Subgroups — #13061627

Order 1344.

580 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×#66728	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621422	C₂	1	2	yes	maximal, complemented
5	#10621413	C₂	1	1	yes	maximal, characteristic, complemented
6	#10621407	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621379	C₂	1	2	yes	maximal, complemented
8	#10621368	C₂	1	2	yes	maximal, complemented
9	#10621364	C₂	1	2	yes	maximal, complemented
10	#10621358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#90243	—	3	1	no	maximal, Hall
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂×#4268	C₂²	1	2	yes	—
17	C₂×#4268	C₂²	1	2	yes	—
18	C₂²×Dic₂₁	C₂²	1	2	yes	—
19	#66728	—	2	4	no	—
20	C₂×D₈₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₁	C₂²	1	2	yes	—
22	C₂×Dic₄₂	—	2	4	no	—
23	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
24	C₂×Dic₄₂	C₂²	1	2	yes	—
25	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
26	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
30	#66636	C₂²	1	2	yes	—
31	#66631	C₂²	1	2	yes	—
32	#66631	C₂²	1	2	yes	—
33	#66631	C₂²	1	2	yes	—
34	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
35	#66629	C₂²	1	2	yes	—
36	#66629	C₂²	1	2	yes	—
37	C₄×Dic₂₁	C₂²	1	2	yes	—
38	C₂×#1157	—	3	1	no	—
39	C₂²×Dic₁₄	—	3	1	no	—
40	C₄×C₇×D₄	D₃	1	1	yes	characteristic, complemented
41	#6502	—	3	2	no	—
42	#6493	—	3	1	no	—
43	#6487	—	3	1	no	—
44	#6459	—	3	2	no	—
45	#6448	—	3	2	no	—
46	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
47	#6438	—	3	1	no	—
48	C₄×Dic₁₄	—	3	1	no	—
49	#5552	—	7	1	no	maximal, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
52	C₂²×D₂₁	C₂³	1	2	yes	—
53	C₃×C₇×D₄	—	2	2	no	—
54	C₂×C₃×C₄×C₇	—	2	2	no	—
55	C₂×C₃×C₄×C₇	—	2	2	no	—
56	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
61	#4268	—	2	4	no	—
62	#4268	—	2	4	no	—
63	C₂×Dic₂₁	—	2	4	no	—
64	C₂×Dic₂₁	C₂³	1	2	yes	—
65	C₂×Dic₂₁	C₂³	1	2	yes	—
66	C₂×Dic₂₁	C₂³	1	2	yes	—
67	D₈₄	—	2	2	no	—
68	C₄×D₂₁	—	2	4	no	—
69	Dic₄₂	—	2	2	no	—
70	Dic₄₂	—	2	4	no	—
71	Dic₄₂	D₄	1	4	yes	complemented
72	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
75	C₂×C₂²⋊D₇	—	3	2	no	—
76	C₂×C₂²⋊D₇	—	3	2	no	—
77	C₂²×Dic₇	—	3	2	no	—
78	#1157	—	6	4	no	—
79	C₂×D₂₈	—	3	1	no	—
80	C₂×C₄×D₇	—	3	2	no	—
81	C₂×Dic₁₄	—	3	1	no	—
82	C₂×Dic₁₄	—	3	1	no	—
83	C₂×Dic₁₄	—	3	2	no	—
84	C₂×Dic₁₄	—	6	4	no	—
85	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
86	C₇×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
87	C₇×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
88	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
89	C₂²⋊Dic₇	—	3	2	no	—
90	(C₂×D₇)⋊C₄	—	3	2	no	—
91	(C₂×D₇)⋊C₄	—	3	2	no	—
92	(C₂×D₇)⋊C₄	—	3	2	no	—
93	C₄⋊Dic₇	—	3	1	no	—
94	Dic₇⋊C₄	—	3	2	no	—
95	Dic₇⋊C₄	—	3	2	no	—
96	C₄×Dic₇	—	3	2	no	—
97	C₂×#235	—	7	1	no	—
98	C₂²×Dic₆	—	7	1	no	—
99	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
100	#939	—	7	2	no	—
101	#930	—	7	1	no	—
102	#924	—	7	1	no	—
103	#896	—	7	2	no	—
104	#885	—	7	2	no	—
105	#67⋊C₄	—	7	2	no	—
106	#875	—	7	1	no	—
107	C₄×Dic₆	—	7	1	no	—
108	C₂²×C₃×C₇	—	2	1	no	—
109	C₂²×C₃×C₇	—	2	1	no	—
110	C₂²×C₃×C₇	—	2	1	no	—
111	C₂²×C₃×C₇	—	2	1	no	—
112	C₂²×C₃×C₇	—	2	1	no	—
113	C₂²×C₃×C₇	#41	1	2	yes	complemented
114	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
115	C₂×D₂₁	—	2	2	no	—
116	C₂×D₂₁	—	2	2	no	—
117	C₂×D₂₁	—	2	2	no	—
118	C₃×C₄×C₇	—	2	1	no	—
119	C₃×C₄×C₇	—	2	1	no	—
120	C₃×C₄×C₇	—	2	1	no	—
121	C₃×C₄×C₇	—	2	1	no	—
122	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
123	Dic₂₁	—	2	2	no	—
124	Dic₂₁	—	2	2	no	—
125	Dic₂₁	C₂×D₄	1	4	yes	—
126	#548	—	21	1	no	Sylow
127	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
128	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
129	C₂²×D₇	—	3	2	no	—
130	C₇×D₄	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
138	C₂²⋊D₇	—	6	4	no	—
139	C₂²⋊D₇	—	6	4	no	—
140	C₂×Dic₇	—	3	2	no	—
141	C₂×Dic₇	—	3	2	no	—
142	C₂×Dic₇	—	3	2	no	—
143	C₂×Dic₇	—	6	4	no	—
144	D₂₈	—	6	2	no	—
145	C₄×D₇	—	6	4	no	—
146	Dic₁₄	—	3	4	no	—
147	Dic₁₄	—	6	4	no	—
148	Dic₁₄	—	6	2	no	—
149	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
152	C₂×#67	—	7	2	no	—
153	C₂×#67	—	7	2	no	—
154	C₂²×Dic₃	—	7	2	no	—
155	#235	—	14	4	no	—
156	C₂×D₁₂	—	7	1	no	—
157	C₂×C₄×D₃	—	7	2	no	—
158	C₂×Dic₆	—	7	1	no	—
159	C₂×Dic₆	—	7	1	no	—
160	C₂×Dic₆	—	7	2	no	—
161	C₂×Dic₆	—	14	4	no	—
162	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
163	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
164	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
165	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
166	#217	—	7	2	no	—
167	(C₂×D₃)⋊C₄	—	7	2	no	—
168	(C₂×D₃)⋊C₄	—	7	2	no	—
169	(C₂×D₃)⋊C₄	—	7	2	no	—
170	C₄⋊Dic₃	—	7	1	no	—
171	Dic₃⋊C₄	—	7	2	no	—
172	Dic₃⋊C₄	—	7	2	no	—
173	C₄×Dic₃	—	7	2	no	—
174	C₂×C₃×C₇	—	2	2	no	—
175	C₂×C₃×C₇	—	4	1	no	—
176	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
178	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
179	D₂₁	—	4	2	no	—
180	C₂×#41	—	21	1	no	—
181	C₂²×Q₈	—	21	1	no	—
182	#124	—	21	2	no	—
183	#124	—	21	1	no	—
184	C₂²⋊Q₈	—	21	1	no	—
185	C₂²⋊Q₈	—	21	2	no	—
186	#121	—	21	2	no	—
187	#121	—	21	1	no	—
188	C₄×Q₈	—	21	1	no	—
189	C₄×D₄	—	21	2	no	—
190	C₄×D₄	D₂₁	1	1	yes	characteristic, complemented
191	C₂²×C₇	—	2	1	no	—
192	C₂²×C₇	—	2	1	no	—
193	C₂²×C₇	—	2	1	no	—
194	C₂²×C₇	—	2	1	no	—
195	C₂²×C₇	—	2	1	no	—
196	C₂²×C₇	#235	1	2	yes	complemented
197	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
198	C₂×D₇	—	6	2	no	—
199	C₂×D₇	—	6	2	no	—
200	C₂×D₇	—	6	2	no	—
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	—	2	1	no	—
204	C₄×C₇	—	2	1	no	—
205	C₄×C₇	D₃×D₄	1	2	yes	—
206	Dic₇	—	3	4	no	—
207	Dic₇	—	6	2	no	—
208	Dic₇	—	6	2	no	—
209	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
210	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
211	C₂²×D₃	—	7	2	no	—
212	C₃×D₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
220	#67	—	14	4	no	—
221	#67	—	14	4	no	—
222	C₂×Dic₃	—	7	2	no	—
223	C₂×Dic₃	—	7	2	no	—
224	C₂×Dic₃	—	7	2	no	—
225	C₂×Dic₃	—	14	4	no	—
226	D₁₂	—	14	2	no	—
227	C₄×D₃	—	14	4	no	—
228	Dic₆	—	7	4	no	—
229	Dic₆	—	14	4	no	—
230	Dic₆	—	14	2	no	—
231	C₃×C₇	#548	1	1	yes	characteristic, complemented, Hall
232	#41	—	42	4	no	—
233	C₂×Q₈	—	21	1	no	—
234	C₂×Q₈	—	21	2	no	—
235	C₂×Q₈	—	21	1	no	—
236	C₂×Q₈	—	42	4	no	—
237	C₂×D₄	—	21	2	no	—
238	C₂×D₄	—	21	2	no	—
239	C₂×D₄	—	21	1	no	—
240	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
241	C₂²×C₄	—	21	2	no	—
242	C₂²×C₄	—	21	2	no	—
243	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
245	C₄⋊C₄	—	21	2	no	—
246	C₄⋊C₄	—	21	2	no	—
247	C₄⋊C₄	—	21	1	no	—
248	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
249	C₂²⋊C₄	—	21	2	no	—
250	C₂²⋊C₄	—	21	2	no	—
251	C₂²⋊C₄	—	21	2	no	—
252	C₂²⋊C₄	—	21	2	no	—
253	C₂²⋊C₄	C₂×D₂₁	1	1	yes	characteristic
254	C₂²⋊C₄	C₂×D₂₁	1	1	yes	characteristic
255	C₄²	—	21	2	no	—
256	C₄²	C₂×D₂₁	1	1	yes	characteristic
257	C₂×C₇	—	2	2	no	—
258	C₂×C₇	—	4	1	no	—
259	C₂×C₇	C₂×#235	1	1	yes	characteristic
260	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
261	C₂×C₇	#1010	1	1	yes	characteristic
262	D₇	—	12	2	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	#1157	1	2	yes	complemented
269	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
270	C₂×D₃	—	14	2	no	—
271	C₂×D₃	—	14	2	no	—
272	C₂×D₃	—	14	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	D₄×D₇	1	2	yes	—
278	Dic₃	—	7	4	no	—
279	Dic₃	—	14	2	no	—
280	Dic₃	—	14	2	no	—
281	C₂³	—	21	2	no	—
282	C₂³	C₂²×D₂₁	1	1	yes	characteristic
283	C₂³	C₂²×D₂₁	1	1	yes	characteristic
284	Q₈	—	21	4	no	—
285	Q₈	—	42	4	no	—
286	Q₈	—	42	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	42	4	no	—
289	D₄	—	42	4	no	—
290	D₄	—	42	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	21	2	no	—
294	C₂×C₄	—	21	2	no	—
295	C₂×C₄	—	21	2	no	—
296	C₂×C₄	—	42	4	no	—
297	C₂×C₄	—	42	4	no	—
298	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
303	C₇	#5552	1	1	yes	characteristic, complemented, Sylow
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	4	1	no	—
306	C₂×C₃	C₂×#1157	1	1	yes	characteristic
307	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
308	C₂×C₃	#6554	1	1	yes	characteristic
309	D₃	—	28	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	42	2	no	—
316	C₂²	—	42	2	no	—
317	C₂²	—	42	2	no	—
318	C₂²	#66728	1	2	yes	complemented
319	C₂²	C₂³×D₂₁	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	21	4	no	—
325	C₄	—	42	2	no	—
326	C₄	—	42	2	no	—
327	C₄	D₄×D₂₁	1	2	yes	—
328	C₃	#90243	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	4	1	no	—
331	C₂	—	84	2	no	—
332	C₂	C₂×#66728	1	1	yes	characteristic
333	C₂	C₂×D₄×D₂₁	1	1	yes	characteristic
334	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₄×C₇×D₄	C₂
hypercenter	C₄×D₄	D₂₁
hypocenter	C₃×C₇	#548
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#66728	C₂	semidirect
2. C₂²×Dic₄₂	C₂	semidirect
3. C₃×C₄×C₇×D₄	C₂	semidirect
4. #10621422	C₂	semidirect
5. #10621413	C₂	semidirect
6. #10621407	C₂	semidirect
7. #10621379	C₂	semidirect
8. #10621368	C₂	semidirect
9. #10621364	C₂	semidirect
10. #10621358	C₂	semidirect
11. C₄×Dic₄₂	C₂	semidirect
40. C₄×C₇×D₄	D₃	semidirect
71. Dic₄₂	D₄	semidirect
99. C₃×C₄×D₄	D₇	semidirect
113. C₂²×C₃×C₇	#41	semidirect
190. C₄×D₄	D₂₁	semidirect
196. C₂²×C₇	#235	semidirect
231. C₃×C₇	#548	semidirect
268. C₂²×C₃	#1157	semidirect
303. C₇	#5552	semidirect
318. C₂²	#66728	semidirect
328. C₃	#90243	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66728	2
C₂²×Dic₄₂	2
C₃×C₄×C₇×D₄	2
#10621422	2
#10621413	2
#10621407	2
#10621379	2
#10621368	2
#10621364	2
#10621358	2
C₄×Dic₄₂	2
#90243	3
#5552	7