Subgroups — #13059599

Order 1344.

468 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	#10621413	C₂	1	1	yes	maximal, characteristic
2	C₇×#924	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621114	C₂	1	2	yes	maximal, complemented
5	#10621108	C₂	1	2	yes	maximal, complemented
6	#10620980	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620964	C₂	1	2	yes	maximal, complemented
8	#10620950	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620949	C₂	1	2	yes	maximal, complemented
10	#10620937	C₂	1	1	yes	maximal, characteristic, complemented
11	Dic₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#90315	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×#4268	C₂²	1	2	yes	—
15	C₂×D₈₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
19	C₂×D₂₁⋊C₄	C₂²	1	2	yes	—
20	#66631	C₂²	1	2	yes	—
21	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
22	C₇×#217	C₂²	1	2	yes	—
23	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
26	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
27	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	Dic₃⋊Dic₇	C₂²	1	2	yes	—
31	#66577	C₂²	1	2	yes	—
32	#66575	C₂²	1	2	yes	—
33	#66575	C₂²	1	2	yes	—
34	#66575	C₂²	1	2	yes	—
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	C₇×C₂²⋊Q₈	—	3	1	no	—
37	#6510	—	3	1	no	—
38	Q₈×Dic₇	—	3	1	no	—
39	#6493	—	3	1	no	—
40	C₂×C₄×Dic₇	D₃	1	1	yes	characteristic, complemented
41	#6461	—	3	2	no	—
42	#6456	—	3	1	no	—
43	#6455	—	3	2	no	—
44	#6453	—	3	1	no	—
45	#6449	—	3	2	no	—
46	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
47	#5479	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×D₂₁	C₂³	1	2	yes	—
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	1	yes	characteristic
53	#4268	—	2	4	no	—
54	C₂×Dic₂₁	C₂³	1	2	yes	—
55	D₈₄	—	2	2	no	—
56	C₂×C₇×Dic₃	C₂³	1	2	yes	—
57	C₂×C₇×Dic₃	C₂³	1	2	yes	—
58	C₇×Dic₆	—	2	2	no	—
59	C₂×C₃×Dic₇	—	2	2	no	—
60	C₂×C₃×Dic₇	—	2	2	no	—
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
65	D₂₁⋊C₄	—	2	4	no	—
66	C₂×C₇×Q₈	—	3	1	no	—
67	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
68	C₂×C₂²⋊D₇	—	3	2	no	—
69	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
70	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
71	C₂×D₂₈	—	3	1	no	—
72	C₂×C₄×D₇	—	3	2	no	—
73	C₇×C₄⋊C₄	—	3	2	no	—
74	C₇×C₄⋊C₄	—	3	1	no	—
75	C₇×C₂²⋊C₄	—	3	2	no	—
76	(C₂×D₇)⋊C₄	—	3	2	no	—
77	(C₂×D₇)⋊C₄	—	3	2	no	—
78	(C₂×D₇)⋊C₄	—	3	2	no	—
79	(C₂×D₇)⋊C₄	—	3	2	no	—
80	C₄⋊Dic₇	—	3	1	no	—
81	C₄⋊Dic₇	—	3	2	no	—
82	C₄⋊Dic₇	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	C₄×Dic₇	—	3	2	no	—
85	C₄×Dic₇	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	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
89	C₂×C₃×C₄²	—	7	1	no	—
90	#930	—	7	1	no	—
91	#929	—	7	2	no	—
92	C₄×#67	—	7	2	no	—
93	#924	D₇	1	1	yes	characteristic, complemented
94	#876	—	7	2	no	—
95	#875	—	7	1	no	—
96	C₄×D₁₂	—	7	1	no	—
97	#872	—	7	2	no	—
98	#870	—	7	1	no	—
99	C₄×Dic₆	—	7	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	—	2	1	no	—
102	C₂²×C₃×C₇	#41	1	2	yes	—
103	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×D₂₁	—	2	2	no	—
105	C₂×D₂₁	—	2	2	no	—
106	C₂×D₂₁	—	2	2	no	—
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×C₄×C₇	#41	1	2	yes	—
109	Dic₂₁	—	2	2	no	—
110	C₃×Dic₇	—	2	1	no	—
111	C₃×Dic₇	—	2	1	no	—
112	C₃×Dic₇	—	2	1	no	—
113	C₃×Dic₇	#41	1	2	yes	—
114	C₇×Dic₃	—	2	2	no	—
115	C₇×Dic₃	—	2	2	no	—
116	#529	—	21	1	no	Sylow
117	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂²×D₇	—	3	2	no	—
119	C₇×Q₈	—	6	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	3	2	no	—
122	C₂×C₄×C₇	—	3	2	no	—
123	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂²⋊D₇	—	6	4	no	—
126	C₂×Dic₇	—	2	2	no	—
127	C₂×Dic₇	—	2	2	no	—
128	C₂×Dic₇	—	3	2	no	—
129	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
130	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
131	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
132	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
133	D₂₈	—	6	2	no	—
134	C₄×D₇	—	6	4	no	—
135	C₂²×C₃×C₄	—	7	1	no	—
136	C₂²×C₃×C₄	—	7	1	no	—
137	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
138	C₂×#67	—	7	2	no	—
139	C₂×D₁₂	—	7	1	no	—
140	C₂×C₄×D₃	—	7	2	no	—
141	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
142	C₃×C₄²	—	7	1	no	—
143	C₃×C₄²	—	7	1	no	—
144	C₃×C₄²	—	7	1	no	—
145	C₃×C₄²	—	7	1	no	—
146	#217	C₂×D₇	1	2	yes	—
147	(C₂×D₃)⋊C₄	—	7	2	no	—
148	(C₂×D₃)⋊C₄	—	7	2	no	—
149	(C₂×D₃)⋊C₄	—	7	2	no	—
150	(C₂×D₃)⋊C₄	—	7	2	no	—
151	C₄⋊Dic₃	—	7	1	no	—
152	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
153	Dic₃⋊C₄	—	7	2	no	—
154	Dic₃⋊C₄	—	7	2	no	—
155	Dic₃⋊C₄	—	7	2	no	—
156	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
157	C₄×Dic₃	—	7	2	no	—
158	C₂×C₃×C₇	—	2	2	no	—
159	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
160	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
162	D₂₁	—	4	2	no	—
163	#126	—	21	2	no	—
164	#125	—	21	1	no	—
165	#124	—	21	1	no	—
166	#123	—	21	2	no	—
167	C₂²⋊Q₈	—	3	1	no	—
168	#121	—	21	1	no	—
169	C₄×Q₈	—	21	1	no	—
170	C₄×D₄	—	21	1	no	—
171	C₄×D₄	—	21	2	no	—
172	#117	—	21	2	no	—
173	C₂×C₄²	—	7	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	#235	1	2	yes	—
177	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
178	C₂×D₇	—	6	2	no	—
179	C₂×D₇	—	6	2	no	—
180	C₂×D₇	—	6	2	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	6	2	no	—
183	C₄×C₇	—	6	2	no	—
184	C₄×C₇	#235	1	2	yes	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	2	1	no	—
187	Dic₇	—	2	1	no	—
188	Dic₇	—	6	2	no	—
189	Dic₇	#235	1	2	yes	—
190	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
191	C₂²×D₃	—	7	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	7	1	no	—
194	C₂×C₃×C₄	—	7	1	no	—
195	C₂×C₃×C₄	—	7	1	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	14	2	no	—
198	C₂×C₃×C₄	—	14	2	no	—
199	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
201	#67	—	14	4	no	—
202	C₂×Dic₃	—	7	2	no	—
203	C₂×Dic₃	C₂²×D₇	1	2	yes	—
204	C₂×Dic₃	C₂²×D₇	1	2	yes	—
205	D₁₂	—	14	2	no	—
206	C₄×D₃	—	14	4	no	—
207	Dic₆	—	2	2	no	—
208	C₃×C₇	#529	1	1	yes	characteristic, complemented, Hall
209	C₂×Q₈	—	3	1	no	—
210	C₂×D₄	—	21	2	no	—
211	C₂×D₄	—	21	1	no	—
212	C₂²×C₄	—	7	1	no	—
213	C₂²×C₄	—	7	1	no	—
214	C₂²×C₄	—	21	2	no	—
215	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
216	C₄⋊C₄	—	3	2	no	—
217	C₄⋊C₄	—	3	1	no	—
218	C₄⋊C₄	—	21	2	no	—
219	C₄⋊C₄	—	21	2	no	—
220	C₄⋊C₄	—	21	2	no	—
221	C₄⋊C₄	—	21	1	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₂²⋊C₄	—	21	2	no	—
224	C₂²⋊C₄	—	21	2	no	—
225	C₂²⋊C₄	—	21	2	no	—
226	C₂²⋊C₄	—	21	2	no	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	1	no	—
229	C₄²	—	7	1	no	—
230	C₄²	—	7	1	no	—
231	C₄²	—	21	2	no	—
232	C₂×C₇	—	2	2	no	—
233	C₂×C₇	C₂×#235	1	1	yes	characteristic
234	C₂×C₇	C₂×#235	1	1	yes	characteristic
235	C₂×C₇	C₂×#235	1	1	yes	characteristic
236	D₇	—	12	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	#1161	1	2	yes	—
240	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
241	C₂×D₃	—	14	2	no	—
242	C₂×D₃	—	14	2	no	—
243	C₂×D₃	—	14	2	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	7	2	no	—
246	C₃×C₄	—	14	1	no	—
247	C₃×C₄	—	14	1	no	—
248	C₃×C₄	—	14	1	no	—
249	C₃×C₄	#1159	1	2	yes	—
250	Dic₃	—	2	2	no	—
251	Dic₃	—	2	2	no	—
252	Dic₃	—	14	2	no	—
253	C₂³	—	21	2	no	—
254	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
255	Q₈	—	6	2	no	—
256	D₄	—	42	4	no	—
257	D₄	—	42	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	—	3	2	no	—
261	C₂×C₄	—	7	1	no	—
262	C₂×C₄	—	7	1	no	—
263	C₂×C₄	—	7	1	no	—
264	C₂×C₄	—	7	1	no	—
265	C₂×C₄	—	14	2	no	—
266	C₂×C₄	—	14	2	no	—
267	C₂×C₄	—	21	2	no	—
268	C₂×C₄	—	42	4	no	—
269	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
270	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
271	C₇	#5479	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×#1161	1	1	yes	characteristic
275	C₂×C₃	D₇×#41	1	1	yes	characteristic
276	D₃	—	28	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	42	2	no	—
280	C₂²	—	42	2	no	—
281	C₂²	—	42	2	no	—
282	C₂²	#66677	1	2	yes	—
283	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	6	2	no	—
286	C₄	—	6	2	no	—
287	C₄	—	7	2	no	—
288	C₄	—	14	1	no	—
289	C₄	—	14	1	no	—
290	C₄	—	14	1	no	—
291	C₄	—	42	2	no	—
292	C₄	#66686	1	2	yes	—
293	C₃	#90315	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	84	2	no	—
296	C₂	C₂×#66677	1	1	yes	characteristic
297	C₂	D₇×#235	1	1	yes	characteristic
298	C₂	C₂×#66686	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₂²×C₃×C₄×C₇	C₂²
prime radical	#10621413	C₂
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#529
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₇×#924	C₂	semidirect
3. C₂×C₃×C₄×Dic₇	C₂	semidirect
4. #10621114	C₂	semidirect
5. #10621108	C₂	semidirect
6. #10620980	C₂	semidirect
7. #10620964	C₂	semidirect
8. #10620950	C₂	semidirect
9. #10620949	C₂	semidirect
10. #10620937	C₂	semidirect
11. Dic₆×Dic₇	C₂	semidirect
40. C₂×C₄×Dic₇	D₃	semidirect
93. #924	D₇	semidirect
208. C₃×C₇	#529	semidirect
271. C₇	#5479	semidirect
293. C₃	#90315	semidirect

Maximal subgroups

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

subgroup	index
#10621413	2
C₇×#924	2
C₂×C₃×C₄×Dic₇	2
#10621114	2
#10621108	2
#10620980	2
#10620964	2
#10620950	2
#10620949	2
#10620937	2
Dic₆×Dic₇	2
#90315	3
#5479	7