Subgroups — C₇×#22522 #13199738

Order 1792.

1110 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1992	C₂	1	2	yes	maximal, complemented
5	C₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1989	C₂	1	2	yes	maximal, complemented
7	C₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1802	C₂	1	2	yes	maximal
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₇×#1762	C₂	1	2	yes	maximal
11	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic
12	C₇×#554	—	2	4	no	—
13	C₇×#554	C₄	1	8	yes	complemented
14	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×#125	C₂²	1	2	yes	—
16	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
17	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
18	C₂×C₄×C₇×Q₈	—	2	1	no	—
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
22	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
23	C₇×#446	—	2	4	no	—
24	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂×C₇×#102	C₂²	1	1	yes	characteristic
27	C₂×C₇×#102	C₂²	1	2	yes	—
28	C₂×C₇×#102	C₂²	1	1	yes	characteristic
29	C₂×C₇×#97	C₂²	1	2	yes	—
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#389	—	2	1	no	—
32	C₇×#389	—	2	2	no	—
33	C₇×#384	C₂²	1	1	yes	characteristic
34	C₇×#384	C₂²	1	2	yes	—
35	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
37	C₇×#340	C₂²	1	2	yes	—
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	1	yes	characteristic
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	2	yes	—
42	C₇×#340	C₂²	1	2	yes	—
43	C₇×#340	C₂²	1	1	yes	characteristic
44	#22522	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	—	2	1	no	—
46	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
47	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₇	—	2	1	no	—
49	C₂³×C₄×C₇	—	2	2	no	—
50	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₇×C₈	C₂³	1	2	yes	—
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₇×C₄⋊Q₈	—	2	2	no	—
54	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
55	C₇×#125	—	2	4	no	—
56	C₇×#125	C₂×C₄	1	8	yes	—
57	C₇×C₂²⋊Q₈	—	2	8	no	—
58	C₇×C₂²⋊Q₈	—	2	16	no	—
59	C₄×C₇×Q₈	—	4	4	no	—
60	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
61	C₄×C₇×Q₈	D₄	1	4	yes	—
62	C₄×C₇×D₄	—	2	2	no	—
63	C₄×C₇×D₄	—	2	4	no	—
64	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
65	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
66	C₂×C₇×C₄⋊C₄	—	2	2	no	—
67	C₂×C₇×C₄⋊C₄	—	2	1	no	—
68	C₂×C₇×C₄⋊C₄	—	2	2	no	—
69	C₂×C₇×C₄⋊C₄	—	2	8	no	—
70	C₂×C₇×C₄⋊C₄	—	2	2	no	—
71	C₂×C₇×C₄⋊C₄	—	2	1	no	—
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	—	2	2	no	—
74	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
75	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
76	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
78	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
80	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
82	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
83	C₂×C₄²×C₇	—	2	2	no	—
84	C₂×C₄²×C₇	—	2	1	no	—
85	C₂×C₄²×C₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
87	C₇×C₄⋊C₈	—	2	2	no	—
88	C₇×C₄⋊C₈	—	2	4	no	—
89	C₇×#102	—	2	2	no	—
90	C₇×#102	—	2	4	no	—
91	C₇×#102	Q₈	1	4	yes	complemented
92	C₇×#97	—	2	4	no	—
93	C₄×C₇×C₈	—	2	2	no	—
94	C₇×#95	—	2	1	no	—
95	C₇×#95	—	2	2	no	—
96	C₂×#554	C₂×C₇	1	1	yes	characteristic
97	C₂×#446	C₂×C₇	1	1	yes	characteristic, complemented
98	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
99	#1992	C₂×C₇	1	2	yes	complemented
100	#1990	C₂×C₇	1	1	yes	characteristic, complemented
101	#1989	C₂×C₇	1	2	yes	complemented
102	#1987	C₂×C₇	1	1	yes	characteristic, complemented
103	#1802	C₂×C₇	1	1	yes	characteristic
104	#1802	C₂×C₇	1	2	yes	—
105	#1762	C₂×C₇	1	2	yes	—
106	C₄×#102	C₂×C₇	1	1	yes	characteristic
107	C₂⁴×C₇	—	2	1	no	—
108	C₂×C₇×Q₈	—	2	4	no	—
109	C₂×C₇×Q₈	—	2	2	no	—
110	C₂×C₇×Q₈	—	2	2	no	—
111	C₂×C₇×Q₈	—	8	1	no	—
112	C₂×C₇×Q₈	—	8	1	no	—
113	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
114	C₂×C₇×D₄	—	2	1	no	—
115	C₂×C₇×D₄	—	2	1	no	—
116	C₂×C₇×D₄	—	2	1	no	—
117	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
118	C₂×C₇×D₄	C₂×Q₈	1	2	yes	—
119	C₂×C₇×D₄	#41	1	2	yes	—
120	C₂²×C₄×C₇	—	2	1	no	—
121	C₂²×C₄×C₇	—	2	1	no	—
122	C₂²×C₄×C₇	—	2	2	no	—
123	C₂²×C₄×C₇	—	2	4	no	—
124	C₂²×C₄×C₇	—	2	8	no	—
125	C₂²×C₄×C₇	—	2	2	no	—
126	C₂²×C₄×C₇	—	2	1	no	—
127	C₂²×C₄×C₇	—	4	2	no	—
128	C₂²×C₄×C₇	—	4	2	no	—
129	C₂²×C₄×C₇	—	8	2	no	—
130	C₂²×C₄×C₇	—	8	2	no	—
131	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
136	C₂×C₇×C₈	—	2	1	no	—
137	C₂×C₇×C₈	—	2	1	no	—
138	C₂×C₇×C₈	—	2	1	no	—
139	C₂×C₇×C₈	—	2	2	no	—
140	C₂×C₇×C₈	—	2	1	no	—
141	C₂×C₇×C₈	—	2	2	no	—
142	C₂×C₇×C₈	—	2	1	no	—
143	C₂×C₇×C₈	—	2	2	no	—
144	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
145	C₇×C₄⋊C₄	—	2	8	no	—
146	C₇×C₄⋊C₄	—	2	4	no	—
147	C₇×C₄⋊C₄	—	2	8	no	—
148	C₇×C₄⋊C₄	—	2	1	no	—
149	C₇×C₄⋊C₄	—	2	2	no	—
150	C₇×C₄⋊C₄	—	2	1	no	—
151	C₇×C₄⋊C₄	—	2	1	no	—
152	C₇×C₄⋊C₄	—	2	2	no	—
153	C₇×C₄⋊C₄	—	2	1	no	—
154	C₇×C₄⋊C₄	—	2	2	no	—
155	C₇×C₄⋊C₄	—	4	4	no	—
156	C₇×C₄⋊C₄	—	4	2	no	—
157	C₇×C₄⋊C₄	—	8	2	no	—
158	C₇×C₄⋊C₄	—	8	1	no	—
159	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
160	C₇×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
161	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
162	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
163	C₇×C₄⋊C₄	C₂×Q₈	1	2	yes	—
164	C₇×C₂²⋊C₄	—	2	4	no	—
165	C₇×C₂²⋊C₄	—	2	8	no	—
166	C₄²×C₇	—	4	4	no	—
167	C₄²×C₇	—	4	2	no	—
168	C₄²×C₇	C₂²×C₄	1	4	yes	—
169	C₄²×C₇	C₂²×C₄	1	2	yes	—
170	C₄²×C₇	C₂×D₄	1	2	yes	—
171	C₄²×C₇	C₂×D₄	1	4	yes	—
172	#554	—	2	4	no	—
173	#554	C₄×C₇	1	8	yes	complemented
174	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
175	C₂×#125	C₂²×C₇	1	2	yes	—
176	C₂×C₂²⋊Q₈	—	2	1	no	—
177	C₂×C₂²⋊Q₈	—	2	2	no	—
178	C₂×C₄×Q₈	—	2	1	no	—
179	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
180	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
181	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
182	C₂²×C₄⋊C₄	—	2	1	no	—
183	#446	—	2	4	no	—
184	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
185	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
186	C₂×#102	C₂²×C₇	1	1	yes	characteristic
187	C₂×#102	C₂²×C₇	1	2	yes	—
188	C₂×#102	C₂²×C₇	1	1	yes	characteristic
189	C₂×#97	C₂²×C₇	1	2	yes	—
190	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
191	#389	—	2	1	no	—
192	#389	—	2	2	no	—
193	#384	C₂²×C₇	1	1	yes	characteristic
194	#384	C₂²×C₇	1	2	yes	—
195	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
196	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
197	#340	C₂²×C₇	1	2	yes	—
198	#340	C₂²×C₇	1	2	yes	—
199	#340	C₂²×C₇	1	2	yes	—
200	#340	C₂²×C₇	1	2	yes	—
201	#340	C₂²×C₇	1	1	yes	characteristic
202	#340	C₂²×C₇	1	2	yes	—
203	#340	C₂²×C₇	1	1	yes	characteristic
204	C₂³×C₇	—	2	2	no	—
205	C₂³×C₇	—	2	2	no	—
206	C₂³×C₇	—	2	2	no	—
207	C₂³×C₇	—	4	1	no	—
208	C₂³×C₇	—	4	1	no	—
209	C₂³×C₇	—	4	1	no	—
210	C₂³×C₇	—	4	1	no	—
211	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
212	C₇×Q₈	—	2	8	no	—
213	C₇×Q₈	—	8	4	no	—
214	C₇×D₄	—	2	2	no	—
215	C₇×D₄	—	2	2	no	—
216	C₇×D₄	—	2	2	no	—
217	C₇×D₄	C₄×Q₈	1	4	yes	complemented
218	C₂×C₄×C₇	—	2	2	no	—
219	C₂×C₄×C₇	—	2	2	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	2	1	no	—
222	C₂×C₄×C₇	—	2	2	no	—
223	C₂×C₄×C₇	—	2	1	no	—
224	C₂×C₄×C₇	—	2	2	no	—
225	C₂×C₄×C₇	—	2	4	no	—
226	C₂×C₄×C₇	—	2	2	no	—
227	C₂×C₄×C₇	—	4	1	no	—
228	C₂×C₄×C₇	—	4	1	no	—
229	C₂×C₄×C₇	—	4	1	no	—
230	C₂×C₄×C₇	—	4	2	no	—
231	C₂×C₄×C₇	—	4	1	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	8	no	—
234	C₂×C₄×C₇	—	4	2	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	2	no	—
237	C₂×C₄×C₇	—	4	1	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	—	8	8	no	—
241	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
242	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
243	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
244	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
245	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
246	C₂×C₄×C₇	#137	1	2	yes	—
247	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
250	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
251	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
252	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
253	C₇×C₈	—	2	2	no	—
254	C₇×C₈	—	4	1	no	—
255	C₇×C₈	—	4	2	no	—
256	C₂²×Q₈	—	2	1	no	—
257	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
258	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
259	C₂³×C₄	—	2	1	no	—
260	C₂³×C₄	—	2	2	no	—
261	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₇	1	2	yes	—
263	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
266	#125	—	2	4	no	—
267	#125	C₂×C₄×C₇	1	8	yes	—
268	C₂²⋊Q₈	—	2	8	no	—
269	C₂²⋊Q₈	—	2	16	no	—
270	C₄×Q₈	—	4	4	no	—
271	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
272	C₄×Q₈	C₇×D₄	1	4	yes	—
273	C₄×D₄	—	2	2	no	—
274	C₄×D₄	—	2	4	no	—
275	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
276	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	8	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₄²	—	2	2	no	—
295	C₂×C₄²	—	2	1	no	—
296	C₂×C₄²	C₂³×C₇	1	2	yes	—
297	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
298	C₄⋊C₈	—	2	2	no	—
299	C₄⋊C₈	—	2	4	no	—
300	#102	—	2	2	no	—
301	#102	—	2	4	no	—
302	#102	C₇×Q₈	1	4	yes	complemented
303	#97	—	2	4	no	—
304	C₄×C₈	—	2	2	no	—
305	#95	—	2	1	no	—
306	#95	—	2	2	no	—
307	C₂²×C₇	—	2	4	no	—
308	C₂²×C₇	—	4	2	no	—
309	C₂²×C₇	—	4	2	no	—
310	C₂²×C₇	—	4	2	no	—
311	C₂²×C₇	—	4	2	no	—
312	C₂²×C₇	—	4	2	no	—
313	C₂²×C₇	—	4	2	no	—
314	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
315	C₂²×C₇	#548	1	1	yes	characteristic
316	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
317	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂²×C₇	C₂×#137	1	1	yes	characteristic
319	C₂²×C₇	#576	1	1	yes	characteristic
320	C₂²×C₇	#577	1	1	yes	characteristic
321	C₄×C₇	—	2	4	no	—
322	C₄×C₇	—	2	4	no	—
323	C₄×C₇	—	2	2	no	—
324	C₄×C₇	—	4	2	no	—
325	C₄×C₇	—	4	2	no	—
326	C₄×C₇	—	4	4	no	—
327	C₄×C₇	—	8	2	no	—
328	C₄×C₇	—	8	1	no	—
329	C₄×C₇	#417	1	4	yes	—
330	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
331	C₄×C₇	#519	1	1	yes	characteristic
332	C₄×C₇	#548	1	1	yes	characteristic
333	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
334	C₂⁴	—	2	1	no	—
335	C₂×Q₈	—	2	4	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	8	1	no	—
339	C₂×Q₈	—	8	1	no	—
340	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
345	C₂×D₄	C₂×C₇×Q₈	1	2	yes	—
346	C₂×D₄	C₇×#41	1	2	yes	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	4	2	no	—
355	C₂²×C₄	—	4	2	no	—
356	C₂²×C₄	—	8	2	no	—
357	C₂²×C₄	—	8	2	no	—
358	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
359	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
360	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
361	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
362	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
372	C₄⋊C₄	—	2	8	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	4	4	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	8	2	no	—
385	C₄⋊C₄	—	8	1	no	—
386	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
387	C₄⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
388	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
389	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
390	C₄⋊C₄	C₂×C₇×Q₈	1	2	yes	—
391	C₂²⋊C₄	—	2	4	no	—
392	C₂²⋊C₄	—	2	8	no	—
393	C₄²	—	4	4	no	—
394	C₄²	—	4	2	no	—
395	C₄²	C₂²×C₄×C₇	1	4	yes	—
396	C₄²	C₂²×C₄×C₇	1	2	yes	—
397	C₄²	C₂×C₇×D₄	1	2	yes	—
398	C₄²	C₂×C₇×D₄	1	4	yes	—
399	C₂×C₇	—	4	4	no	—
400	C₂×C₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
401	C₂×C₇	C₂×#417	1	1	yes	characteristic
402	C₂×C₇	#2897	1	1	yes	characteristic
403	C₂×C₇	#3366	1	1	yes	characteristic
404	C₂×C₇	#3372	1	1	yes	characteristic
405	C₂×C₇	#3384	1	1	yes	characteristic
406	C₂×C₇	#3385	1	1	yes	characteristic
407	C₂³	—	2	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	4	1	no	—
413	C₂³	—	4	1	no	—
414	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
415	Q₈	—	2	8	no	—
416	Q₈	—	8	4	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	C₄×C₇×Q₈	1	4	yes	complemented
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	8	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	8	8	no	—
444	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
445	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
446	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
447	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
448	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
449	C₂×C₄	C₇×#137	1	2	yes	—
450	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
451	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
452	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
453	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
454	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
455	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
456	C₈	—	2	2	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	2	no	—
459	C₇	#22522	1	1	yes	characteristic, direct product factor, Sylow
460	C₂²	—	2	4	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
468	C₂²	C₇×#548	1	1	yes	characteristic
469	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
470	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
471	C₂²	C₂×C₇×#137	1	1	yes	characteristic
472	C₂²	C₇×#576	1	1	yes	characteristic
473	C₂²	C₇×#577	1	1	yes	characteristic
474	C₄	—	2	4	no	—
475	C₄	—	2	4	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	4	no	—
480	C₄	—	8	2	no	—
481	C₄	—	8	1	no	—
482	C₄	C₇×#417	1	4	yes	—
483	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
484	C₄	C₇×#519	1	1	yes	characteristic
485	C₄	C₇×#548	1	1	yes	characteristic
486	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
487	C₂	—	4	4	no	—
488	C₂	C₇×Q₈×C₂²⋊C₄	1	1	yes	characteristic
489	C₂	C₂×C₇×#417	1	1	yes	characteristic
490	C₂	C₇×#2897	1	1	yes	characteristic
491	C₂	C₇×#3366	1	1	yes	characteristic
492	C₂	C₇×#3372	1	1	yes	characteristic
493	C₂	C₇×#3384	1	1	yes	characteristic
494	C₂	C₇×#3385	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#446	C₂	semidirect
3. C₇×Q₈×C₄⋊C₄	C₂	semidirect
4. C₇×#1992	C₂	semidirect
5. C₇×#1990	C₂	semidirect
6. C₇×#1989	C₂	semidirect
7. C₇×#1987	C₂	semidirect
13. C₇×#554	C₄	semidirect
44. #22522	C₇	direct (N × H)
91. C₇×#102	Q₈	semidirect
97. C₂×#446	C₂×C₇	semidirect
98. Q₈×C₄⋊C₄	C₂×C₇	semidirect
99. #1992	C₂×C₇	semidirect
100. #1990	C₂×C₇	semidirect
101. #1989	C₂×C₇	semidirect
102. #1987	C₂×C₇	semidirect
173. #554	C₄×C₇	semidirect
217. C₇×D₄	C₄×Q₈	semidirect
302. #102	C₇×Q₈	semidirect
420. D₄	C₄×C₇×Q₈	semidirect
459. C₇	#22522	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#554	2
C₂×C₇×#446	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1992	2
C₇×#1990	2
C₇×#1989	2
C₇×#1987	2
C₇×#1802	2
C₇×#1762	2
C₄×C₇×#102	2
#22522	7

Subgroups — C7×#22522 #13199738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22522 #13199738