Subgroups — C₇×#7625 #13184841

Order 1792.

462 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1839	C₂	1	1	yes	maximal, characteristic
3	C₇×#1767	C₂	1	1	yes	maximal, characteristic
4	C₇×#1271	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇×Q₈	—	2	1	no	—
7	C₂×C₇×C₄⋊C₈	—	2	1	no	—
8	C₂×C₇×#97	—	2	1	no	—
9	C₇×#391	—	2	1	no	—
10	C₇×#389	—	2	1	no	—
11	C₇×#389	C₂²	1	2	yes	—
12	C₇×#384	C₄	1	1	yes	characteristic
13	C₄×C₇×C₄⋊C₄	—	2	1	no	—
14	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
15	C₄³×C₇	C₂²	1	1	yes	characteristic
16	C₇×#336	—	2	1	no	—
17	C₇×#336	—	2	1	no	—
18	C₇×#336	C₂²	1	2	yes	—
19	C₇×#336	C₂²	1	2	yes	—
20	#7625	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	1	no	—
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	—	2	2	no	—
24	C₇×C₄⋊Q₈	—	2	4	no	—
25	C₄×C₇×Q₈	—	4	2	no	—
26	C₄×C₇×Q₈	—	4	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	1	no	—
28	C₂×C₇×C₄⋊C₄	—	2	1	no	—
29	C₂×C₇×C₄⋊C₄	—	2	1	no	—
30	C₂×C₇×C₄⋊C₄	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	—	2	2	no	—
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₇×C₄⋊C₈	—	2	4	no	—
42	C₇×#97	—	4	2	no	—
43	C₇×#95	—	2	2	no	—
44	C₇×#95	—	2	2	no	—
45	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
46	#1839	C₂×C₇	1	1	yes	characteristic
47	#1767	C₂×C₇	1	1	yes	characteristic
48	#1271	C₂×C₇	1	4	yes	—
49	C₂×C₇×Q₈	—	2	4	no	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	—	2	2	no	—
54	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₇×C₈	—	2	4	no	—
62	C₂×C₇×C₈	—	4	2	no	—
63	C₂×C₇×C₈	—	4	2	no	—
64	C₂×C₇×C₈	—	4	2	no	—
65	C₂×C₇×C₈	—	4	2	no	—
66	C₂×C₇×C₈	—	4	2	no	—
67	C₇×C₄⋊C₄	—	2	4	no	—
68	C₇×C₄⋊C₄	—	2	2	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	4	2	no	—
71	C₇×C₄⋊C₄	—	4	2	no	—
72	C₇×C₄⋊C₄	—	4	2	no	—
73	C₇×C₄⋊C₄	—	4	2	no	—
74	C₇×C₄⋊C₄	M₁₆	1	4	yes	—
75	C₄²×C₇	—	2	2	no	—
76	C₄²×C₇	—	2	2	no	—
77	C₄²×C₇	—	2	2	no	—
78	C₄²×C₇	—	2	2	no	—
79	C₄²×C₇	—	4	1	no	—
80	C₄²×C₇	—	4	1	no	—
81	C₄²×C₇	—	4	4	no	—
82	C₄²×C₇	D₈	1	2	yes	—
83	C₄²×C₇	SD₁₆	1	2	yes	—
84	C₂×C₄⋊Q₈	C₄×C₇	1	1	yes	characteristic
85	C₂×C₄×Q₈	—	2	1	no	—
86	C₂×C₄⋊C₈	—	2	1	no	—
87	C₂×#97	—	2	1	no	—
88	#391	—	2	1	no	—
89	#389	—	2	1	no	—
90	#389	C₂²×C₇	1	2	yes	—
91	#384	C₄×C₇	1	1	yes	characteristic
92	C₄×C₄⋊C₄	—	2	1	no	—
93	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
94	C₄³	C₂²×C₇	1	1	yes	characteristic
95	#336	—	2	1	no	—
96	#336	—	2	1	no	—
97	#336	C₂²×C₇	1	2	yes	—
98	#336	C₂²×C₇	1	2	yes	—
99	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₇×Q₈	—	4	8	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	4	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	2	no	—
118	C₂×C₄×C₇	—	4	2	no	—
119	C₂×C₄×C₇	—	4	2	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	#101	1	2	yes	—
122	C₂×C₄×C₇	#102	1	2	yes	—
123	C₂×C₄×C₇	#102	1	2	yes	—
124	C₂×C₄×C₇	#104	1	2	yes	—
125	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₇	#137	1	2	yes	—
129	C₇×C₈	—	4	4	no	—
130	C₇×C₈	—	8	2	no	—
131	C₂²×Q₈	—	2	1	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₄⋊Q₈	—	2	4	no	—
135	C₄×Q₈	—	4	2	no	—
136	C₄×Q₈	—	4	2	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
146	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
147	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
148	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
149	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
150	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	#97	—	4	2	no	—
153	#95	—	2	2	no	—
154	#95	—	2	2	no	—
155	C₂²×C₇	#407	1	1	yes	characteristic
156	C₂²×C₇	#410	1	1	yes	characteristic
157	C₂²×C₇	C₂×#101	1	1	yes	characteristic
158	C₂²×C₇	C₂×#102	1	1	yes	characteristic
159	C₂²×C₇	#419	1	1	yes	characteristic
160	C₂²×C₇	C₂×#104	1	1	yes	characteristic
161	C₂²×C₇	#421	1	1	yes	characteristic
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	2	2	no	—
166	C₄×C₇	—	2	1	no	—
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	4	4	no	—
171	C₄×C₇	—	8	2	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	C₇×M₁₆	1	4	yes	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	4	no	—
205	C₄²	C₇×D₈	1	2	yes	—
206	C₄²	C₇×SD₁₆	1	2	yes	—
207	C₂×C₇	#1467	1	1	yes	characteristic
208	C₂×C₇	#1486	1	1	yes	characteristic
209	C₂×C₇	#1491	1	1	yes	characteristic
210	C₂×C₇	#1502	1	1	yes	characteristic
211	C₂×C₇	#1515	1	1	yes	characteristic
212	C₂×C₇	#1531	1	1	yes	characteristic
213	C₂×C₇	#1546	1	1	yes	characteristic
214	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
215	Q₈	—	4	8	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₇×#101	1	2	yes	—
237	C₂×C₄	C₇×#102	1	2	yes	—
238	C₂×C₄	C₇×#102	1	2	yes	—
239	C₂×C₄	C₇×#104	1	2	yes	—
240	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
241	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
242	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
243	C₂×C₄	C₇×#137	1	2	yes	—
244	C₈	—	4	4	no	—
245	C₈	—	8	2	no	—
246	C₇	#7625	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	C₇×#407	1	1	yes	characteristic
248	C₂²	C₇×#410	1	1	yes	characteristic
249	C₂²	C₂×C₇×#101	1	1	yes	characteristic
250	C₂²	C₂×C₇×#102	1	1	yes	characteristic
251	C₂²	C₇×#419	1	1	yes	characteristic
252	C₂²	C₂×C₇×#104	1	1	yes	characteristic
253	C₂²	C₇×#421	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₂	C₇×#1467	1	1	yes	characteristic
265	C₂	C₇×#1486	1	1	yes	characteristic
266	C₂	C₇×#1491	1	1	yes	characteristic
267	C₂	C₇×#1502	1	1	yes	characteristic
268	C₂	C₇×#1515	1	1	yes	characteristic
269	C₂	C₇×#1531	1	1	yes	characteristic
270	C₂	C₇×#1546	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
20. #7625	C₇	direct (N × H)
246. C₇	#7625	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×C₄⋊Q₈	2
C₇×#1839	2
C₇×#1767	2
C₇×#1271	2
#7625	7

Subgroups — C7×#7625 #13184841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7625 #13184841