Subgroups — C₇×#21984 #13199200

Order 1792.

850 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₇×#1949	C₂	1	4	yes	maximal, complemented
5	C₇×#1854	C₂	1	4	yes	maximal
6	C₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#117	—	2	1	no	—
11	C₂×C₇×#117	C₂²	1	2	yes	—
12	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₇×#431	C₄	1	8	yes	complemented
14	C₂×C₇×#107	C₂²	1	4	yes	—
15	C₂×C₇×#106	C₂²	1	4	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#384	C₂²	1	1	yes	characteristic
21	C₇×#384	C₂²	1	1	yes	characteristic
22	C₇×#384	C₂²	1	1	yes	characteristic
23	C₇×#384	C₂²	1	1	yes	characteristic
24	C₇×#382	—	2	1	no	—
25	C₇×#382	C₂²	1	2	yes	—
26	C₇×#380	—	2	1	no	—
27	C₇×#380	—	2	1	no	—
28	C₇×#380	—	2	1	no	—
29	C₇×#380	C₂²	1	2	yes	—
30	C₄×C₇×C₄⋊C₄	—	2	1	no	—
31	C₄×C₇×C₄⋊C₄	—	2	1	no	—
32	C₄×C₇×C₄⋊C₄	—	2	1	no	—
33	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
34	C₇×#340	C₂²	1	4	yes	—
35	C₇×#340	C₂²	1	4	yes	—
36	#21984	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	2	yes	—
41	C₂²×C₇×C₈	C₂³	1	2	yes	—
42	C₇×#117	—	4	4	no	—
43	C₇×#117	—	4	4	no	—
44	C₂×C₇×C₄⋊C₄	—	2	1	no	—
45	C₂×C₇×C₄⋊C₄	—	2	1	no	—
46	C₂×C₇×C₄⋊C₄	—	2	1	no	—
47	C₂×C₇×C₄⋊C₄	—	2	1	no	—
48	C₂×C₇×C₄⋊C₄	—	2	1	no	—
49	C₂×C₇×C₄⋊C₄	—	2	1	no	—
50	C₂×C₇×C₄⋊C₄	—	2	1	no	—
51	C₂×C₇×C₄⋊C₄	—	2	1	no	—
52	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
57	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₄²×C₇	—	2	2	no	—
59	C₂×C₄²×C₇	—	2	2	no	—
60	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₇	Q₈	1	4	yes	—
64	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
66	C₇×#107	—	4	4	no	—
67	C₇×#106	—	4	4	no	—
68	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
69	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
70	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
71	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
72	C₇×#95	—	2	2	no	—
73	C₇×#95	—	2	2	no	—
74	C₇×#95	—	2	2	no	—
75	C₇×#95	—	2	2	no	—
76	C₂×#431	C₂×C₇	1	1	yes	characteristic
77	#2285	C₂×C₇	1	1	yes	characteristic
78	#2285	C₂×C₇	1	1	yes	characteristic
79	#1949	C₂×C₇	1	4	yes	complemented
80	#1854	C₂×C₇	1	4	yes	—
81	#1845	C₂×C₇	1	1	yes	characteristic, complemented
82	#1844	C₂×C₇	1	1	yes	characteristic, complemented
83	#1834	C₂×C₇	1	1	yes	characteristic, complemented
84	#1834	C₂×C₇	1	1	yes	characteristic, complemented
85	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	—	2	2	no	—
87	C₂²×C₄×C₇	—	2	2	no	—
88	C₂²×C₄×C₇	—	2	2	no	—
89	C₂²×C₄×C₇	—	2	2	no	—
90	C₂²×C₄×C₇	—	2	2	no	—
91	C₂²×C₄×C₇	—	2	2	no	—
92	C₂²×C₄×C₇	—	2	2	no	—
93	C₂²×C₄×C₇	—	2	2	no	—
94	C₂²×C₄×C₇	—	2	2	no	—
95	C₂²×C₄×C₇	—	2	2	no	—
96	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
97	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
98	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
106	C₂×C₇×C₈	—	2	2	no	—
107	C₂×C₇×C₈	—	2	2	no	—
108	C₂×C₇×C₈	—	2	2	no	—
109	C₂×C₇×C₈	—	2	2	no	—
110	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
111	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
112	C₇×C₄⋊C₄	—	2	4	no	—
113	C₇×C₄⋊C₄	—	2	4	no	—
114	C₇×C₄⋊C₄	—	2	4	no	—
115	C₇×C₄⋊C₄	—	2	4	no	—
116	C₇×C₄⋊C₄	—	4	2	no	—
117	C₇×C₄⋊C₄	—	4	2	no	—
118	C₇×C₄⋊C₄	—	4	2	no	—
119	C₇×C₄⋊C₄	—	4	2	no	—
120	C₇×C₄⋊C₄	—	4	2	no	—
121	C₇×C₄⋊C₄	—	4	2	no	—
122	C₇×C₄⋊C₄	—	4	2	no	—
123	C₇×C₄⋊C₄	—	4	2	no	—
124	C₇×C₂²⋊C₄	—	4	4	no	—
125	C₇×C₂²⋊C₄	—	4	4	no	—
126	C₄²×C₇	—	4	4	no	—
127	C₄²×C₇	—	4	4	no	—
128	C₄²×C₇	C₄⋊C₄	1	4	yes	—
129	C₄²×C₇	C₄⋊C₄	1	4	yes	—
130	C₄²×C₇	C₂²×C₄	1	2	yes	—
131	C₄²×C₇	C₂²×C₄	1	2	yes	—
132	C₄²×C₇	C₂×Q₈	1	2	yes	—
133	C₄²×C₇	C₂×Q₈	1	2	yes	—
134	C₂×#117	—	2	1	no	—
135	C₂×#117	C₂²×C₇	1	2	yes	—
136	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
137	#431	C₄×C₇	1	8	yes	complemented
138	C₂×#107	C₂²×C₇	1	4	yes	—
139	C₂×#106	C₂²×C₇	1	4	yes	—
140	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
141	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
144	#384	C₂²×C₇	1	1	yes	characteristic
145	#384	C₂²×C₇	1	1	yes	characteristic
146	#384	C₂²×C₇	1	1	yes	characteristic
147	#384	C₂²×C₇	1	1	yes	characteristic
148	#382	—	2	1	no	—
149	#382	C₂²×C₇	1	2	yes	—
150	#380	—	2	1	no	—
151	#380	—	2	1	no	—
152	#380	—	2	1	no	—
153	#380	C₂²×C₇	1	2	yes	—
154	C₄×C₄⋊C₄	—	2	1	no	—
155	C₄×C₄⋊C₄	—	2	1	no	—
156	C₄×C₄⋊C₄	—	2	1	no	—
157	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
158	#340	C₂²×C₇	1	4	yes	—
159	#340	C₂²×C₇	1	4	yes	—
160	C₂³×C₇	—	2	1	no	—
161	C₂³×C₇	—	2	1	no	—
162	C₂³×C₇	—	2	1	no	—
163	C₂³×C₇	—	2	1	no	—
164	C₂³×C₇	—	2	1	no	—
165	C₂³×C₇	—	2	1	no	—
166	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₇	—	2	1	no	—
169	C₂×C₄×C₇	—	2	1	no	—
170	C₂×C₄×C₇	—	2	1	no	—
171	C₂×C₄×C₇	—	2	1	no	—
172	C₂×C₄×C₇	—	2	1	no	—
173	C₂×C₄×C₇	—	2	1	no	—
174	C₂×C₄×C₇	—	2	4	no	—
175	C₂×C₄×C₇	—	2	4	no	—
176	C₂×C₄×C₇	—	2	8	no	—
177	C₂×C₄×C₇	—	2	8	no	—
178	C₂×C₄×C₇	—	4	2	no	—
179	C₂×C₄×C₇	—	4	2	no	—
180	C₂×C₄×C₇	—	4	2	no	—
181	C₂×C₄×C₇	—	4	2	no	—
182	C₂×C₄×C₇	—	4	2	no	—
183	C₂×C₄×C₇	—	4	2	no	—
184	C₂×C₄×C₇	—	4	2	no	—
185	C₂×C₄×C₇	—	4	2	no	—
186	C₂×C₄×C₇	—	4	2	no	—
187	C₂×C₄×C₇	—	4	2	no	—
188	C₂×C₄×C₇	—	4	2	no	—
189	C₂×C₄×C₇	—	4	2	no	—
190	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
191	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
192	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
193	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
194	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
195	C₂×C₄×C₇	#135	1	2	yes	—
196	C₂×C₄×C₇	#135	1	2	yes	—
197	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
198	C₂×C₄×C₇	#137	1	2	yes	—
199	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
201	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
202	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
203	C₇×C₈	—	2	4	no	—
204	C₇×C₈	—	2	4	no	—
205	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
206	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
207	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₇	1	2	yes	—
209	C₂²×C₈	C₂³×C₇	1	2	yes	—
210	#117	—	4	4	no	—
211	#117	—	4	4	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
229	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
230	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
231	C₂×C₄²	C₇×Q₈	1	4	yes	—
232	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
234	#107	—	4	4	no	—
235	#106	—	4	4	no	—
236	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
237	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
238	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
239	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	C₂²×C₇	—	2	2	no	—
245	C₂²×C₇	—	2	2	no	—
246	C₂²×C₇	—	2	2	no	—
247	C₂²×C₇	—	4	1	no	—
248	C₂²×C₇	—	4	1	no	—
249	C₂²×C₇	—	4	1	no	—
250	C₂²×C₇	—	4	1	no	—
251	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
252	C₂²×C₇	#535	1	1	yes	characteristic
253	C₂²×C₇	#536	1	1	yes	characteristic
254	C₂²×C₇	C₂×#135	1	1	yes	characteristic
255	C₂²×C₇	C₂×#135	1	1	yes	characteristic
256	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂²×C₇	C₂×#137	1	1	yes	characteristic
258	C₄×C₇	—	2	2	no	—
259	C₄×C₇	—	2	2	no	—
260	C₄×C₇	—	2	2	no	—
261	C₄×C₇	—	8	2	no	—
262	C₄×C₇	—	8	2	no	—
263	C₄×C₇	—	8	2	no	—
264	C₄×C₇	—	8	2	no	—
265	C₄×C₇	#427	1	4	yes	—
266	C₄×C₇	#428	1	4	yes	—
267	C₄×C₇	#520	1	2	yes	—
268	C₄×C₇	#544	1	2	yes	—
269	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
281	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
282	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
283	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
289	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
295	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	4	4	no	—
309	C₂²⋊C₄	—	4	4	no	—
310	C₄²	—	4	4	no	—
311	C₄²	—	4	4	no	—
312	C₄²	C₇×C₄⋊C₄	1	4	yes	—
313	C₄²	C₇×C₄⋊C₄	1	4	yes	—
314	C₄²	C₂²×C₄×C₇	1	2	yes	—
315	C₄²	C₂²×C₄×C₇	1	2	yes	—
316	C₄²	C₂×C₇×Q₈	1	2	yes	—
317	C₄²	C₂×C₇×Q₈	1	2	yes	—
318	C₂×C₇	—	4	2	no	—
319	C₂×C₇	#2317	1	1	yes	characteristic
320	C₂×C₇	C₂×#427	1	1	yes	characteristic
321	C₂×C₇	C₂×#428	1	1	yes	characteristic
322	C₂×C₇	#3207	1	1	yes	characteristic
323	C₂×C₇	#3210	1	1	yes	characteristic
324	C₂×C₇	#3228	1	1	yes	characteristic
325	C₂×C₇	#3231	1	1	yes	characteristic
326	C₂³	—	2	1	no	—
327	C₂³	—	2	1	no	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	—	2	1	no	—
331	C₂³	—	2	1	no	—
332	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
333	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
357	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
358	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
359	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
360	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
361	C₂×C₄	C₇×#135	1	2	yes	—
362	C₂×C₄	C₇×#135	1	2	yes	—
363	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
364	C₂×C₄	C₇×#137	1	2	yes	—
365	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
367	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
368	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
369	C₈	—	2	4	no	—
370	C₈	—	2	4	no	—
371	C₇	#21984	1	1	yes	characteristic, direct product factor, Sylow
372	C₂²	—	2	2	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	4	1	no	—
378	C₂²	—	4	1	no	—
379	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
380	C₂²	C₇×#535	1	1	yes	characteristic
381	C₂²	C₇×#536	1	1	yes	characteristic
382	C₂²	C₂×C₇×#135	1	1	yes	characteristic
383	C₂²	C₂×C₇×#135	1	1	yes	characteristic
384	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
385	C₂²	C₂×C₇×#137	1	1	yes	characteristic
386	C₄	—	2	2	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	2	no	—
389	C₄	—	8	2	no	—
390	C₄	—	8	2	no	—
391	C₄	—	8	2	no	—
392	C₄	—	8	2	no	—
393	C₄	C₇×#427	1	4	yes	—
394	C₄	C₇×#428	1	4	yes	—
395	C₄	C₇×#520	1	2	yes	—
396	C₄	C₇×#544	1	2	yes	—
397	C₂	—	4	2	no	—
398	C₂	C₇×#2317	1	1	yes	characteristic
399	C₂	C₂×C₇×#427	1	1	yes	characteristic
400	C₂	C₂×C₇×#428	1	1	yes	characteristic
401	C₂	C₇×#3207	1	1	yes	characteristic
402	C₂	C₇×#3210	1	1	yes	characteristic
403	C₂	C₇×#3228	1	1	yes	characteristic
404	C₂	C₇×#3231	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	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1949	C₂	semidirect
6. C₇×#1845	C₂	semidirect
7. C₇×#1844	C₂	semidirect
8. C₇×#1834	C₂	semidirect
9. C₇×#1834	C₂	semidirect
13. C₇×#431	C₄	semidirect
36. #21984	C₇	direct (N × H)
79. #1949	C₂×C₇	semidirect
81. #1845	C₂×C₇	semidirect
82. #1844	C₂×C₇	semidirect
83. #1834	C₂×C₇	semidirect
84. #1834	C₂×C₇	semidirect
137. #431	C₄×C₇	semidirect
371. C₇	#21984	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×#431	2
C₇×#2285	2
C₇×#1949	2
C₇×#1854	2
C₇×#1845	2
C₇×#1844	2
C₇×#1834	2
#21984	7

Subgroups — C7×#21984 #13199200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21984 #13199200