Subgroups — #13059535

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66683	C₂	1	2	yes	maximal, complemented
2	#10621413	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6493	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621131	C₂	1	2	yes	maximal, complemented
6	#10621031	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621020	C₂	1	1	yes	maximal, characteristic, complemented
8	#10620990	C₂	1	1	yes	maximal, characteristic, complemented
9	Dic₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620957	C₂	1	2	yes	maximal, complemented
11	#10620915	C₂	1	2	yes	maximal, complemented
12	#90168	—	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	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
17	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
19	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
20	C₂×#4248	C₂²	1	2	yes	—
21	C₂×#4246	C₂²	1	2	yes	—
22	C₂×#4246	C₂²	1	2	yes	—
23	C₂×D₂₁⋊C₄	C₂²	1	2	yes	—
24	#66683	—	2	8	no	—
25	C₂×Dic₃×D₇	C₂²	1	2	yes	—
26	#66631	C₂²	1	2	yes	—
27	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
28	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
29	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
30	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
31	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
32	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
33	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
34	Dic₃⋊Dic₇	C₂²	1	2	yes	—
35	#66575	C₂²	1	2	yes	—
36	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
37	C₂×#1157	—	3	2	no	—
38	C₂×C₄²×C₇	—	3	1	no	—
39	#6493	—	3	1	no	—
40	#6493	—	3	2	no	—
41	#6493	D₃	1	1	yes	characteristic, complemented
42	#6438	—	3	2	no	—
43	#6437	—	3	1	no	—
44	C₄×D₂₈	—	3	1	no	—
45	C₄×D₂₈	—	3	2	no	—
46	C₄×D₂₈	—	3	1	no	—
47	#6432	—	3	1	no	—
48	#5596	—	7	1	no	maximal, Hall
49	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
50	C₂²×D₂₁	C₂³	1	2	yes	—
51	C₂²×C₃×D₇	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₇	—	2	2	no	—
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	#4268	—	2	4	no	—
56	C₂×Dic₂₁	C₂³	1	2	yes	—
57	D₈₄	—	2	2	no	—
58	C₂×C₇×Dic₃	—	2	2	no	—
59	C₂×C₇×Dic₃	D₄	1	4	yes	complemented
60	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
63	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
64	C₃×C₂²⋊D₇	—	2	4	no	—
65	C₂×C₃×Dic₇	C₂³	1	2	yes	—
66	C₃×D₂₈	—	2	2	no	—
67	#4248	—	2	4	no	—
68	#4246	—	2	4	no	—
69	#4246	—	2	4	no	—
70	D₂₁⋊C₄	—	2	4	no	—
71	Dic₃×D₇	—	2	4	no	—
72	C₂²×C₄×C₇	—	3	1	no	—
73	C₂²×C₄×C₇	—	3	1	no	—
74	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
75	C₂×C₂²⋊D₇	—	3	2	no	—
76	C₂×C₂²⋊D₇	C₂×D₃	1	2	yes	—
77	#1157	—	6	8	no	—
78	C₂×D₂₈	—	3	2	no	—
79	C₂×D₂₈	—	3	2	no	—
80	C₂×D₂₈	—	3	1	no	—
81	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
82	C₂×C₄×D₇	—	3	2	no	—
83	C₂×C₄×D₇	—	3	2	no	—
84	C₂×Dic₁₄	—	3	2	no	—
85	C₄²×C₇	—	3	1	no	—
86	C₄²×C₇	—	3	1	no	—
87	C₄²×C₇	—	3	1	no	—
88	C₄²×C₇	—	3	1	no	—
89	(C₂×D₇)⋊C₄	—	3	2	no	—
90	(C₂×D₇)⋊C₄	—	3	2	no	—
91	(C₂×D₇)⋊C₄	—	3	2	no	—
92	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	—
93	C₄⋊Dic₇	—	3	1	no	—
94	C₄⋊Dic₇	—	3	2	no	—
95	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
96	C₂×#237	—	7	2	no	—
97	C₃×#121	—	7	1	no	—
98	#940	—	7	1	no	—
99	#939	—	7	2	no	—
100	D₄×Dic₃	—	7	1	no	—
101	#930	—	7	1	no	—
102	C₂×C₄×Dic₃	D₇	1	1	yes	characteristic, complemented
103	#893	—	7	1	no	—
104	#888	—	7	1	no	—
105	#885	—	7	2	no	—
106	#67⋊C₄	—	7	2	no	—
107	C₂²×C₃×C₇	—	2	1	no	—
108	C₂²×C₃×C₇	—	2	1	no	—
109	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
110	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
111	C₂×D₂₁	—	2	2	no	—
112	C₂×D₂₁	—	2	2	no	—
113	C₂×D₂₁	—	2	2	no	—
114	C₂×C₃×D₇	—	2	2	no	—
115	C₂×C₃×D₇	—	2	2	no	—
116	C₂×C₃×D₇	—	2	2	no	—
117	C₃×C₄×C₇	—	2	1	no	—
118	C₃×C₄×C₇	#41	1	2	yes	complemented
119	Dic₂₁	—	2	2	no	—
120	C₃×Dic₇	—	2	2	no	—
121	C₇×Dic₃	—	2	1	no	—
122	C₇×Dic₃	C₂×D₄	1	2	yes	—
123	C₇×Dic₃	C₂×D₄	1	2	yes	—
124	C₇×Dic₃	#41	1	2	yes	complemented
125	#532	—	21	1	no	Sylow
126	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
127	C₂²×D₇	—	3	2	no	—
128	C₂²×D₇	C₂²×D₃	1	2	yes	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	3	4	no	—
131	C₂×C₄×C₇	—	3	1	no	—
132	C₂×C₄×C₇	—	3	1	no	—
133	C₂×C₄×C₇	—	3	1	no	—
134	C₂×C₄×C₇	—	3	1	no	—
135	C₂×C₄×C₇	—	6	2	no	—
136	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
138	C₂²⋊D₇	—	2	4	no	—
139	C₂²⋊D₇	—	6	4	no	—
140	C₂×Dic₇	—	3	2	no	—
141	C₂×Dic₇	C₂²×D₃	1	2	yes	—
142	D₂₈	—	2	2	no	—
143	D₂₈	—	6	4	no	—
144	D₂₈	—	6	4	no	—
145	D₂₈	—	6	2	no	—
146	C₄×D₇	—	6	4	no	—
147	C₄×D₇	—	6	4	no	—
148	Dic₁₄	—	6	4	no	—
149	C₂×C₃×D₄	—	7	2	no	—
150	C₂×C₃×D₄	—	7	1	no	—
151	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
152	C₂×#67	—	7	2	no	—
153	C₂×#67	—	7	2	no	—
154	C₂×#67	—	7	2	no	—
155	C₂²×Dic₃	—	7	2	no	—
156	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
157	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
158	#237	—	14	8	no	—
159	C₂×D₁₂	—	7	1	no	—
160	C₂×C₄×D₃	—	7	2	no	—
161	C₂×Dic₆	—	7	2	no	—
162	C₃×C₄⋊C₄	—	7	1	no	—
163	C₃×C₂²⋊C₄	—	7	2	no	—
164	#217	—	7	2	no	—
165	(C₂×D₃)⋊C₄	—	7	2	no	—
166	(C₂×D₃)⋊C₄	—	7	2	no	—
167	C₄⋊Dic₃	—	7	1	no	—
168	Dic₃⋊C₄	—	7	2	no	—
169	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
170	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
171	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
172	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
173	C₂×C₃×C₇	—	2	2	no	—
174	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
177	D₂₁	—	4	2	no	—
178	C₃×D₇	—	4	2	no	—
179	C₂×#41	—	21	2	no	—
180	C₄⋊Q₈	—	21	1	no	—
181	#127	—	21	1	no	—
182	#124	—	21	2	no	—
183	#121	—	7	1	no	—
184	#121	—	21	2	no	—
185	#121	—	21	1	no	—
186	C₄×D₄	—	21	2	no	—
187	C₄×D₄	—	21	1	no	—
188	C₄×D₄	—	21	1	no	—
189	C₂×C₄²	—	3	1	no	—
190	C₂²×C₇	—	2	1	no	—
191	C₂²×C₇	—	2	1	no	—
192	C₂²×C₇	D₃×D₄	1	2	yes	—
193	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
194	C₂×D₇	—	2	2	no	—
195	C₂×D₇	—	2	2	no	—
196	C₂×D₇	—	2	2	no	—
197	C₂×D₇	—	6	2	no	—
198	C₂×D₇	—	6	2	no	—
199	C₂×D₇	—	6	2	no	—
200	C₄×C₇	—	2	1	no	—
201	C₄×C₇	—	3	2	no	—
202	C₄×C₇	—	3	2	no	—
203	C₄×C₇	—	3	2	no	—
204	C₄×C₇	—	6	1	no	—
205	C₄×C₇	#237	1	2	yes	complemented
206	Dic₇	—	2	2	no	—
207	Dic₇	—	6	2	no	—
208	C₂³×C₃	—	7	2	no	—
209	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
210	C₂²×D₃	—	7	2	no	—
211	C₃×D₄	—	14	4	no	—
212	C₃×D₄	—	14	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	7	2	no	—
215	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
217	#67	—	14	4	no	—
218	#67	—	14	4	no	—
219	#67	—	14	4	no	—
220	C₂×Dic₃	—	2	2	no	—
221	C₂×Dic₃	—	7	2	no	—
222	C₂×Dic₃	—	14	4	no	—
223	C₂×Dic₃	D₂₈	1	4	yes	complemented
224	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
225	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
226	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
227	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
228	D₁₂	—	14	2	no	—
229	C₄×D₃	—	14	4	no	—
230	Dic₆	—	14	4	no	—
231	C₃×C₇	#532	1	1	yes	characteristic, complemented, Hall
232	#41	—	42	8	no	—
233	C₂×Q₈	—	21	2	no	—
234	C₂×D₄	—	7	2	no	—
235	C₂×D₄	—	7	1	no	—
236	C₂×D₄	—	21	2	no	—
237	C₂×D₄	—	21	2	no	—
238	C₂×D₄	—	21	2	no	—
239	C₂×D₄	—	21	1	no	—
240	C₂²×C₄	—	3	1	no	—
241	C₂²×C₄	—	3	1	no	—
242	C₂²×C₄	—	21	2	no	—
243	C₂²×C₄	—	21	2	no	—
244	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
245	C₄⋊C₄	—	7	1	no	—
246	C₄⋊C₄	—	21	2	no	—
247	C₄⋊C₄	—	21	1	no	—
248	C₂²⋊C₄	—	7	2	no	—
249	C₂²⋊C₄	—	21	2	no	—
250	C₂²⋊C₄	—	21	2	no	—
251	C₂²⋊C₄	—	21	2	no	—
252	C₄²	—	3	1	no	—
253	C₄²	—	3	1	no	—
254	C₄²	—	3	1	no	—
255	C₄²	—	3	1	no	—
256	C₂×C₇	—	2	2	no	—
257	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
258	C₂×C₇	C₂×#237	1	1	yes	characteristic
259	C₂×C₇	D₃×#41	1	1	yes	characteristic
260	D₇	—	4	2	no	—
261	D₇	—	12	2	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	14	2	no	—
265	C₂²×C₃	—	14	2	no	—
266	C₂²×C₃	—	14	2	no	—
267	C₂²×C₃	C₂×D₂₈	1	2	yes	—
268	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
269	C₂×D₃	—	14	2	no	—
270	C₂×D₃	—	14	2	no	—
271	C₂×D₃	—	14	2	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	14	2	no	—
274	C₃×C₄	#1157	1	2	yes	complemented
275	Dic₃	—	2	1	no	—
276	Dic₃	—	14	2	no	—
277	Dic₃	C₂×D₂₈	1	2	yes	—
278	Dic₃	C₂×D₂₈	1	2	yes	—
279	Dic₃	#1157	1	2	yes	complemented
280	C₂³	—	7	2	no	—
281	C₂³	—	21	2	no	—
282	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
283	Q₈	—	42	4	no	—
284	D₄	—	14	4	no	—
285	D₄	—	14	2	no	—
286	D₄	—	42	4	no	—
287	D₄	—	42	4	no	—
288	D₄	—	42	4	no	—
289	D₄	—	42	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	3	4	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	—	7	2	no	—
298	C₂×C₄	—	21	2	no	—
299	C₂×C₄	—	42	4	no	—
300	C₂×C₄	—	42	4	no	—
301	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
303	C₇	#5596	1	1	yes	characteristic, complemented, Sylow
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	28	2	no	—
306	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
307	C₂×C₃	C₂×#1157	1	1	yes	characteristic
308	C₂×C₃	C₂×#1157	1	1	yes	characteristic
309	D₃	—	28	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	14	2	no	—
313	C₂²	—	14	2	no	—
314	C₂²	—	14	2	no	—
315	C₂²	—	42	2	no	—
316	C₂²	—	42	2	no	—
317	C₂²	—	42	2	no	—
318	C₂²	D₃×D₂₈	1	2	yes	—
319	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	3	2	no	—
322	C₄	—	3	2	no	—
323	C₄	—	3	2	no	—
324	C₄	—	6	1	no	—
325	C₄	—	14	2	no	—
326	C₄	—	42	2	no	—
327	C₄	#66683	1	2	yes	complemented
328	C₃	#90168	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	28	2	no	—
331	C₂	—	84	2	no	—
332	C₂	C₂×D₃×D₂₈	1	1	yes	characteristic
333	C₂	D₃×#1157	1	1	yes	characteristic
334	C₂	C₂×#66683	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
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#532
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
1. C₂×#66683	C₂	semidirect
2. #10621413	C₂	semidirect
3. C₂×C₄×C₇×Dic₃	C₂	semidirect
4. C₃×#6493	C₂	semidirect
5. #10621131	C₂	semidirect
6. #10621031	C₂	semidirect
7. #10621020	C₂	semidirect
8. #10620990	C₂	semidirect
9. Dic₃×D₂₈	C₂	semidirect
10. #10620957	C₂	semidirect
11. #10620915	C₂	semidirect
41. #6493	D₃	semidirect
59. C₂×C₇×Dic₃	D₄	semidirect
102. C₂×C₄×Dic₃	D₇	semidirect
118. C₃×C₄×C₇	#41	semidirect
124. C₇×Dic₃	#41	semidirect
205. C₄×C₇	#237	semidirect
223. C₂×Dic₃	D₂₈	semidirect
231. C₃×C₇	#532	semidirect
274. C₃×C₄	#1157	semidirect
279. Dic₃	#1157	semidirect
303. C₇	#5596	semidirect
327. C₄	#66683	semidirect
328. C₃	#90168	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66683	2
#10621413	2
C₂×C₄×C₇×Dic₃	2
C₃×#6493	2
#10621131	2
#10621031	2
#10621020	2
#10620990	2
Dic₃×D₂₈	2
#10620957	2
#10620915	2
#90168	3
#5596	7