Subgroups — C₇×#21413 #13198629

Order 1792.

1070 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#541	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
3	C₇×#2314	C₂	1	1	yes	maximal, characteristic
4	C₇×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1894	C₂	1	2	yes	maximal
9	C₇×#1892	C₂	1	2	yes	maximal
10	C₇×#1763	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×#103	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×#541	—	2	4	no	—
15	C₇×#541	C₄	1	8	yes	complemented
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×#126	—	2	2	no	—
18	C₂×C₇×#125	—	2	2	no	—
19	C₂×C₇×#125	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
22	C₂×C₇×#117	C₂²	1	1	yes	characteristic
23	C₂×C₇×#103	C₂²	1	2	yes	—
24	C₂×C₇×#103	C₂²	1	2	yes	—
25	C₂×C₇×#103	C₂²	1	2	yes	—
26	C₂×C₇×#103	C₂²	1	2	yes	—
27	C₇×#405	—	2	4	no	—
28	C₂×C₇×#97	C₂²	1	1	yes	characteristic
29	C₂×C₇×#97	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#388	—	2	2	no	—
33	C₇×#387	—	2	2	no	—
34	C₇×#384	C₂²	1	1	yes	characteristic
35	C₇×#382	C₂²	1	1	yes	characteristic
36	C₇×#380	C₂²	1	2	yes	—
37	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	2	yes	—
42	#21413	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×Q₈	C₂³	1	2	yes	—
44	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂×C₇×M₁₆	—	2	2	no	—
46	C₂×C₇×M₁₆	—	2	2	no	—
47	C₂×C₇×M₁₆	—	2	2	no	—
48	C₂²×C₇×C₈	—	2	2	no	—
49	C₂²×C₇×C₈	—	2	2	no	—
50	C₂²×C₇×C₈	—	2	2	no	—
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₇×C₄⋊Q₈	—	2	2	no	—
56	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
57	C₇×#126	—	2	16	no	—
58	C₇×#125	—	2	16	no	—
59	C₇×#125	—	2	2	no	—
60	C₇×#125	C₂×C₄	1	4	yes	—
61	C₇×C₂²⋊Q₈	—	2	4	no	—
62	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
63	C₄×C₇×Q₈	—	2	4	no	—
64	C₄×C₇×Q₈	C₂×C₄	1	8	yes	—
65	C₇×#117	C₂×C₄	1	4	yes	—
66	C₇×#117	D₄	1	4	yes	—
67	C₂×C₇×C₄⋊C₄	—	2	2	no	—
68	C₂×C₇×C₄⋊C₄	—	2	2	no	—
69	C₂×C₇×C₄⋊C₄	—	2	2	no	—
70	C₂×C₇×C₄⋊C₄	—	2	2	no	—
71	C₂×C₇×C₄⋊C₄	—	2	2	no	—
72	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
73	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
74	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
75	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
76	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
78	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
79	C₂×C₄²×C₇	—	2	2	no	—
80	C₂×C₄²×C₇	—	2	2	no	—
81	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
83	C₇×#103	—	2	4	no	—
84	C₇×#103	—	2	4	no	—
85	C₇×#103	—	2	4	no	—
86	C₇×#103	—	2	4	no	—
87	C₇×#97	—	2	2	no	—
88	C₇×#97	—	2	2	no	—
89	C₄×C₇×C₈	—	2	2	no	—
90	C₄×C₇×C₈	—	2	2	no	—
91	C₇×#95	—	2	2	no	—
92	C₇×#95	—	2	2	no	—
93	C₇×#95	—	2	2	no	—
94	C₂×#541	C₂×C₇	1	1	yes	characteristic
95	C₂×#405	C₂×C₇	1	1	yes	characteristic
96	#2314	C₂×C₇	1	1	yes	characteristic
97	#1992	C₂×C₇	1	1	yes	characteristic, complemented
98	#1991	C₂×C₇	1	1	yes	characteristic, complemented
99	#1960	C₂×C₇	1	1	yes	characteristic, complemented
100	#1956	C₂×C₇	1	1	yes	characteristic, complemented
101	#1894	C₂×C₇	1	2	yes	—
102	#1892	C₂×C₇	1	2	yes	—
103	#1763	C₂×C₇	1	2	yes	complemented
104	C₄×#103	C₂×C₇	1	2	yes	complemented
105	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₇×Q₈	—	2	2	no	—
107	C₂×C₇×Q₈	—	2	2	no	—
108	C₂×C₇×Q₈	—	2	2	no	—
109	C₂×C₇×Q₈	—	2	2	no	—
110	C₂×C₇×Q₈	—	2	2	no	—
111	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
112	C₂²×C₄×C₇	—	2	1	no	—
113	C₂²×C₄×C₇	—	2	1	no	—
114	C₂²×C₄×C₇	—	2	1	no	—
115	C₂²×C₄×C₇	—	2	1	no	—
116	C₂²×C₄×C₇	—	2	2	no	—
117	C₂²×C₄×C₇	—	2	2	no	—
118	C₂²×C₄×C₇	—	2	2	no	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
126	C₇×M₁₆	—	2	8	no	—
127	C₂×C₇×C₈	—	2	8	no	—
128	C₂×C₇×C₈	—	2	1	no	—
129	C₂×C₇×C₈	—	2	1	no	—
130	C₂×C₇×C₈	—	2	1	no	—
131	C₂×C₇×C₈	—	2	1	no	—
132	C₂×C₇×C₈	—	2	1	no	—
133	C₂×C₇×C₈	—	2	1	no	—
134	C₂×C₇×C₈	—	2	1	no	—
135	C₂×C₇×C₈	—	2	1	no	—
136	C₂×C₇×C₈	#41	1	2	yes	—
137	C₂×C₇×C₈	#41	1	2	yes	—
138	C₂×C₇×C₈	#41	1	2	yes	—
139	C₂×C₇×C₈	#41	1	2	yes	—
140	C₇×C₄⋊C₄	—	2	8	no	—
141	C₇×C₄⋊C₄	—	2	8	no	—
142	C₇×C₄⋊C₄	—	2	8	no	—
143	C₇×C₄⋊C₄	—	2	8	no	—
144	C₇×C₄⋊C₄	—	2	2	no	—
145	C₇×C₄⋊C₄	—	2	2	no	—
146	C₇×C₄⋊C₄	—	2	2	no	—
147	C₇×C₄⋊C₄	—	2	2	no	—
148	C₇×C₄⋊C₄	—	2	2	no	—
149	C₇×C₄⋊C₄	—	8	2	no	—
150	C₇×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
151	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
152	C₇×C₂²⋊C₄	—	2	8	no	—
153	C₇×C₂²⋊C₄	—	8	2	no	—
154	C₇×C₂²⋊C₄	C₂²⋊C₄	1	8	yes	—
155	C₄²×C₇	—	2	8	no	—
156	C₄²×C₇	—	8	2	no	—
157	C₄²×C₇	C₂²×C₄	1	2	yes	—
158	C₄²×C₇	C₂²×C₄	1	2	yes	—
159	C₄²×C₇	C₂×D₄	1	2	yes	—
160	C₄²×C₇	C₂×D₄	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
162	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
163	#541	—	2	4	no	—
164	#541	C₄×C₇	1	8	yes	complemented
165	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
166	C₂×#126	—	2	2	no	—
167	C₂×#125	—	2	2	no	—
168	C₂×#125	C₂²×C₇	1	1	yes	characteristic
169	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
170	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
171	C₂×#117	C₂²×C₇	1	1	yes	characteristic
172	C₂×#103	C₂²×C₇	1	2	yes	—
173	C₂×#103	C₂²×C₇	1	2	yes	—
174	C₂×#103	C₂²×C₇	1	2	yes	—
175	C₂×#103	C₂²×C₇	1	2	yes	—
176	#405	—	2	4	no	—
177	C₂×#97	C₂²×C₇	1	1	yes	characteristic
178	C₂×#97	C₂²×C₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
181	#388	—	2	2	no	—
182	#387	—	2	2	no	—
183	#384	C₂²×C₇	1	1	yes	characteristic
184	#382	C₂²×C₇	1	1	yes	characteristic
185	#380	C₂²×C₇	1	2	yes	—
186	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
187	#340	C₂²×C₇	1	2	yes	—
188	#340	C₂²×C₇	1	2	yes	—
189	#340	C₂²×C₇	1	2	yes	—
190	#340	C₂²×C₇	1	2	yes	—
191	C₂³×C₇	—	2	1	no	—
192	C₂³×C₇	—	2	1	no	—
193	C₂³×C₇	—	2	2	no	—
194	C₂³×C₇	—	2	2	no	—
195	C₂³×C₇	C₂×C₂²⋊C₄	1	2	yes	—
196	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
197	C₇×Q₈	—	2	4	no	—
198	C₇×Q₈	—	2	4	no	—
199	C₇×Q₈	—	4	2	no	—
200	C₇×Q₈	—	4	2	no	—
201	C₂×C₄×C₇	—	2	1	no	—
202	C₂×C₄×C₇	—	2	1	no	—
203	C₂×C₄×C₇	—	2	2	no	—
204	C₂×C₄×C₇	—	2	2	no	—
205	C₂×C₄×C₇	—	2	2	no	—
206	C₂×C₄×C₇	—	2	2	no	—
207	C₂×C₄×C₇	—	2	2	no	—
208	C₂×C₄×C₇	—	2	2	no	—
209	C₂×C₄×C₇	—	2	2	no	—
210	C₂×C₄×C₇	—	2	2	no	—
211	C₂×C₄×C₇	—	2	4	no	—
212	C₂×C₄×C₇	—	2	4	no	—
213	C₂×C₄×C₇	—	4	2	no	—
214	C₂×C₄×C₇	—	4	2	no	—
215	C₂×C₄×C₇	—	4	2	no	—
216	C₂×C₄×C₇	—	4	2	no	—
217	C₂×C₄×C₇	—	4	2	no	—
218	C₂×C₄×C₇	—	4	2	no	—
219	C₂×C₄×C₇	—	4	2	no	—
220	C₂×C₄×C₇	—	4	2	no	—
221	C₂×C₄×C₇	—	4	2	no	—
222	C₂×C₄×C₇	—	4	2	no	—
223	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
224	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
225	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
226	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
227	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
232	C₇×C₈	—	2	2	no	—
233	C₇×C₈	—	2	2	no	—
234	C₇×C₈	—	2	2	no	—
235	C₇×C₈	—	2	2	no	—
236	C₂²×Q₈	C₂³×C₇	1	2	yes	—
237	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂²×C₈	—	2	2	no	—
242	C₂²×C₈	—	2	2	no	—
243	C₂²×C₈	—	2	2	no	—
244	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	2	2	no	—
249	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
250	#126	—	2	16	no	—
251	#125	—	2	16	no	—
252	#125	—	2	2	no	—
253	#125	C₂×C₄×C₇	1	4	yes	—
254	C₂²⋊Q₈	—	2	4	no	—
255	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	—
256	C₄×Q₈	—	2	4	no	—
257	C₄×Q₈	C₂×C₄×C₇	1	8	yes	—
258	#117	C₂×C₄×C₇	1	4	yes	—
259	#117	C₇×D₄	1	4	yes	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
266	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
267	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
268	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	—	2	2	no	—
274	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
276	#103	—	2	4	no	—
277	#103	—	2	4	no	—
278	#103	—	2	4	no	—
279	#103	—	2	4	no	—
280	#97	—	2	2	no	—
281	#97	—	2	2	no	—
282	C₄×C₈	—	2	2	no	—
283	C₄×C₈	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	#95	—	2	2	no	—
287	C₂²×C₇	—	2	4	no	—
288	C₂²×C₇	—	2	2	no	—
289	C₂²×C₇	—	4	1	no	—
290	C₂²×C₇	—	4	1	no	—
291	C₂²×C₇	—	4	2	no	—
292	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
293	C₂²×C₇	#532	1	1	yes	characteristic
294	C₂²×C₇	#532	1	1	yes	characteristic
295	C₂²×C₇	#577	1	2	yes	—
296	C₂²×C₇	#578	1	2	yes	—
297	C₄×C₇	—	2	4	no	—
298	C₄×C₇	—	2	4	no	—
299	C₄×C₇	—	2	2	no	—
300	C₄×C₇	—	4	4	no	—
301	C₄×C₇	—	4	4	no	—
302	C₄×C₇	—	8	2	no	—
303	C₄×C₇	—	8	2	no	—
304	C₄×C₇	C₄×#41	1	1	yes	characteristic
305	C₄×C₇	C₄×#41	1	1	yes	characteristic
306	C₄×C₇	#525	1	1	yes	characteristic
307	C₄×C₇	#525	1	1	yes	characteristic
308	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
324	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
325	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
326	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
327	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
328	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
329	M₁₆	—	2	8	no	—
330	C₂×C₈	—	2	8	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	C₇×#41	1	2	yes	—
340	C₂×C₈	C₇×#41	1	2	yes	—
341	C₂×C₈	C₇×#41	1	2	yes	—
342	C₂×C₈	C₇×#41	1	2	yes	—
343	C₄⋊C₄	—	2	8	no	—
344	C₄⋊C₄	—	2	8	no	—
345	C₄⋊C₄	—	2	8	no	—
346	C₄⋊C₄	—	2	8	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	8	2	no	—
353	C₄⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
354	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
355	C₂²⋊C₄	—	2	8	no	—
356	C₂²⋊C₄	—	8	2	no	—
357	C₂²⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
358	C₄²	—	2	8	no	—
359	C₄²	—	8	2	no	—
360	C₄²	C₂²×C₄×C₇	1	2	yes	—
361	C₄²	C₂²×C₄×C₇	1	2	yes	—
362	C₄²	C₂×C₇×D₄	1	2	yes	—
363	C₄²	C₂×C₇×D₄	1	2	yes	—
364	C₂×C₇	—	4	2	no	—
365	C₂×C₇	#2297	1	1	yes	characteristic
366	C₂×C₇	#2898	1	2	yes	—
367	C₂×C₇	#3174	1	1	yes	characteristic
368	C₂×C₇	#3177	1	1	yes	characteristic
369	C₂×C₇	#3178	1	2	yes	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	C₂×C₇×C₂²⋊C₄	1	2	yes	—
375	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
376	Q₈	—	2	4	no	—
377	Q₈	—	2	4	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	4	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
403	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
404	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
405	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
406	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
407	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
408	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
409	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
410	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
411	C₈	—	2	2	no	—
412	C₈	—	2	2	no	—
413	C₈	—	2	2	no	—
414	C₈	—	2	2	no	—
415	C₇	#21413	1	1	yes	characteristic, direct product factor, Sylow
416	C₂²	—	2	2	no	—
417	C₂²	—	2	4	no	—
418	C₂²	—	4	1	no	—
419	C₂²	—	4	1	no	—
420	C₂²	—	4	2	no	—
421	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
422	C₂²	C₇×#532	1	1	yes	characteristic
423	C₂²	C₇×#532	1	1	yes	characteristic
424	C₂²	C₇×#577	1	2	yes	—
425	C₂²	C₇×#578	1	2	yes	—
426	C₄	—	2	2	no	—
427	C₄	—	2	4	no	—
428	C₄	—	2	4	no	—
429	C₄	—	4	4	no	—
430	C₄	—	4	4	no	—
431	C₄	—	8	2	no	—
432	C₄	—	8	2	no	—
433	C₄	C₄×C₇×#41	1	1	yes	characteristic
434	C₄	C₄×C₇×#41	1	1	yes	characteristic
435	C₄	C₇×#525	1	1	yes	characteristic
436	C₄	C₇×#525	1	1	yes	characteristic
437	C₂	—	4	2	no	—
438	C₂	C₇×#2297	1	1	yes	characteristic
439	C₂	C₇×#2898	1	2	yes	—
440	C₂	C₇×#3174	1	1	yes	characteristic
441	C₂	C₇×#3177	1	1	yes	characteristic
442	C₂	C₇×#3178	1	2	yes	—

	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 ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1992	C₂	semidirect
5. C₇×#1991	C₂	semidirect
6. C₇×#1960	C₂	semidirect
7. C₇×#1956	C₂	semidirect
10. C₇×#1763	C₂	semidirect
11. C₄×C₇×#103	C₂	semidirect
15. C₇×#541	C₄	semidirect
42. #21413	C₇	direct (N × H)
97. #1992	C₂×C₇	semidirect
98. #1991	C₂×C₇	semidirect
99. #1960	C₂×C₇	semidirect
100. #1956	C₂×C₇	semidirect
103. #1763	C₂×C₇	semidirect
104. C₄×#103	C₂×C₇	semidirect
164. #541	C₄×C₇	semidirect
415. C₇	#21413	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×#541	2
C₂×C₇×#405	2
C₇×#2314	2
C₇×#1992	2
C₇×#1991	2
C₇×#1960	2
C₇×#1956	2
C₇×#1894	2
C₇×#1892	2
C₇×#1763	2
C₄×C₇×#103	2
#21413	7

Subgroups — C7×#21413 #13198629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21413 #13198629