Subgroups — #13060554

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1010	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621566	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621432	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6513	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621082	C₂	1	2	yes	maximal, complemented
6	D₃×#1143	C₂	1	2	yes	maximal, complemented
7	#10621057	C₂	1	2	yes	maximal, complemented
8	#10621042	C₂	1	2	yes	maximal, complemented
9	C₂×#66568	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620869	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620848	C₂	1	1	yes	maximal, characteristic, complemented
12	#90528	—	3	1	no	maximal, Hall
13	C₃×C₇×#41	C₂²	1	1	yes	characteristic, complemented
14	C₂×D₈₄	C₂²	1	1	yes	characteristic
15	D₃×C₇×Q₈	C₂²	1	2	yes	complemented
16	C₇×#237	—	2	2	no	—
17	C₇×#237	C₂²	1	2	yes	complemented
18	C₇×#235	—	2	1	no	—
19	C₇×#235	C₂²	1	1	yes	characteristic
20	C₂×C₇×Dic₆	—	2	1	no	—
21	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic, complemented
22	C₃×#1157	C₂²	1	1	yes	characteristic
23	#66694	—	2	2	no	—
24	D₃×D₂₈	C₂²	1	2	yes	—
25	D₇×D₁₂	—	2	1	no	—
26	#66677	—	2	1	no	—
27	#66673	C₂²	1	2	yes	—
28	#66634	C₂²	1	2	yes	complemented
29	#66632	C₂²	1	2	yes	complemented
30	#66627	C₂²	1	1	yes	characteristic, complemented
31	C₃×#1144	C₂²	1	1	yes	characteristic, complemented
32	C₃×#1143	C₂²	1	1	yes	characteristic, complemented
33	C₃×#1142	C₂²	1	1	yes	characteristic, complemented
34	C₃×#1141	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₃×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	#66569	C₂²	1	2	yes	complemented
37	#66568	C₂²	1	2	yes	complemented
38	#66568	C₂²	1	2	yes	complemented
39	#66563	C₂²	1	2	yes	complemented
40	#66558	C₂²	1	2	yes	complemented
41	D₃×C₇⋊C₈	C₂²	1	2	yes	complemented
42	C₇×2₋¹⁺⁴	—	3	1	no	—
43	#6553	—	3	1	no	—
44	#6513	—	3	2	no	—
45	#6513	D₃	1	1	yes	characteristic, complemented
46	#6512	—	3	1	no	—
47	#6512	—	3	2	no	—
48	#6511	—	3	1	no	—
49	#6505	—	3	1	no	—
50	#6505	—	3	2	no	—
51	C₂×#1143	—	3	2	no	—
52	C₂×#1143	—	3	1	no	—
53	#5770	—	7	1	no	maximal, Hall
54	C₂²×D₂₁	—	2	1	no	—
55	C₂×D₃×D₇	—	2	2	no	—
56	C₃×C₇×Q₈	C₂³	1	1	yes	characteristic
57	C₃×C₇×D₄	—	2	1	no	—
58	C₃×C₇×D₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	—	2	1	no	—
60	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
61	D₈₄	—	2	1	no	—
62	D₈₄	C₂³	1	2	yes	—
63	C₇×#67	—	2	2	no	—
64	C₇×#67	D₄	1	2	yes	—
65	C₂×C₇×Dic₃	—	2	1	no	—
66	C₂×C₇×Dic₃	—	2	2	no	—
67	C₇×D₁₂	D₄	1	1	yes	characteristic
68	C₄×D₃×C₇	—	2	2	no	—
69	C₄×D₃×C₇	C₂³	1	2	yes	—
70	C₇×Dic₆	—	2	1	no	—
71	C₇×Dic₆	—	2	2	no	—
72	C₇×Dic₆	D₄	1	1	yes	characteristic
73	C₇×Dic₆	C₂³	1	2	yes	—
74	C₃×C₂²⋊D₇	—	2	1	no	—
75	C₃×D₂₈	C₂³	1	1	yes	characteristic
76	C₃×C₄×D₇	—	2	1	no	—
77	C₃×Dic₁₄	C₂³	1	1	yes	characteristic
78	#4247	—	2	2	no	—
79	#4246	—	2	2	no	—
80	D₂₁⋊C₄	—	2	2	no	—
81	C₃⋊(C₇⋊C₈)	C₂³	1	2	yes	—
82	C₃×C₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₃×C₇⋊C₈	C₂³	1	1	yes	characteristic
84	C₇×#41	—	3	1	no	—
85	C₇×#41	—	3	2	no	—
86	C₇×#41	—	6	1	no	—
87	C₇×#41	—	6	2	no	—
88	C₇×#41	C₂×D₃	1	1	yes	characteristic, complemented
89	C₂×C₇×Q₈	—	3	2	no	—
90	C₂×C₇×Q₈	—	3	1	no	—
91	C₂×C₇×Q₈	—	6	1	no	—
92	#1161	—	6	1	no	—
93	D₄×D₇	—	6	1	no	—
94	D₄×D₇	—	6	2	no	—
95	#1157	—	3	2	no	—
96	#1157	C₂×D₃	1	1	yes	characteristic
97	C₂×D₂₈	—	3	1	no	—
98	C₂×D₂₈	—	3	2	no	—
99	#1144	—	3	2	no	—
100	#1144	C₂×D₃	1	1	yes	characteristic, complemented
101	#1143	—	3	2	no	—
102	#1143	—	3	2	no	—
103	#1143	—	3	2	no	—
104	#1143	—	3	2	no	—
105	#1143	C₂×D₃	1	1	yes	characteristic, complemented
106	#1142	C₂×D₃	1	1	yes	characteristic, complemented
107	#1141	—	3	2	no	—
108	#1141	C₂×D₃	1	1	yes	characteristic, complemented
109	#1136	—	3	1	no	—
110	#1136	—	3	2	no	—
111	C₂×C₇⋊C₈	—	3	2	no	—
112	C₂×C₇⋊C₈	C₂×D₃	1	1	yes	characteristic, complemented
113	#1010	D₇	1	1	yes	characteristic, complemented
114	#1009	—	7	1	no	—
115	C₃×#135	—	7	1	no	—
116	#951	—	7	1	no	—
117	#949	—	7	1	no	—
118	#918	—	7	2	no	—
119	#916	—	7	2	no	—
120	D₃×SD₁₆	—	7	2	no	—
121	#911	—	7	2	no	—
122	C₂×#204	—	7	1	no	—
123	#901	—	7	1	no	—
124	C₂²×C₃×C₇	—	2	1	no	—
125	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
126	C₂×D₂₁	—	2	2	no	—
127	C₂×D₂₁	—	8	1	no	—
128	C₂×D₃×C₇	C₂×D₄	1	2	yes	—
129	C₂×C₃×D₇	—	2	1	no	—
130	D₃×D₇	—	8	2	no	—
131	C₃×C₄×C₇	—	2	1	no	—
132	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
134	C₃×Dic₇	—	2	1	no	—
135	C₇×Dic₃	—	2	2	no	—
136	C₇×Dic₃	C₂×D₄	1	2	yes	—
137	#577	—	21	1	no	Sylow
138	C₂²×D₇	—	6	2	no	—
139	C₂²×D₇	—	6	1	no	—
140	C₇×Q₈	—	3	1	no	—
141	C₇×Q₈	—	3	2	no	—
142	C₇×Q₈	—	6	2	no	—
143	C₇×Q₈	—	6	1	no	—
144	C₇×Q₈	C₂²×D₃	1	1	yes	characteristic
145	C₇×D₄	—	2	1	no	—
146	C₇×D₄	—	3	2	no	—
147	C₇×D₄	—	3	1	no	—
148	C₇×D₄	—	6	2	no	—
149	C₇×D₄	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₄×C₇	—	2	1	no	—
151	C₂×C₄×C₇	—	3	2	no	—
152	C₂×C₄×C₇	—	6	2	no	—
153	C₂×C₄×C₇	—	6	2	no	—
154	C₂×C₄×C₇	—	6	1	no	—
155	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
156	C₂²⋊D₇	—	2	1	no	—
157	C₂²⋊D₇	—	6	2	no	—
158	D₂₈	—	3	2	no	—
159	D₂₈	—	6	2	no	—
160	D₂₈	—	6	1	no	—
161	D₂₈	C₂²×D₃	1	1	yes	characteristic
162	C₄×D₇	—	2	1	no	—
163	C₄×D₇	—	6	2	no	—
164	Dic₁₄	C₂²×D₃	1	1	yes	characteristic
165	C₇⋊C₈	—	3	2	no	—
166	C₇⋊C₈	C₂²×D₃	1	1	yes	characteristic
167	C₇⋊C₈	C₂²×D₃	1	1	yes	characteristic
168	C₃×#41	—	7	1	no	—
169	C₃×#41	C₂×D₇	1	1	yes	characteristic, complemented
170	#239	—	7	2	no	—
171	D₃×Q₈	C₂×D₇	1	2	yes	complemented
172	#237	—	2	2	no	—
173	#237	C₂×D₇	1	2	yes	complemented
174	D₃×D₄	—	7	2	no	—
175	D₃×D₄	—	14	2	no	—
176	#235	—	2	1	no	—
177	#235	—	14	1	no	—
178	#235	C₂×D₇	1	1	yes	characteristic
179	C₂×D₁₂	—	7	1	no	—
180	C₂×D₁₂	—	14	1	no	—
181	C₂×Dic₆	—	2	1	no	—
182	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic, complemented
183	C₃×Q₁₆	—	7	1	no	—
184	C₃×SD₁₆	—	7	1	no	—
185	C₃×SD₁₆	—	7	1	no	—
186	C₃×D₈	—	7	1	no	—
187	C₂×C₃×C₈	—	7	1	no	—
188	#216	—	7	2	no	—
189	#215	—	7	2	no	—
190	#214	—	7	2	no	—
191	#213	—	7	2	no	—
192	#208	—	7	1	no	—
193	#204	—	7	2	no	—
194	#204	—	7	2	no	—
195	ΓL(1,𝔽₂₅)	—	7	2	no	—
196	D₃×C₈	—	7	2	no	—
197	C₂×C₃×C₇	—	2	1	no	—
198	C₂×C₃×C₇	—	4	1	no	—
199	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
200	D₂₁	—	4	2	no	—
201	C₃×D₇	—	4	1	no	—
202	D₃×C₇	—	2	2	no	—
203	2₋¹⁺⁴	—	3	1	no	—
204	H(5,𝔽₂)	—	21	1	no	—
205	#137	—	21	2	no	—
206	#137	—	21	1	no	—
207	AGL(1,ℤ/8)	—	21	2	no	—
208	AGL(1,ℤ/8)	—	21	1	no	—
209	#135	—	7	1	no	—
210	#135	—	21	2	no	—
211	C₂×SD₁₆	—	21	1	no	—
212	C₂×SD₁₆	—	21	2	no	—
213	#131	—	21	1	no	—
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	3	2	no	—
216	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
217	C₂×D₇	—	2	1	no	—
218	C₂×D₇	—	6	2	no	—
219	C₂×D₇	—	24	2	no	—
220	C₂×D₇	—	24	1	no	—
221	C₄×C₇	—	2	1	no	—
222	C₄×C₇	—	3	2	no	—
223	C₄×C₇	—	6	2	no	—
224	C₄×C₇	D₃×D₄	1	1	yes	characteristic
225	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
226	Dic₇	—	2	1	no	—
227	C₂²×D₃	—	14	1	no	—
228	C₂²×D₃	—	14	2	no	—
229	C₃×Q₈	—	7	1	no	—
230	C₃×Q₈	C₂²×D₇	1	1	yes	characteristic
231	C₃×D₄	—	2	1	no	—
232	C₃×D₄	—	7	1	no	—
233	C₃×D₄	—	14	1	no	—
234	C₃×D₄	C₂²×D₇	1	1	yes	characteristic
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	14	1	no	—
237	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
238	#67	—	2	2	no	—
239	#67	—	14	2	no	—
240	#67	C₂²⋊D₇	1	2	yes	—
241	C₂×Dic₃	—	2	1	no	—
242	C₂×Dic₃	—	2	2	no	—
243	D₁₂	—	7	2	no	—
244	D₁₂	—	14	2	no	—
245	D₁₂	—	14	1	no	—
246	D₁₂	C₂²⋊D₇	1	1	yes	characteristic
247	C₄×D₃	—	2	2	no	—
248	C₄×D₃	—	14	2	no	—
249	C₄×D₃	C₂²×D₇	1	2	yes	—
250	Dic₆	—	2	1	no	—
251	Dic₆	—	2	2	no	—
252	Dic₆	C₂²⋊D₇	1	1	yes	characteristic
253	Dic₆	C₂²×D₇	1	2	yes	—
254	C₃×C₈	—	7	1	no	—
255	C₃×C₈	—	7	1	no	—
256	C₃⋊C₈	—	7	2	no	—
257	C₃×C₇	#577	1	1	yes	characteristic, complemented, Hall
258	#41	—	3	1	no	—
259	#41	—	3	2	no	—
260	#41	—	6	1	no	—
261	#41	—	6	2	no	—
262	#41	—	7	1	no	—
263	#41	—	21	2	no	—
264	#41	—	42	1	no	—
265	#41	D₃×D₇	1	1	yes	characteristic, complemented
266	C₂×Q₈	—	3	2	no	—
267	C₂×Q₈	—	3	1	no	—
268	C₂×Q₈	—	6	1	no	—
269	C₂×D₄	—	21	2	no	—
270	C₂×D₄	—	21	1	no	—
271	C₂×D₄	—	42	1	no	—
272	C₂×D₄	—	42	2	no	—
273	Q₁₆	—	7	1	no	—
274	Q₁₆	—	21	2	no	—
275	SD₁₆	—	7	1	no	—
276	SD₁₆	—	7	1	no	—
277	SD₁₆	—	21	2	no	—
278	SD₁₆	—	21	2	no	—
279	SD₁₆	—	21	2	no	—
280	SD₁₆	—	21	2	no	—
281	D₈	—	7	1	no	—
282	D₈	—	21	2	no	—
283	M₁₆	—	21	2	no	—
284	M₁₆	—	21	1	no	—
285	C₂×C₈	—	7	1	no	—
286	C₂×C₈	—	21	2	no	—
287	C₂×C₇	—	2	1	no	—
288	C₂×C₇	—	4	1	no	—
289	C₂×C₇	—	6	2	no	—
290	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
291	D₇	—	4	1	no	—
292	D₇	—	12	2	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	14	1	no	—
295	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
296	C₂×D₃	—	14	2	no	—
297	C₂×D₃	—	56	2	no	—
298	C₂×D₃	—	56	1	no	—
299	C₂×D₃	C₂×C₂²⋊D₇	1	2	yes	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	14	1	no	—
302	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
303	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
304	Dic₃	—	2	2	no	—
305	Dic₃	C₂×C₂²⋊D₇	1	2	yes	—
306	C₂³	—	42	2	no	—
307	C₂³	—	42	1	no	—
308	Q₈	—	3	1	no	—
309	Q₈	—	3	2	no	—
310	Q₈	—	6	2	no	—
311	Q₈	—	6	1	no	—
312	Q₈	—	7	1	no	—
313	Q₈	C₂×D₃×D₇	1	1	yes	characteristic
314	D₄	—	2	1	no	—
315	D₄	—	3	2	no	—
316	D₄	—	3	1	no	—
317	D₄	—	6	2	no	—
318	D₄	—	7	1	no	—
319	D₄	—	14	1	no	—
320	D₄	—	21	2	no	—
321	D₄	—	42	2	no	—
322	D₄	—	42	2	no	—
323	D₄	—	42	1	no	—
324	D₄	C₂×D₃×D₇	1	1	yes	characteristic
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	3	2	no	—
327	C₂×C₄	—	6	2	no	—
328	C₂×C₄	—	6	2	no	—
329	C₂×C₄	—	6	1	no	—
330	C₂×C₄	—	14	1	no	—
331	C₂×C₄	—	42	2	no	—
332	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
333	C₈	—	7	1	no	—
334	C₈	—	7	1	no	—
335	C₈	—	21	2	no	—
336	C₇	#5770	1	1	yes	characteristic, complemented, Sylow
337	C₂×C₃	—	2	1	no	—
338	C₂×C₃	—	4	1	no	—
339	C₂×C₃	—	28	1	no	—
340	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
341	D₃	—	2	2	no	—
342	D₃	—	28	2	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	3	2	no	—
345	C₂²	—	14	1	no	—
346	C₂²	—	42	2	no	—
347	C₂²	—	168	1	no	—
348	C₂²	—	168	2	no	—
349	C₂²	D₃×C₂²⋊D₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	3	2	no	—
352	C₄	—	6	2	no	—
353	C₄	—	14	1	no	—
354	C₄	D₃×C₂²⋊D₇	1	1	yes	characteristic
355	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
356	C₃	#90528	1	1	yes	characteristic, complemented, Sylow
357	C₂	—	2	1	no	—
358	C₂	—	4	1	no	—
359	C₂	—	6	2	no	—
360	C₂	—	28	1	no	—
361	C₂	—	84	2	no	—
362	C₂	C₂×D₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×C₂²⋊D₇
commutator, cosocle	C₃×C₄×C₇	C₂⁴
Frattini	C₄	C₂²×D₃×D₇
Fitting	C₃×C₇×#41	C₂²
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	#41	D₃×D₇
hypocenter	C₃×C₇	#577
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₇×#1010	C₂	semidirect
2. #10621566	C₂	semidirect
3. #10621432	C₂	semidirect
4. C₃×#6513	C₂	semidirect
5. #10621082	C₂	semidirect
6. D₃×#1143	C₂	semidirect
7. #10621057	C₂	semidirect
8. #10621042	C₂	semidirect
9. C₂×#66568	C₂	semidirect
10. #10620869	C₂	semidirect
11. #10620848	C₂	semidirect
13. C₃×C₇×#41	C₂²	semidirect
15. D₃×C₇×Q₈	C₂²	semidirect
17. C₇×#237	C₂²	semidirect
21. C₂×C₇×Dic₆	C₂²	semidirect
28. #66634	C₂²	semidirect
29. #66632	C₂²	semidirect
30. #66627	C₂²	semidirect
31. C₃×#1144	C₂²	semidirect
32. C₃×#1143	C₂²	semidirect
33. C₃×#1142	C₂²	semidirect
34. C₃×#1141	C₂²	semidirect
35. C₂×C₃×C₇⋊C₈	C₂²	semidirect
36. #66569	C₂²	semidirect
37. #66568	C₂²	semidirect
38. #66568	C₂²	semidirect
39. #66563	C₂²	semidirect
40. #66558	C₂²	semidirect
41. D₃×C₇⋊C₈	C₂²	semidirect
45. #6513	D₃	semidirect
88. C₇×#41	C₂×D₃	semidirect
100. #1144	C₂×D₃	semidirect
105. #1143	C₂×D₃	semidirect
106. #1142	C₂×D₃	semidirect
108. #1141	C₂×D₃	semidirect
112. C₂×C₇⋊C₈	C₂×D₃	semidirect
113. #1010	D₇	semidirect
169. C₃×#41	C₂×D₇	semidirect
171. D₃×Q₈	C₂×D₇	semidirect
173. #237	C₂×D₇	semidirect
182. C₂×Dic₆	C₂×D₇	semidirect
257. C₃×C₇	#577	semidirect
265. #41	D₃×D₇	semidirect
336. C₇	#5770	semidirect
356. C₃	#90528	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1010	2
#10621566	2
#10621432	2
C₃×#6513	2
#10621082	2
D₃×#1143	2
#10621057	2
#10621042	2
C₂×#66568	2
#10620869	2
#10620848	2
#90528	3
#5770	7