Subgroups — D₇×#888 #13058558

Order 1344.

580 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×Dic₆	C₂	1	2	yes	maximal
2	#10621371	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#888	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#10621032	C₂	1	1	yes	maximal, characteristic, complemented
6	#10621031	C₂	1	1	yes	maximal, characteristic, complemented
7	D₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
8	D₇×Dic₃⋊C₄	C₂	1	2	yes	maximal
9	C₄×Dic₃×D₇	C₂	1	1	yes	maximal, characteristic
10	#10620891	C₂	1	2	yes	maximal, complemented
11	#10620890	C₂	1	2	yes	maximal, complemented
12	D₇×C₄⋊Q₈	—	3	1	no	maximal, Hall
13	C₂×Dic₄₂	C₂²	1	2	yes	—
14	C₂×C₇×Dic₆	C₂²	1	2	yes	—
15	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
16	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
17	C₂×#4248	C₂²	1	2	yes	—
18	C₂×#4248	C₂²	1	2	yes	—
19	C₂×Dic₃×D₇	C₂²	1	2	yes	—
20	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
22	D₇×Dic₆	—	2	8	no	—
23	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
25	#66629	C₂²	1	2	yes	—
26	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
27	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
31	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
32	Dic₃⋊Dic₇	C₂²	1	2	yes	—
33	#66577	C₂²	1	2	yes	—
34	Dic₇⋊Dic₃	C₂²	1	2	yes	—
35	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
36	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
37	C₂×Q₈×D₇	—	3	2	no	—
38	C₇×C₄⋊Q₈	—	3	1	no	—
39	#6507	—	3	2	no	—
40	D₇×C₄⋊C₄	—	3	2	no	—
41	D₇×C₄⋊C₄	—	3	1	no	—
42	D₇×C₄⋊C₄	D₃	1	1	yes	characteristic
43	#6451	—	3	1	no	—
44	#6451	—	3	1	no	—
45	#6451	—	3	2	no	—
46	C₄²×D₇	—	3	1	no	—
47	#6432	—	3	1	no	—
48	C₂×#888	—	7	1	no	maximal, Hall
49	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	2	yes	—
53	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
55	Dic₄₂	—	2	4	no	—
56	C₂×C₇×Dic₃	C₂³	1	2	yes	—
57	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
59	C₇×Dic₆	—	2	4	no	—
60	C₂×C₃×Dic₇	C₂³	1	2	yes	—
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₇	—	2	4	no	—
63	C₃×C₄×D₇	Q₈	1	4	yes	complemented
64	#4248	—	2	4	no	—
65	#4248	—	2	4	no	—
66	Dic₃×D₇	—	2	4	no	—
67	Dic₃×D₇	D₄	1	4	yes	—
68	Dic₃×D₇	Q₈	1	4	yes	complemented
69	C₂×C₇×Q₈	—	3	2	no	—
70	Q₈×D₇	—	6	8	no	—
71	C₂×C₄×D₇	—	3	1	no	—
72	C₂×C₄×D₇	—	3	1	no	—
73	C₂×C₄×D₇	—	3	2	no	—
74	C₂×C₄×D₇	C₂×D₃	1	2	yes	—
75	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
76	C₂×Dic₁₄	—	3	2	no	—
77	C₂×Dic₁₄	—	3	2	no	—
78	C₂×Dic₁₄	—	3	2	no	—
79	C₇×C₄⋊C₄	—	3	2	no	—
80	C₇×C₄⋊C₄	—	3	1	no	—
81	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
82	C₄²×C₇	—	3	1	no	—
83	C₄⋊Dic₇	—	3	1	no	—
84	C₄⋊Dic₇	—	3	2	no	—
85	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
86	Dic₇⋊C₄	—	3	2	no	—
87	Dic₇⋊C₄	—	3	2	no	—
88	Dic₇⋊C₄	—	3	2	no	—
89	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
90	C₄×Dic₇	—	3	1	no	—
91	C₄×Dic₇	—	3	1	no	—
92	C₄×Dic₇	—	3	1	no	—
93	C₂²×Dic₆	—	7	2	no	—
94	C₂×C₃×C₄⋊C₄	—	7	1	no	—
95	C₂×C₄⋊Dic₃	—	7	1	no	—
96	C₂×Dic₃⋊C₄	—	7	2	no	—
97	C₂×C₄×Dic₃	—	7	1	no	—
98	#888	—	7	2	no	—
99	#888	—	7	1	no	—
100	#888	—	7	1	no	—
101	#888	—	7	2	no	—
102	#888	—	7	1	no	—
103	#888	D₇	1	1	yes	characteristic, direct product factor
104	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
106	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
107	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
108	C₃×C₄×C₇	—	2	2	no	—
109	C₃×C₄×C₇	C₂×Q₈	1	2	yes	complemented
110	Dic₂₁	—	2	2	no	—
111	Dic₂₁	C₂×D₄	1	2	yes	—
112	Dic₂₁	C₂×Q₈	1	2	yes	complemented
113	C₃×Dic₇	—	2	2	no	—
114	C₃×Dic₇	C₂×Q₈	1	2	yes	complemented
115	C₇×Dic₃	—	2	2	no	—
116	C₇×Dic₃	C₂×D₄	1	2	yes	—
117	C₇×Dic₃	C₂×Q₈	1	2	yes	complemented
118	C₂×C₄⋊Q₈	—	21	1	no	Sylow
119	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
120	C₇×Q₈	—	6	4	no	—
121	C₂×C₄×C₇	—	3	1	no	—
122	C₂×C₄×C₇	—	3	1	no	—
123	C₂×C₄×C₇	—	3	2	no	—
124	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
125	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
126	C₂×Dic₇	—	3	1	no	—
127	C₂×Dic₇	—	3	1	no	—
128	C₂×Dic₇	—	3	2	no	—
129	C₂×Dic₇	C₂²×D₃	1	2	yes	—
130	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
131	C₄×D₇	—	2	4	no	—
132	C₄×D₇	—	3	4	no	—
133	C₄×D₇	—	3	4	no	—
134	C₄×D₇	—	6	4	no	—
135	C₄×D₇	Dic₆	1	4	yes	complemented
136	Dic₁₄	—	6	4	no	—
137	Dic₁₄	—	6	4	no	—
138	Dic₁₄	—	6	4	no	—
139	C₂²×C₃×C₄	—	7	1	no	—
140	C₂²×C₃×C₄	—	7	2	no	—
141	C₂²×Dic₃	—	7	1	no	—
142	C₂²×Dic₃	—	7	1	no	—
143	C₂²×Dic₃	—	7	2	no	—
144	C₂×Dic₆	—	7	2	no	—
145	C₂×Dic₆	—	7	2	no	—
146	C₂×Dic₆	—	7	2	no	—
147	C₂×Dic₆	—	14	8	no	—
148	C₂×Dic₆	C₂×D₇	1	2	yes	—
149	C₃×C₄⋊C₄	—	7	2	no	—
150	C₃×C₄⋊C₄	—	7	1	no	—
151	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
152	C₄⋊Dic₃	—	7	1	no	—
153	C₄⋊Dic₃	—	7	2	no	—
154	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
155	Dic₃⋊C₄	—	7	2	no	—
156	Dic₃⋊C₄	—	7	2	no	—
157	Dic₃⋊C₄	—	7	2	no	—
158	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
159	C₄×Dic₃	—	7	1	no	—
160	C₄×Dic₃	—	7	1	no	—
161	C₄×Dic₃	—	7	1	no	—
162	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
163	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
166	C₃×D₇	C₄⋊Q₈	1	4	yes	complemented
167	C₂²×Q₈	—	21	2	no	—
168	C₄⋊Q₈	—	3	1	no	—
169	C₄⋊Q₈	—	21	1	no	—
170	C₄⋊Q₈	—	21	2	no	—
171	C₄⋊Q₈	—	21	1	no	—
172	C₄⋊Q₈	—	21	2	no	—
173	C₄⋊Q₈	—	21	1	no	—
174	C₂×C₄⋊C₄	—	7	1	no	—
175	C₂×C₄⋊C₄	—	21	2	no	—
176	C₂×C₄⋊C₄	—	21	1	no	—
177	C₂×C₄²	—	21	1	no	—
178	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
179	C₂×D₇	C₂×Dic₆	1	2	yes	—
180	C₂×D₇	D₃×D₄	1	2	yes	—
181	C₂×D₇	D₃×Q₈	1	2	yes	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	3	2	no	—
184	C₄×C₇	—	3	2	no	—
185	C₄×C₇	—	6	2	no	—
186	C₄×C₇	C₂×Dic₆	1	2	yes	complemented
187	Dic₇	—	2	2	no	—
188	Dic₇	—	3	2	no	—
189	Dic₇	—	3	2	no	—
190	Dic₇	—	6	2	no	—
191	Dic₇	C₂×Dic₆	1	2	yes	complemented
192	C₂³×C₃	—	7	1	no	—
193	C₂×C₃×C₄	—	7	1	no	—
194	C₂×C₃×C₄	—	7	4	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	14	4	no	—
197	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
198	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
199	C₂×Dic₃	—	7	1	no	—
200	C₂×Dic₃	—	7	1	no	—
201	C₂×Dic₃	—	7	4	no	—
202	C₂×Dic₃	—	7	4	no	—
203	C₂×Dic₃	—	7	2	no	—
204	C₂×Dic₃	—	14	4	no	—
205	C₂×Dic₃	C₂²×D₇	1	2	yes	—
206	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
207	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
208	Dic₆	—	2	4	no	—
209	Dic₆	—	14	4	no	—
210	Dic₆	—	14	4	no	—
211	Dic₆	—	14	4	no	—
212	C₃×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
213	C₂×Q₈	—	3	2	no	—
214	C₂×Q₈	—	21	2	no	—
215	C₂×Q₈	—	21	2	no	—
216	C₂×Q₈	—	21	2	no	—
217	C₂×Q₈	—	42	8	no	—
218	C₂²×C₄	—	7	1	no	—
219	C₂²×C₄	—	7	2	no	—
220	C₂²×C₄	—	21	1	no	—
221	C₂²×C₄	—	21	1	no	—
222	C₂²×C₄	—	21	2	no	—
223	C₄⋊C₄	—	3	2	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	7	2	no	—
226	C₄⋊C₄	—	7	1	no	—
227	C₄⋊C₄	—	21	2	no	—
228	C₄⋊C₄	—	21	2	no	—
229	C₄⋊C₄	—	21	2	no	—
230	C₄⋊C₄	—	21	2	no	—
231	C₄⋊C₄	—	21	1	no	—
232	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
233	C₄²	—	3	1	no	—
234	C₄²	—	21	1	no	—
235	C₄²	—	21	1	no	—
236	C₄²	—	21	1	no	—
237	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
238	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
239	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
240	D₇	#888	1	4	yes	direct product factor
241	C₂²×C₃	—	7	2	no	—
242	C₂²×C₃	—	7	2	no	—
243	C₂²×C₃	—	7	2	no	—
244	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	7	2	no	—
247	C₃×C₄	—	14	2	no	—
248	C₃×C₄	Q₈×D₇	1	2	yes	complemented
249	Dic₃	—	2	2	no	—
250	Dic₃	—	7	2	no	—
251	Dic₃	—	7	2	no	—
252	Dic₃	—	14	2	no	—
253	Dic₃	D₄×D₇	1	2	yes	—
254	Dic₃	Q₈×D₇	1	2	yes	complemented
255	C₂³	—	7	1	no	—
256	Q₈	—	6	4	no	—
257	Q₈	—	42	4	no	—
258	Q₈	—	42	4	no	—
259	Q₈	—	42	4	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	7	1	no	—
264	C₂×C₄	—	7	4	no	—
265	C₂×C₄	—	7	2	no	—
266	C₂×C₄	—	14	4	no	—
267	C₂×C₄	—	21	4	no	—
268	C₂×C₄	—	21	4	no	—
269	C₂×C₄	—	21	1	no	—
270	C₂×C₄	—	21	1	no	—
271	C₂×C₄	—	21	2	no	—
272	C₂×C₄	—	42	4	no	—
273	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
274	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
275	C₇	C₂×#888	1	1	yes	characteristic, complemented, Sylow
276	C₂×C₃	—	7	4	no	—
277	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
278	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
279	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
280	C₂²	—	7	2	no	—
281	C₂²	—	7	2	no	—
282	C₂²	—	7	2	no	—
283	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	3	2	no	—
286	C₄	—	3	2	no	—
287	C₄	—	6	2	no	—
288	C₄	—	7	2	no	—
289	C₄	—	14	2	no	—
290	C₄	—	21	2	no	—
291	C₄	—	21	2	no	—
292	C₄	—	42	2	no	—
293	C₄	D₇×Dic₆	1	2	yes	complemented
294	C₃	D₇×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	7	4	no	—
296	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
297	C₂	D₃×D₄×D₇	1	1	yes	characteristic
298	C₂	D₃×Q₈×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×C₄⋊C₄	C₂²
prime radical	C₂²×C₃×D₇	C₂³
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #10621371	C₂	semidirect
3. C₇×#888	C₂	semidirect
5. #10621032	C₂	semidirect
6. #10621031	C₂	semidirect
10. #10620891	C₂	semidirect
11. #10620890	C₂	semidirect
63. C₃×C₄×D₇	Q₈	semidirect
68. Dic₃×D₇	Q₈	semidirect
103. #888	D₇	direct (N × H)
109. C₃×C₄×C₇	C₂×Q₈	semidirect
112. Dic₂₁	C₂×Q₈	semidirect
114. C₃×Dic₇	C₂×Q₈	semidirect
117. C₇×Dic₃	C₂×Q₈	semidirect
135. C₄×D₇	Dic₆	semidirect
166. C₃×D₇	C₄⋊Q₈	semidirect
186. C₄×C₇	C₂×Dic₆	semidirect
191. Dic₇	C₂×Dic₆	semidirect
212. C₃×C₇	C₂×C₄⋊Q₈	semidirect
240. D₇	#888	direct (N × H)
248. C₃×C₄	Q₈×D₇	semidirect
254. Dic₃	Q₈×D₇	semidirect
275. C₇	C₂×#888	semidirect
293. C₄	D₇×Dic₆	semidirect
294. C₃	D₇×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₇×Dic₆	2
#10621371	2
C₇×#888	2
C₃×D₇×C₄⋊C₄	2
#10621032	2
#10621031	2
D₇×C₄⋊Dic₃	2
D₇×Dic₃⋊C₄	2
C₄×Dic₃×D₇	2
#10620891	2
#10620890	2
D₇×C₄⋊Q₈	3
C₂×#888	7

Subgroups — D7×#888 #13058558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#888 #13058558