Subgroups — D₇×#924 #13059480

Order 1344.

644 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×D₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621407	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#924	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621136	C₂	1	2	yes	maximal, complemented
6	D₇×#217	C₂	1	2	yes	maximal
7	D₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	D₇×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
9	#10620922	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620921	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620911	C₂	1	2	yes	maximal, complemented
12	D₇×C₂²⋊Q₈	—	3	1	no	maximal, Hall
13	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₇	—	2	4	no	—
19	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
23	C₂×#4248	C₂²	1	2	yes	complemented
24	C₂×Dic₃×D₇	C₂²	1	2	yes	—
25	C₂×Dic₃×D₇	C₂²	1	2	yes	—
26	D₇×Dic₆	—	2	4	no	—
27	#66636	C₂²	1	2	yes	—
28	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic, complemented
29	#66629	C₂²	1	2	yes	complemented
30	C₇×#217	C₂²	1	2	yes	—
31	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
32	C₇×Dic₃⋊C₄	C₂²	1	2	yes	complemented
33	Dic₃⋊Dic₇	C₂²	1	2	yes	complemented
34	#66577	C₂²	1	2	yes	complemented
35	Dic₇⋊Dic₃	C₂²	1	2	yes	complemented
36	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
37	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
38	C₂×Q₈×D₇	—	3	1	no	—
39	C₂²×C₄×D₇	D₃	1	1	yes	characteristic
40	C₇×C₂²⋊Q₈	—	3	1	no	—
41	#6509	—	3	1	no	—
42	#6487	—	3	1	no	—
43	#6460	—	3	1	no	—
44	#6459	—	3	2	no	—
45	D₇×C₄⋊C₄	—	3	2	no	—
46	D₇×C₄⋊C₄	—	3	1	no	—
47	D₇×C₂²⋊C₄	—	3	2	no	—
48	#6441	—	3	2	no	—
49	C₂×#924	—	7	1	no	maximal, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₇	—	2	2	no	—
52	C₂²×C₃×D₇	—	2	2	no	—
53	C₂²×C₃×D₇	Q₈	1	4	yes	complemented
54	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
55	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₇	—	2	2	no	—
57	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂×Dic₂₁	C₂³	1	2	yes	—
60	C₂×Dic₂₁	C₂³	1	2	yes	—
61	Dic₄₂	—	2	2	no	—
62	C₂×C₇×Dic₃	C₂³	1	2	yes	—
63	C₂×C₇×Dic₃	C₂³	1	2	yes	—
64	C₇×Dic₆	—	2	2	no	—
65	C₂×C₃×Dic₇	—	2	2	no	—
66	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
68	C₃×C₄×D₇	—	2	2	no	—
69	C₃×C₄×D₇	—	2	2	no	—
70	C₃×C₄×D₇	—	2	2	no	—
71	C₃×C₄×D₇	D₄	1	4	yes	—
72	#4248	—	2	4	no	—
73	Dic₃×D₇	—	2	4	no	—
74	Dic₃×D₇	—	2	4	no	—
75	C₂³×D₇	C₂×D₃	1	1	yes	characteristic
76	C₂×C₇×Q₈	—	3	1	no	—
77	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
78	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
79	Q₈×D₇	—	6	4	no	—
80	C₂×C₄×D₇	—	2	4	no	—
81	C₂×C₄×D₇	—	3	2	no	—
82	C₂×C₄×D₇	—	3	2	no	—
83	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
84	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
85	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
86	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
87	C₂×Dic₁₄	—	3	1	no	—
88	C₂×Dic₁₄	—	3	2	no	—
89	C₇×C₄⋊C₄	—	3	2	no	—
90	C₇×C₄⋊C₄	—	3	1	no	—
91	C₇×C₂²⋊C₄	—	3	2	no	—
92	C₂²⋊Dic₇	—	3	2	no	—
93	(C₂×D₇)⋊C₄	—	3	2	no	—
94	(C₂×D₇)⋊C₄	—	3	2	no	—
95	C₄⋊Dic₇	—	3	1	no	—
96	C₄⋊Dic₇	—	3	2	no	—
97	Dic₇⋊C₄	—	3	2	no	—
98	Dic₇⋊C₄	—	3	2	no	—
99	Dic₇⋊C₄	—	3	2	no	—
100	C₂³×C₃×C₄	—	7	1	no	—
101	C₂²×Dic₆	—	7	1	no	—
102	C₂×#217	—	7	2	no	—
103	C₂×C₄⋊Dic₃	—	7	1	no	—
104	#924	—	7	1	no	—
105	#924	—	7	1	no	—
106	#924	—	7	2	no	—
107	#924	—	7	1	no	—
108	#924	—	7	2	no	—
109	#924	D₇	1	1	yes	characteristic, direct product factor
110	C₂×Dic₃⋊C₄	—	7	2	no	—
111	C₂²×C₃×C₇	—	2	1	no	—
112	C₂²×C₃×C₇	—	2	1	no	—
113	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	complemented
114	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
115	C₂×C₃×D₇	—	2	1	no	—
116	C₂×C₃×D₇	—	2	1	no	—
117	C₂×C₃×D₇	—	2	8	no	—
118	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
119	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
120	C₂×C₃×D₇	C₂×Q₈	1	2	yes	complemented
121	C₂×C₃×D₇	#41	1	2	yes	—
122	C₃×C₄×C₇	—	2	1	no	—
123	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
124	Dic₂₁	—	2	2	no	—
125	Dic₂₁	—	2	2	no	—
126	C₃×Dic₇	—	2	1	no	—
127	C₃×Dic₇	C₂×D₄	1	2	yes	—
128	C₇×Dic₃	—	2	2	no	—
129	C₇×Dic₃	—	2	2	no	—
130	C₂×C₂²⋊Q₈	—	21	1	no	Sylow
131	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
132	C₂²×D₇	—	2	2	no	—
133	C₂²×D₇	—	2	2	no	—
134	C₂²×D₇	Dic₆	1	4	yes	complemented
135	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
136	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
137	C₇×Q₈	—	6	2	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	3	2	no	—
140	C₂×C₄×C₇	—	3	2	no	—
141	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
143	C₂×Dic₇	—	2	2	no	—
144	C₂×Dic₇	—	3	2	no	—
145	C₂×Dic₇	—	3	2	no	—
146	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
147	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
148	C₄×D₇	—	2	2	no	—
149	C₄×D₇	—	2	2	no	—
150	C₄×D₇	—	2	2	no	—
151	C₄×D₇	—	6	4	no	—
152	C₄×D₇	—	6	4	no	—
153	C₄×D₇	#67	1	4	yes	—
154	Dic₁₄	—	6	2	no	—
155	Dic₁₄	—	6	4	no	—
156	C₂⁴×C₃	—	7	1	no	—
157	C₂²×C₃×C₄	—	7	1	no	—
158	C₂²×C₃×C₄	—	7	1	no	—
159	C₂²×C₃×C₄	—	7	1	no	—
160	C₂²×C₃×C₄	—	7	1	no	—
161	C₂²×C₃×C₄	—	7	1	no	—
162	C₂²×C₃×C₄	—	14	4	no	—
163	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
164	C₂²×Dic₃	—	7	2	no	—
165	C₂²×Dic₃	—	7	2	no	—
166	C₂×Dic₆	—	7	1	no	—
167	C₂×Dic₆	—	7	2	no	—
168	C₂×Dic₆	—	14	4	no	—
169	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic, complemented
170	#217	—	7	2	no	—
171	#217	—	7	2	no	—
172	#217	—	7	2	no	—
173	#217	C₂×D₇	1	2	yes	—
174	C₄⋊Dic₃	—	7	1	no	—
175	C₄⋊Dic₃	—	7	2	no	—
176	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic, complemented
177	Dic₃⋊C₄	—	7	2	no	—
178	Dic₃⋊C₄	—	7	2	no	—
179	Dic₃⋊C₄	—	7	2	no	—
180	Dic₃⋊C₄	C₂×D₇	1	2	yes	complemented
181	C₂×C₃×C₇	—	2	2	no	—
182	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
185	C₃×D₇	—	2	2	no	—
186	C₃×D₇	C₂²⋊Q₈	1	4	yes	complemented
187	C₂²×Q₈	—	21	1	no	—
188	C₂³×C₄	—	7	1	no	—
189	C₂²⋊Q₈	—	3	1	no	—
190	C₂²⋊Q₈	—	21	2	no	—
191	C₂²⋊Q₈	—	21	2	no	—
192	C₂²⋊Q₈	—	21	1	no	—
193	C₂²⋊Q₈	—	21	1	no	—
194	C₂²⋊Q₈	—	21	1	no	—
195	C₂×C₄⋊C₄	—	21	2	no	—
196	C₂×C₄⋊C₄	—	21	1	no	—
197	C₂×C₂²⋊C₄	—	21	2	no	—
198	C₂²×C₇	—	2	1	no	—
199	C₂²×C₇	—	2	1	no	—
200	C₂²×C₇	C₂×Dic₆	1	2	yes	complemented
201	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
202	C₂×D₇	—	2	1	no	—
203	C₂×D₇	—	2	1	no	—
204	C₂×D₇	—	2	8	no	—
205	C₂×D₇	C₂×Dic₆	1	2	yes	—
206	C₂×D₇	C₂×Dic₆	1	2	yes	complemented
207	C₂×D₇	#235	1	2	yes	—
208	C₂×D₇	C₂×#67	1	2	yes	—
209	C₄×C₇	—	2	1	no	—
210	C₄×C₇	—	6	2	no	—
211	C₄×C₇	—	6	2	no	—
212	C₄×C₇	C₂×#67	1	2	yes	—
213	Dic₇	—	2	1	no	—
214	Dic₇	—	6	2	no	—
215	Dic₇	—	6	2	no	—
216	Dic₇	C₂×#67	1	2	yes	—
217	C₂³×C₃	—	7	1	no	—
218	C₂³×C₃	—	7	4	no	—
219	C₂³×C₃	—	7	1	no	—
220	C₂³×C₃	—	14	2	no	—
221	C₂³×C₃	—	14	2	no	—
222	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	7	1	no	—
225	C₂×C₃×C₄	—	7	4	no	—
226	C₂×C₃×C₄	—	7	1	no	—
227	C₂×C₃×C₄	—	14	2	no	—
228	C₂×C₃×C₄	—	14	2	no	—
229	C₂×C₃×C₄	—	14	2	no	—
230	C₂×C₃×C₄	—	14	2	no	—
231	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
233	C₂×Dic₃	—	7	2	no	—
234	C₂×Dic₃	—	7	2	no	—
235	C₂×Dic₃	—	14	4	no	—
236	C₂×Dic₃	—	14	4	no	—
237	C₂×Dic₃	C₂²×D₇	1	2	yes	—
238	C₂×Dic₃	C₂²×D₇	1	2	yes	—
239	Dic₆	—	2	2	no	—
240	Dic₆	—	14	4	no	—
241	Dic₆	—	14	2	no	—
242	C₃×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
243	C₂⁴	—	7	1	no	—
244	C₂×Q₈	—	3	1	no	—
245	C₂×Q₈	—	21	2	no	—
246	C₂×Q₈	—	21	1	no	—
247	C₂×Q₈	—	42	4	no	—
248	C₂²×C₄	—	7	1	no	—
249	C₂²×C₄	—	7	1	no	—
250	C₂²×C₄	—	7	1	no	—
251	C₂²×C₄	—	7	1	no	—
252	C₂²×C₄	—	7	1	no	—
253	C₂²×C₄	—	14	4	no	—
254	C₂²×C₄	—	21	2	no	—
255	C₂²×C₄	—	21	2	no	—
256	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
257	C₄⋊C₄	—	3	2	no	—
258	C₄⋊C₄	—	3	1	no	—
259	C₄⋊C₄	—	21	2	no	—
260	C₄⋊C₄	—	21	2	no	—
261	C₄⋊C₄	—	21	2	no	—
262	C₄⋊C₄	—	21	2	no	—
263	C₄⋊C₄	—	21	1	no	—
264	C₂²⋊C₄	—	3	2	no	—
265	C₂²⋊C₄	—	21	2	no	—
266	C₂²⋊C₄	—	21	2	no	—
267	C₂²⋊C₄	—	21	2	no	—
268	C₂×C₇	—	2	2	no	—
269	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
270	C₂×C₇	C₂×#235	1	1	yes	characteristic
271	C₂×C₇	C₂²×#67	1	1	yes	characteristic
272	D₇	—	2	2	no	—
273	D₇	#924	1	4	yes	direct product factor
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	7	2	no	—
277	C₂²×C₃	—	7	2	no	—
278	C₂²×C₃	—	7	2	no	—
279	C₂²×C₃	—	7	2	no	—
280	C₂²×C₃	—	14	8	no	—
281	C₂²×C₃	—	14	1	no	—
282	C₂²×C₃	—	14	1	no	—
283	C₂²×C₃	Q₈×D₇	1	2	yes	complemented
284	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	7	2	no	—
287	C₃×C₄	—	14	1	no	—
288	C₃×C₄	D₄×D₇	1	2	yes	—
289	Dic₃	—	2	2	no	—
290	Dic₃	—	2	2	no	—
291	Dic₃	—	14	2	no	—
292	Dic₃	—	14	2	no	—
293	C₂³	—	7	1	no	—
294	C₂³	—	7	4	no	—
295	C₂³	—	7	1	no	—
296	C₂³	—	14	2	no	—
297	C₂³	—	14	2	no	—
298	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
299	Q₈	—	6	2	no	—
300	Q₈	—	42	4	no	—
301	Q₈	—	42	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	3	2	no	—
304	C₂×C₄	—	3	2	no	—
305	C₂×C₄	—	7	1	no	—
306	C₂×C₄	—	7	4	no	—
307	C₂×C₄	—	7	1	no	—
308	C₂×C₄	—	14	2	no	—
309	C₂×C₄	—	14	2	no	—
310	C₂×C₄	—	14	2	no	—
311	C₂×C₄	—	14	2	no	—
312	C₂×C₄	—	21	2	no	—
313	C₂×C₄	—	21	2	no	—
314	C₂×C₄	—	42	4	no	—
315	C₂×C₄	—	42	4	no	—
316	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
317	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
318	C₇	C₂×#924	1	1	yes	characteristic, complemented, Sylow
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	7	4	no	—
321	C₂×C₃	—	14	2	no	—
322	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
323	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
324	C₂×C₃	D₇×#41	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	7	2	no	—
328	C₂²	—	7	2	no	—
329	C₂²	—	7	2	no	—
330	C₂²	—	7	2	no	—
331	C₂²	—	14	8	no	—
332	C₂²	—	14	1	no	—
333	C₂²	—	14	1	no	—
334	C₂²	D₇×Dic₆	1	2	yes	complemented
335	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	6	2	no	—
338	C₄	—	6	2	no	—
339	C₄	—	7	2	no	—
340	C₄	—	14	1	no	—
341	C₄	—	42	2	no	—
342	C₄	—	42	2	no	—
343	C₄	D₇×#67	1	2	yes	—
344	C₃	D₇×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
345	C₂	—	2	2	no	—
346	C₂	—	7	4	no	—
347	C₂	—	14	2	no	—
348	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
349	C₂	D₇×#235	1	1	yes	characteristic
350	C₂	C₂×D₇×#67	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	C₂³×C₃×D₇	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×C₂²⋊Q₈
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. C₂×D₇×Dic₆	C₂	semidirect
3. #10621407	C₂	semidirect
4. C₇×#924	C₂	semidirect
5. #10621136	C₂	semidirect
7. D₇×C₄⋊Dic₃	C₂	semidirect
8. D₇×Dic₃⋊C₄	C₂	semidirect
9. #10620922	C₂	semidirect
10. #10620921	C₂	semidirect
11. #10620911	C₂	semidirect
15. C₂×Dic₄₂	C₂²	semidirect
16. C₂×C₇×Dic₆	C₂²	semidirect
23. C₂×#4248	C₂²	semidirect
28. C₄⋊Dic₂₁	C₂²	semidirect
29. #66629	C₂²	semidirect
31. C₇×C₄⋊Dic₃	C₂²	semidirect
32. C₇×Dic₃⋊C₄	C₂²	semidirect
33. Dic₃⋊Dic₇	C₂²	semidirect
34. #66577	C₂²	semidirect
35. Dic₇⋊Dic₃	C₂²	semidirect
53. C₂²×C₃×D₇	Q₈	semidirect
109. #924	D₇	direct (N × H)
113. C₂²×C₃×C₇	C₂×Q₈	semidirect
120. C₂×C₃×D₇	C₂×Q₈	semidirect
134. C₂²×D₇	Dic₆	semidirect
169. C₂×Dic₆	C₂×D₇	semidirect
176. C₄⋊Dic₃	C₂×D₇	semidirect
180. Dic₃⋊C₄	C₂×D₇	semidirect
186. C₃×D₇	C₂²⋊Q₈	semidirect
200. C₂²×C₇	C₂×Dic₆	semidirect
206. C₂×D₇	C₂×Dic₆	semidirect
242. C₃×C₇	C₂×C₂²⋊Q₈	semidirect
273. D₇	#924	direct (N × H)
283. C₂²×C₃	Q₈×D₇	semidirect
318. C₇	C₂×#924	semidirect
334. C₂²	D₇×Dic₆	semidirect
344. C₃	D₇×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×D₇	2
C₂×D₇×Dic₆	2
#10621407	2
C₇×#924	2
#10621136	2
D₇×#217	2
D₇×C₄⋊Dic₃	2
D₇×Dic₃⋊C₄	2
#10620922	2
#10620921	2
#10620911	2
D₇×C₂²⋊Q₈	3
C₂×#924	7

Subgroups — D7×#924 #13059480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#924 #13059480