Subgroups — #13059504

Order 1344.

584 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66684	C₂	1	2	yes	maximal, complemented
2	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#930	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621105	C₂	1	2	yes	maximal, complemented
6	#10621030	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621011	C₂	1	1	yes	maximal, characteristic, complemented
8	#10620979	C₂	1	2	yes	maximal, complemented
9	D₁₂×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620976	C₂	1	2	yes	maximal, complemented
11	D₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#90302	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₁	C₂²	1	2	yes	—
15	C₂×C₇×#67	C₂²	1	2	yes	—
16	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
19	C₂×#4248	C₂²	1	2	yes	—
20	C₂×#4245	C₂²	1	2	yes	—
21	C₂×D₃×Dic₇	C₂²	1	2	yes	—
22	#66684	—	2	8	no	—
23	C₂×Dic₃×D₇	C₂²	1	2	yes	—
24	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₁	C₂²	1	2	yes	—
26	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
27	C₇×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
30	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
31	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
32	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
33	Dic₇⋊Dic₃	C₂²	1	2	yes	—
34	Dic₇⋊Dic₃	C₂²	1	2	yes	—
35	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
36	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
37	C₂×#1159	—	3	2	no	—
38	C₇×#121	—	3	1	no	—
39	#6501	—	3	1	no	—
40	#6498	—	3	2	no	—
41	D₄×Dic₇	—	3	1	no	—
42	C₄×C₂²⋊D₇	D₃	1	1	yes	characteristic, complemented
43	C₂×C₄⋊Dic₇	—	3	1	no	—
44	D₇×C₄⋊C₄	—	3	1	no	—
45	#6451	—	3	1	no	—
46	#6446	—	3	2	no	—
47	#6441	—	3	2	no	—
48	#5555	—	7	1	no	maximal, Hall
49	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
50	C₂²×D₃×C₇	C₂³	1	2	yes	—
51	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
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	C₂×Dic₂₁	—	2	4	no	—
56	C₂×Dic₂₁	C₂³	1	2	yes	—
57	C₂×Dic₂₁	C₂³	1	2	yes	—
58	C₇×#67	—	2	4	no	—
59	C₂×C₇×Dic₃	C₂³	1	2	yes	—
60	C₇×D₁₂	—	2	2	no	—
61	C₃×C₂²⋊D₇	D₄	1	4	yes	complemented
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	C₃×C₄×D₇	—	2	2	no	—
66	#4248	—	2	4	no	—
67	#4245	—	2	4	no	—
68	D₃×Dic₇	—	2	4	no	—
69	Dic₃×D₇	—	2	4	no	—
70	C₂×C₇×D₄	—	3	2	no	—
71	C₂×C₇×D₄	—	3	1	no	—
72	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
73	C₂×C₂²⋊D₇	—	3	2	no	—
74	C₂×C₂²⋊D₇	C₂×D₃	1	1	yes	characteristic
75	C₂²×Dic₇	—	3	2	no	—
76	C₂²×Dic₇	—	3	2	no	—
77	#1159	—	6	8	no	—
78	C₂×C₄×D₇	—	3	2	no	—
79	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
80	C₂×Dic₁₄	—	3	2	no	—
81	C₇×C₄⋊C₄	—	3	1	no	—
82	C₇×C₂²⋊C₄	—	3	2	no	—
83	C₂²⋊Dic₇	—	3	2	no	—
84	C₂²⋊Dic₇	—	3	2	no	—
85	C₂²⋊Dic₇	C₂×D₃	1	1	yes	characteristic
86	(C₂×D₇)⋊C₄	C₂×D₃	1	1	yes	characteristic
87	C₄⋊Dic₇	—	3	2	no	—
88	C₄⋊Dic₇	—	3	1	no	—
89	C₄⋊Dic₇	—	3	1	no	—
90	Dic₇⋊C₄	—	3	2	no	—
91	Dic₇⋊C₄	—	3	2	no	—
92	Dic₇⋊C₄	C₂×D₃	1	1	yes	characteristic
93	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
94	C₂×#237	—	7	2	no	—
95	C₃×C₄×D₄	—	7	1	no	—
96	#930	—	7	1	no	—
97	#930	D₇	1	1	yes	characteristic, complemented
98	C₂×C₄⋊Dic₃	—	7	1	no	—
99	C₂×C₄⋊Dic₃	—	7	1	no	—
100	#897	—	7	2	no	—
101	#886	—	7	2	no	—
102	#886	—	7	2	no	—
103	C₄×D₁₂	—	7	1	no	—
104	#869	—	7	1	no	—
105	C₂²×C₃×C₇	—	2	1	no	—
106	C₂²×C₃×C₇	—	2	1	no	—
107	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
108	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
109	C₂×D₃×C₇	—	2	2	no	—
110	C₂×D₃×C₇	—	2	2	no	—
111	C₂×D₃×C₇	—	2	2	no	—
112	C₂×C₃×D₇	—	2	1	no	—
113	C₂×C₃×D₇	—	2	1	no	—
114	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
115	C₃×C₄×C₇	—	2	1	no	—
116	C₃×C₄×C₇	#41	1	2	yes	complemented
117	Dic₂₁	—	2	2	no	—
118	Dic₂₁	—	2	2	no	—
119	C₃×Dic₇	—	2	1	no	—
120	C₃×Dic₇	—	2	1	no	—
121	C₃×Dic₇	C₂×D₄	1	2	yes	—
122	C₇×Dic₃	—	2	2	no	—
123	#547	—	21	1	no	Sylow
124	C₂³×C₇	—	3	2	no	—
125	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
126	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
127	C₇×D₄	—	6	4	no	—
128	C₇×D₄	—	6	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	3	2	no	—
131	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
133	C₂²⋊D₇	—	6	4	no	—
134	C₂²⋊D₇	D₁₂	1	4	yes	complemented
135	C₂×Dic₇	—	3	2	no	—
136	C₂×Dic₇	—	3	2	no	—
137	C₂×Dic₇	—	6	4	no	—
138	C₂×Dic₇	—	6	4	no	—
139	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
140	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
141	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
142	C₄×D₇	—	2	2	no	—
143	C₄×D₇	—	6	4	no	—
144	Dic₁₄	—	6	4	no	—
145	C₂×C₃×D₄	—	7	1	no	—
146	C₂²×C₃×C₄	—	7	1	no	—
147	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
148	C₂×#67	—	7	2	no	—
149	C₂×#67	C₂×D₇	1	2	yes	—
150	C₂²×Dic₃	—	7	2	no	—
151	C₂²×Dic₃	—	7	2	no	—
152	#237	—	14	8	no	—
153	C₂×D₁₂	C₂×D₇	1	1	yes	characteristic
154	C₂×C₄×D₃	—	7	2	no	—
155	C₂×Dic₆	—	7	2	no	—
156	C₃×C₄⋊C₄	—	7	1	no	—
157	C₃×C₂²⋊C₄	—	7	1	no	—
158	C₃×C₂²⋊C₄	—	7	1	no	—
159	C₃×C₄²	—	7	1	no	—
160	(C₂×D₃)⋊C₄	—	7	2	no	—
161	(C₂×D₃)⋊C₄	—	7	2	no	—
162	(C₂×D₃)⋊C₄	C₂×D₇	1	2	yes	—
163	C₄⋊Dic₃	—	7	1	no	—
164	C₄⋊Dic₃	—	7	1	no	—
165	C₄⋊Dic₃	—	7	2	no	—
166	C₄⋊Dic₃	—	7	2	no	—
167	C₄⋊Dic₃	—	7	2	no	—
168	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
169	C₂×C₃×C₇	—	2	2	no	—
170	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
172	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
173	C₃×D₇	—	2	2	no	—
174	D₃×C₇	—	4	2	no	—
175	C₂×#41	—	21	2	no	—
176	C₄⋊Q₈	—	21	1	no	—
177	#123	—	21	2	no	—
178	#123	—	21	2	no	—
179	C₂²⋊Q₈	—	21	2	no	—
180	#121	—	3	1	no	—
181	#121	—	21	1	no	—
182	C₄×D₄	—	7	1	no	—
183	C₄×D₄	—	21	1	no	—
184	C₂×C₄⋊C₄	—	21	1	no	—
185	C₂×C₄⋊C₄	—	21	1	no	—
186	C₂²×C₇	—	2	1	no	—
187	C₂²×C₇	—	2	1	no	—
188	C₂²×C₇	—	6	2	no	—
189	C₂²×C₇	—	6	2	no	—
190	C₂²×C₇	—	6	2	no	—
191	C₂²×C₇	C₂×D₁₂	1	2	yes	—
192	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
193	C₂×D₇	—	2	1	no	—
194	C₂×D₇	—	2	1	no	—
195	C₂×D₇	C₂×D₁₂	1	2	yes	—
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	6	2	no	—
198	C₄×C₇	#237	1	2	yes	complemented
199	Dic₇	—	2	1	no	—
200	Dic₇	—	2	1	no	—
201	Dic₇	—	6	2	no	—
202	Dic₇	—	6	2	no	—
203	Dic₇	C₂×D₁₂	1	2	yes	—
204	C₂³×C₃	—	7	1	no	—
205	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
206	C₂²×D₃	C₂²×D₇	1	2	yes	—
207	C₃×D₄	—	7	4	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	7	1	no	—
210	C₂×C₃×C₄	—	7	1	no	—
211	C₂×C₃×C₄	—	7	1	no	—
212	C₂×C₃×C₄	—	14	2	no	—
213	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
215	#67	—	2	4	no	—
216	#67	—	14	4	no	—
217	C₂×Dic₃	—	7	2	no	—
218	C₂×Dic₃	—	7	2	no	—
219	C₂×Dic₃	—	14	4	no	—
220	C₂×Dic₃	—	14	4	no	—
221	C₂×Dic₃	C₂²×D₇	1	2	yes	—
222	D₁₂	—	2	2	no	—
223	C₄×D₃	—	14	4	no	—
224	Dic₆	—	14	4	no	—
225	C₃×C₇	#547	1	1	yes	characteristic, complemented, Hall
226	#41	—	42	8	no	—
227	C₂×Q₈	—	21	2	no	—
228	C₂×D₄	—	3	2	no	—
229	C₂×D₄	—	3	1	no	—
230	C₂×D₄	—	7	1	no	—
231	C₂×D₄	—	21	2	no	—
232	C₂²×C₄	—	7	1	no	—
233	C₂²×C₄	—	21	2	no	—
234	C₂²×C₄	—	21	2	no	—
235	C₂²×C₄	—	21	2	no	—
236	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
237	C₄⋊C₄	—	3	1	no	—
238	C₄⋊C₄	—	7	1	no	—
239	C₄⋊C₄	—	21	2	no	—
240	C₄⋊C₄	—	21	2	no	—
241	C₄⋊C₄	—	21	1	no	—
242	C₄⋊C₄	—	21	1	no	—
243	C₄⋊C₄	—	21	2	no	—
244	C₂²⋊C₄	—	3	2	no	—
245	C₂²⋊C₄	—	7	1	no	—
246	C₂²⋊C₄	—	7	1	no	—
247	C₂²⋊C₄	—	21	2	no	—
248	C₂²⋊C₄	—	21	2	no	—
249	C₄²	—	7	1	no	—
250	C₂×C₇	—	2	2	no	—
251	C₂×C₇	—	12	2	no	—
252	C₂×C₇	C₂²×D₁₂	1	1	yes	characteristic
253	C₂×C₇	C₂×#237	1	1	yes	characteristic
254	C₂×C₇	#1010	1	1	yes	characteristic
255	D₇	—	2	2	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	7	2	no	—
259	C₂²×C₃	—	14	1	no	—
260	C₂²×C₃	—	14	1	no	—
261	C₂²×C₃	D₄×D₇	1	2	yes	—
262	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
263	C₂×D₃	—	2	2	no	—
264	C₂×D₃	—	2	2	no	—
265	C₂×D₃	—	2	2	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	7	2	no	—
268	C₃×C₄	—	14	1	no	—
269	C₃×C₄	—	14	1	no	—
270	C₃×C₄	#1159	1	2	yes	complemented
271	Dic₃	—	2	2	no	—
272	Dic₃	—	14	2	no	—
273	Dic₃	—	14	2	no	—
274	C₂³	—	3	2	no	—
275	C₂³	—	7	1	no	—
276	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
277	Q₈	—	42	4	no	—
278	D₄	—	6	4	no	—
279	D₄	—	6	2	no	—
280	D₄	—	7	4	no	—
281	D₄	—	42	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	3	2	no	—
284	C₂×C₄	—	7	1	no	—
285	C₂×C₄	—	7	1	no	—
286	C₂×C₄	—	7	1	no	—
287	C₂×C₄	—	14	2	no	—
288	C₂×C₄	—	21	2	no	—
289	C₂×C₄	—	21	2	no	—
290	C₂×C₄	—	42	4	no	—
291	C₂×C₄	—	42	4	no	—
292	C₂×C₄	—	42	4	no	—
293	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
294	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
295	C₇	#5555	1	1	yes	characteristic, complemented, Sylow
296	C₂×C₃	—	2	2	no	—
297	C₂×C₃	—	14	2	no	—
298	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
299	C₂×C₃	C₂×#1159	1	1	yes	characteristic
300	C₂×C₃	#6554	1	1	yes	characteristic
301	D₃	—	4	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	6	2	no	—
305	C₂²	—	6	2	no	—
306	C₂²	—	6	2	no	—
307	C₂²	—	7	2	no	—
308	C₂²	—	14	1	no	—
309	C₂²	—	14	1	no	—
310	C₂²	D₇×D₁₂	1	2	yes	—
311	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	6	2	no	—
314	C₄	—	7	2	no	—
315	C₄	—	14	1	no	—
316	C₄	—	14	1	no	—
317	C₄	—	42	2	no	—
318	C₄	—	42	2	no	—
319	C₄	#66684	1	2	yes	complemented
320	C₃	#90302	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	2	2	no	—
322	C₂	—	12	2	no	—
323	C₂	—	14	2	no	—
324	C₂	#10621548	1	1	yes	characteristic
325	C₂	C₂×D₇×D₁₂	1	1	yes	characteristic
326	C₂	C₂×#66684	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	#547
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×#66684	C₂	semidirect
2. C₂×C₄⋊Dic₂₁	C₂	semidirect
3. C₇×#930	C₂	semidirect
4. C₃×C₄×C₂²⋊D₇	C₂	semidirect
5. #10621105	C₂	semidirect
6. #10621030	C₂	semidirect
7. #10621011	C₂	semidirect
8. #10620979	C₂	semidirect
9. D₁₂×Dic₇	C₂	semidirect
10. #10620976	C₂	semidirect
11. D₇×C₄⋊Dic₃	C₂	semidirect
42. C₄×C₂²⋊D₇	D₃	semidirect
61. C₃×C₂²⋊D₇	D₄	semidirect
97. #930	D₇	semidirect
116. C₃×C₄×C₇	#41	semidirect
134. C₂²⋊D₇	D₁₂	semidirect
198. C₄×C₇	#237	semidirect
225. C₃×C₇	#547	semidirect
270. C₃×C₄	#1159	semidirect
295. C₇	#5555	semidirect
319. C₄	#66684	semidirect
320. C₃	#90302	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66684	2
C₂×C₄⋊Dic₂₁	2
C₇×#930	2
C₃×C₄×C₂²⋊D₇	2
#10621105	2
#10621030	2
#10621011	2
#10620979	2
D₁₂×Dic₇	2
#10620976	2
D₇×C₄⋊Dic₃	2
#90302	3
#5555	7