall
48	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₁	C₂³	1	2	yes	—
52	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
54	C₂×C₇×Dic₃	C₂³	1	2	yes	—
55	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₇×Dic₆	—	2	4	no	—
58	C₂×C₃×Dic₇	C₂³	1	2	yes	—
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₃×C₄×D₇	Q₈	1	4	yes	complemented
61	#4248	—	2	4	no	—
62	Dic₃×D₇	—	2	2	no	—
63	Dic₃×D₇	—	2	2	no	—
64	C₂×C₇×Q₈	—	3	2	no	—
65	C₂×C₄×D₇	—	3	1	no	—
66	C₂×C₄×D₇	—	3	1	no	—
67	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
68	C₂×Dic₁₄	—	3	2	no	—
69	C₇×C₄⋊C₄	—	3	2	no	—
70	C₇×C₄⋊C₄	—	3	1	no	—
71	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
72	C₄²×C₇	—	3	1	no	—
73	(C₂×D₇)⋊C₄	—	3	2	no	—
74	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	—
75	C₄⋊Dic₇	—	3	1	no	—
76	C₄⋊Dic₇	—	3	2	no	—
77	C₄⋊Dic₇	—	3	2	no	—
78	C₄⋊Dic₇	—	3	2	no	—
79	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	C₄×Dic₇	—	3	1	no	—
83	C₄×Dic₇	—	3	1	no	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	—	3	2	no	—
86	C₄×Dic₇	C₂×D₃	1	2	yes	—
87	C₃×#117	—	7	1	no	—
88	#926	—	7	1	no	—
89	C₂×C₄×Dic₃	—	7	1	no	—
90	#890	—	7	1	no	—
91	#890	—	7	1	no	—
92	#888	—	7	1	no	—
93	#888	D₇	1	1	yes	characteristic, complemented
94	#878	—	7	2	no	—
95	#878	—	7	2	no	—
96	C₄×Dic₆	—	7	2	no	—
97	C₄×Dic₆	—	7	2	no	—
98	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×D₇	—	2	1	no	—
100	C₂×C₃×D₇	—	2	1	no	—
101	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
102	C₃×C₄×C₇	—	2	2	no	—
103	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
104	Dic₂₁	—	2	1	no	—
105	Dic₂₁	—	2	1	no	—
106	Dic₂₁	—	2	2	no	—
107	C₃×Dic₇	—	2	2	no	—
108	C₃×Dic₇	C₂×Q₈	1	2	yes	—
109	C₇×Dic₃	—	2	2	no	—
110	C₇×Dic₃	#41	1	2	yes	—
111	C₇×Dic₃	#41	1	2	yes	—
112	#533	—	21	1	no	Sylow
113	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
114	C₇×Q₈	—	6	4	no	—
115	C₂×C₄×C₇	—	3	1	no	—
116	C₂×C₄×C₇	—	3	1	no	—
117	C₂×C₄×C₇	—	3	2	no	—
118	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×Dic₇	—	3	1	no	—
121	C₂×Dic₇	—	3	1	no	—
122	C₂×Dic₇	—	3	2	no	—
123	C₂×Dic₇	C₂²×D₃	1	2	yes	—
124	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
125	C₄×D₇	—	6	2	no	—
126	C₄×D₇	—	6	2	no	—
127	C₄×D₇	Dic₆	1	4	yes	complemented
128	Dic₁₄	—	6	4	no	—
129	C₂²×C₃×C₄	—	7	1	no	—
130	C₂²×Dic₃	—	7	1	no	—
131	C₂²×Dic₃	—	7	1	no	—
132	C₂×Dic₆	—	7	2	no	—
133	C₂×Dic₆	C₂×D₇	1	2	yes	—
134	C₃×C₄⋊C₄	—	7	1	no	—
135	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
136	C₃×C₂²⋊C₄	—	7	2	no	—
137	C₃×C₄²	—	7	2	no	—
138	#217	—	7	2	no	—
139	C₄⋊Dic₃	—	7	2	no	—
140	C₄⋊Dic₃	—	7	1	no	—
141	C₄⋊Dic₃	—	7	2	no	—
142	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
143	Dic₃⋊C₄	—	7	2	no	—
144	Dic₃⋊C₄	—	7	2	no	—
145	Dic₃⋊C₄	—	7	2	no	—
146	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
147	C₄×Dic₃	—	7	1	no	—
148	C₄×Dic₃	—	7	1	no	—
149	C₄×Dic₃	—	7	1	no	—
150	C₄×Dic₃	—	7	2	no	—
151	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
152	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
155	C₃×D₇	—	2	2	no	—
156	C₄⋊Q₈	—	3	1	no	—
157	C₄⋊Q₈	—	21	1	no	—
158	#125	—	21	1	no	—
159	#125	—	21	1	no	—
160	C₂²⋊Q₈	—	21	2	no	—
161	C₂²⋊Q₈	—	21	2	no	—
162	C₄×Q₈	—	21	2	no	—
163	C₄×Q₈	—	21	2	no	—
164	#117	—	7	1	no	—
165	#117	—	21	1	no	—
166	C₂×C₄²	—	21	1	no	—
167	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
168	C₂×D₇	—	2	1	no	—
169	C₂×D₇	—	2	1	no	—
170	C₂×D₇	C₂×Dic₆	1	2	yes	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	3	2	no	—
173	C₄×C₇	—	3	2	no	—
174	C₄×C₇	—	6	2	no	—
175	C₄×C₇	C₂×Dic₆	1	2	yes	—
176	Dic₇	—	2	2	no	—
177	Dic₇	—	6	2	no	—
178	Dic₇	—	6	1	no	—
179	Dic₇	—	6	1	no	—
180	Dic₇	C₂×Dic₆	1	2	yes	—
181	C₂³×C₃	—	7	1	no	—
182	C₂×C₃×C₄	—	7	1	no	—
183	C₂×C₃×C₄	—	7	4	no	—
184	C₂×C₃×C₄	—	7	2	no	—
185	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
187	C₂×Dic₃	—	7	1	no	—
188	C₂×Dic₃	—	7	1	no	—
189	C₂×Dic₃	—	7	2	no	—
190	C₂×Dic₃	—	14	2	no	—
191	C₂×Dic₃	—	14	2	no	—
192	C₂×Dic₃	C₂²×D₇	1	2	yes	—
193	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
194	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
195	Dic₆	—	2	4	no	—
196	Dic₆	—	14	4	no	—
197	C₃×C₇	#533	1	1	yes	characteristic, complemented, Hall
198	C₂×Q₈	—	3	2	no	—
199	C₂×Q₈	—	21	2	no	—
200	C₂²×C₄	—	7	1	no	—
201	C₂²×C₄	—	21	1	no	—
202	C₂²×C₄	—	21	1	no	—
203	C₄⋊C₄	—	3	1	no	—
204	C₄⋊C₄	—	3	2	no	—
205	C₄⋊C₄	—	7	1	no	—
206	C₄⋊C₄	—	21	2	no	—
207	C₄⋊C₄	—	21	2	no	—
208	C₄⋊C₄	—	21	2	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	1	no	—
211	C₄⋊C₄	—	21	2	no	—
212	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
213	C₂²⋊C₄	—	7	2	no	—
214	C₂²⋊C₄	—	21	2	no	—
215	C₄²	—	3	1	no	—
216	C₄²	—	7	2	no	—
217	C₄²	—	21	1	no	—
218	C₄²	—	21	1	no	—
219	C₄²	—	21	2	no	—
220	C₄²	—	21	1	no	—
221	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
222	C₂×C₇	D₃×#41	1	1	yes	characteristic
223	C₂×C₇	D₃×#41	1	1	yes	characteristic
224	D₇	—	2	2	no	—
225	C₂²×C₃	—	7	2	no	—
226	C₂²×C₃	—	14	1	no	—
227	C₂²×C₃	—	14	1	no	—
228	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	7	2	no	—
231	C₃×C₄	—	14	2	no	—
232	C₃×C₄	Q₈×D₇	1	2	yes	—
233	Dic₃	—	2	2	no	—
234	Dic₃	—	14	1	no	—
235	Dic₃	—	14	2	no	—
236	Dic₃	—	14	1	no	—
237	Dic₃	#1159	1	2	yes	—
238	Dic₃	#1161	1	2	yes	—
239	C₂³	—	7	1	no	—
240	Q₈	—	6	4	no	—
241	Q₈	—	42	4	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	7	1	no	—
246	C₂×C₄	—	7	4	no	—
247	C₂×C₄	—	7	2	no	—
248	C₂×C₄	—	21	1	no	—
249	C₂×C₄	—	21	1	no	—
250	C₂×C₄	—	21	2	no	—
251	C₂×C₄	—	42	2	no	—
252	C₂×C₄	—	42	2	no	—
253	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
254	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
255	C₇	#5517	1	1	yes	characteristic, complemented, Sylow
256	C₂×C₃	—	14	2	no	—
257	C₂×C₃	C₂×#1159	1	1	yes	characteristic
258	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
259	C₂×C₃	C₂×#1161	1	1	yes	characteristic
260	C₂²	—	7	2	no	—
261	C₂²	—	14	1	no	—
262	C₂²	—	14	1	no	—
263	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	3	2	no	—
266	C₄	—	3	2	no	—
267	C₄	—	6	2	no	—
268	C₄	—	7	2	no	—
269	C₄	—	14	2	no	—
270	C₄	—	42	1	no	—
271	C₄	—	42	1	no	—
272	C₄	—	42	2	no	—
273	C₄	D₇×Dic₆	1	2	yes	—
274	C₃	#90419	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	14	2	no	—
276	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
277	C₂	D₃×#1159	1	1	yes	characteristic
278	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	C₂²×C₃×D₇	C₂³
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	#533
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. #10621031	C₂	semidirect
8. #10620946	C₂	semidirect
10. Dic₆×Dic₇	C₂	semidirect
11. #10620885	C₂	semidirect
60. C₃×C₄×D₇	Q₈	semidirect
93. #888	D₇	semidirect
127. C₄×D₇	Dic₆	semidirect
197. C₃×C₇	#533	semidirect
255. C₇	#5517	semidirect
274. C₃	#90419	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×#6455	2
#10621031	2
#10620984	2
#10620983	2
C₄×Dic₃×D₇	2
#10620946	2
#10620920	2
Dic₆×Dic₇	2
#10620885	2
#90419	3
#5517	7