Subgroups — C₇×#22101 #13199317

Order 1792.

736 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#433	C₂	1	1	yes	maximal, characteristic
3	C₇×#2396	C₂	1	1	yes	maximal, characteristic
4	C₇×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1940	C₂	1	2	yes	maximal, complemented
6	C₇×#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1924	C₂	1	2	yes	maximal
8	C₇×#1923	C₂	1	2	yes	maximal
9	C₇×#1916	C₂	1	2	yes	maximal, complemented
10	C₇×#1798	C₂	1	1	yes	maximal, characteristic
11	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×#117	C₂²	1	1	yes	characteristic
14	C₇×C₈×Q₈	—	2	4	no	—
15	C₇×#433	—	2	4	no	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₇×#97	C₂²	1	1	yes	characteristic
26	C₂×C₇×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#400	C₄	1	1	yes	characteristic
30	C₇×#399	C₄	1	1	yes	characteristic
31	C₇×#398	C₄	1	2	yes	—
32	C₇×#396	C₄	1	2	yes	—
33	C₇×#395	C₄	1	1	yes	characteristic
34	C₇×#393	C₄	1	1	yes	characteristic
35	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
37	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
38	C₇×#336	C₂²	1	2	yes	—
39	C₇×#336	C₂²	1	2	yes	—
40	C₇×#336	C₂²	1	2	yes	—
41	C₇×#336	C₂²	1	2	yes	—
42	C₇×#336	C₂²	1	2	yes	—
43	#22101	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇×C₈	C₂³	1	2	yes	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₇×Q₈	—	2	2	no	—
53	C₄×C₇×Q₈	—	2	2	no	—
54	C₇×#117	—	2	2	no	—
55	C₇×#117	—	2	2	no	—
56	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
60	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
62	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
64	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₇	C₂³	1	2	yes	—
69	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
71	C₇×C₄⋊C₈	—	2	2	no	—
72	C₇×C₄⋊C₈	—	2	2	no	—
73	C₇×C₄⋊C₈	—	2	4	no	—
74	C₇×C₄⋊C₈	—	2	2	no	—
75	C₇×C₄⋊C₈	—	2	2	no	—
76	C₇×C₄⋊C₈	—	4	2	no	—
77	C₇×C₂²⋊C₈	—	2	2	no	—
78	C₇×C₂²⋊C₈	—	2	2	no	—
79	C₇×C₂²⋊C₈	—	4	2	no	—
80	C₇×#97	—	2	2	no	—
81	C₇×#97	—	4	2	no	—
82	C₄×C₇×C₈	—	2	4	no	—
83	C₄×C₇×C₈	D₄	1	4	yes	—
84	C₇×#95	C₂×C₄	1	2	yes	—
85	C₇×#95	C₂×C₄	1	2	yes	—
86	C₇×#95	C₂×C₄	1	2	yes	—
87	C₇×#95	C₂×C₄	1	1	yes	characteristic
88	C₇×#95	C₂×C₄	1	1	yes	characteristic
89	C₇×#95	C₂×C₄	1	2	yes	—
90	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
91	C₂×#433	C₂×C₇	1	1	yes	characteristic
92	#2396	C₂×C₇	1	1	yes	characteristic
93	#1985	C₂×C₇	1	1	yes	characteristic, complemented
94	#1940	C₂×C₇	1	2	yes	complemented
95	#1940	C₂×C₇	1	1	yes	characteristic, complemented
96	#1924	C₂×C₇	1	2	yes	—
97	#1923	C₂×C₇	1	2	yes	—
98	#1916	C₂×C₇	1	2	yes	complemented
99	#1798	C₂×C₇	1	1	yes	characteristic
100	#1770	C₂×C₇	1	1	yes	characteristic, complemented
101	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×Q₈	—	2	2	no	—
103	C₂×C₇×Q₈	—	2	4	no	—
104	C₂²×C₄×C₇	—	2	1	no	—
105	C₂²×C₄×C₇	—	2	1	no	—
106	C₂²×C₄×C₇	—	2	1	no	—
107	C₂²×C₄×C₇	—	2	1	no	—
108	C₂²×C₄×C₇	—	2	1	no	—
109	C₂²×C₄×C₇	—	2	1	no	—
110	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
120	C₂×C₇×C₈	—	2	1	no	—
121	C₂×C₇×C₈	—	2	1	no	—
122	C₂×C₇×C₈	—	2	1	no	—
123	C₂×C₇×C₈	—	2	1	no	—
124	C₂×C₇×C₈	—	2	2	no	—
125	C₂×C₇×C₈	—	2	1	no	—
126	C₂×C₇×C₈	—	2	1	no	—
127	C₂×C₇×C₈	—	2	2	no	—
128	C₂×C₇×C₈	—	2	2	no	—
129	C₂×C₇×C₈	—	2	2	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	2	no	—
134	C₂×C₇×C₈	—	2	1	no	—
135	C₂×C₇×C₈	—	2	2	no	—
136	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
137	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
138	C₇×C₄⋊C₄	—	2	2	no	—
139	C₇×C₄⋊C₄	—	2	2	no	—
140	C₇×C₄⋊C₄	—	2	4	no	—
141	C₇×C₄⋊C₄	—	2	2	no	—
142	C₇×C₄⋊C₄	—	2	4	no	—
143	C₇×C₄⋊C₄	—	4	2	no	—
144	C₇×C₄⋊C₄	—	4	2	no	—
145	C₇×C₂²⋊C₄	—	2	2	no	—
146	C₇×C₂²⋊C₄	—	2	2	no	—
147	C₇×C₂²⋊C₄	—	2	4	no	—
148	C₇×C₂²⋊C₄	—	4	2	no	—
149	C₄²×C₇	—	2	2	no	—
150	C₄²×C₇	—	2	1	no	—
151	C₄²×C₇	—	2	1	no	—
152	C₄²×C₇	—	2	2	no	—
153	C₄²×C₇	—	2	2	no	—
154	C₄²×C₇	—	2	2	no	—
155	C₄²×C₇	C₂×D₄	1	2	yes	—
156	C₄²×C₇	#41	1	2	yes	—
157	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
158	C₂×#117	C₂²×C₇	1	1	yes	characteristic
159	C₈×Q₈	—	2	4	no	—
160	#433	—	2	4	no	—
161	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
167	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
168	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
170	C₂×#97	C₂²×C₇	1	1	yes	characteristic
171	C₂×#97	C₂²×C₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
174	#400	C₄×C₇	1	1	yes	characteristic
175	#399	C₄×C₇	1	1	yes	characteristic
176	#398	C₄×C₇	1	2	yes	—
177	#396	C₄×C₇	1	2	yes	—
178	#395	C₄×C₇	1	1	yes	characteristic
179	#393	C₄×C₇	1	1	yes	characteristic
180	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
182	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
183	#336	C₂²×C₇	1	2	yes	—
184	#336	C₂²×C₇	1	2	yes	—
185	#336	C₂²×C₇	1	2	yes	—
186	#336	C₂²×C₇	1	2	yes	—
187	#336	C₂²×C₇	1	2	yes	—
188	C₂³×C₇	—	2	1	no	—
189	C₂³×C₇	—	2	1	no	—
190	C₂³×C₇	—	2	1	no	—
191	C₂³×C₇	—	2	2	no	—
192	C₂³×C₇	—	2	2	no	—
193	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
194	C₇×Q₈	—	2	8	no	—
195	C₂×C₄×C₇	—	2	2	no	—
196	C₂×C₄×C₇	—	2	2	no	—
197	C₂×C₄×C₇	—	2	2	no	—
198	C₂×C₄×C₇	—	2	2	no	—
199	C₂×C₄×C₇	—	2	1	no	—
200	C₂×C₄×C₇	—	2	1	no	—
201	C₂×C₄×C₇	—	2	2	no	—
202	C₂×C₄×C₇	—	2	1	no	—
203	C₂×C₄×C₇	—	2	1	no	—
204	C₂×C₄×C₇	—	2	2	no	—
205	C₂×C₄×C₇	—	2	2	no	—
206	C₂×C₄×C₇	—	2	1	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	2	no	—
212	C₂×C₄×C₇	—	2	2	no	—
213	C₂×C₄×C₇	—	2	2	no	—
214	C₂×C₄×C₇	—	2	2	no	—
215	C₂×C₄×C₇	—	4	1	no	—
216	C₂×C₄×C₇	—	4	1	no	—
217	C₂×C₄×C₇	—	4	1	no	—
218	C₂×C₄×C₇	—	4	1	no	—
219	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
220	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
221	C₂×C₄×C₇	#131	1	4	yes	—
222	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
226	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
227	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
228	C₇×C₈	—	2	2	no	—
229	C₇×C₈	—	2	2	no	—
230	C₇×C₈	—	4	1	no	—
231	C₇×C₈	—	4	2	no	—
232	C₇×C₈	—	4	1	no	—
233	C₇×C₈	—	4	2	no	—
234	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
235	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
236	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
237	C₂²×C₈	C₂³×C₇	1	2	yes	—
238	C₂²×C₈	C₂³×C₇	1	2	yes	—
239	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
240	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
241	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	2	2	no	—
244	#117	—	2	2	no	—
245	#117	—	2	2	no	—
246	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
249	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
250	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
251	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
252	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
253	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
254	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
255	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
256	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
257	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
258	C₂×C₄²	C₂³×C₇	1	2	yes	—
259	C₂×C₄²	C₂³×C₇	1	2	yes	—
260	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	2	no	—
263	C₄⋊C₈	—	2	4	no	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	4	2	no	—
267	C₂²⋊C₈	—	2	2	no	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	—	4	2	no	—
270	#97	—	2	2	no	—
271	#97	—	4	2	no	—
272	C₄×C₈	—	2	4	no	—
273	C₄×C₈	C₇×D₄	1	4	yes	—
274	#95	C₂×C₄×C₇	1	2	yes	—
275	#95	C₂×C₄×C₇	1	2	yes	—
276	#95	C₂×C₄×C₇	1	2	yes	—
277	#95	C₂×C₄×C₇	1	1	yes	characteristic
278	#95	C₂×C₄×C₇	1	1	yes	characteristic
279	#95	C₂×C₄×C₇	1	2	yes	—
280	C₂²×C₇	—	4	1	no	—
281	C₂²×C₇	—	4	1	no	—
282	C₂²×C₇	—	4	1	no	—
283	C₂²×C₇	—	4	2	no	—
284	C₂²×C₇	—	4	2	no	—
285	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
286	C₂²×C₇	#519	1	1	yes	characteristic
287	C₂²×C₇	#520	1	1	yes	characteristic
288	C₂²×C₇	C₂×#131	1	2	yes	—
289	C₂²×C₇	#568	1	2	yes	—
290	C₄×C₇	—	2	2	no	—
291	C₄×C₇	—	2	4	no	—
292	C₄×C₇	—	2	2	no	—
293	C₄×C₇	—	4	1	no	—
294	C₄×C₇	—	4	1	no	—
295	C₄×C₇	—	4	1	no	—
296	C₄×C₇	—	4	2	no	—
297	C₄×C₇	—	4	2	no	—
298	C₄×C₇	—	4	2	no	—
299	C₄×C₇	#536	1	1	yes	characteristic
300	C₄×C₇	#537	1	1	yes	characteristic
301	C₄×C₇	#541	1	1	yes	characteristic
302	C₄×C₇	#542	1	1	yes	characteristic
303	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
313	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
316	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
317	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
318	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
319	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
320	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
321	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
339	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	2	4	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	C₂×C₇×D₄	1	2	yes	—
358	C₄²	C₇×#41	1	2	yes	—
359	C₂×C₇	—	8	1	no	—
360	C₂×C₇	#2323	1	1	yes	characteristic
361	C₂×C₇	#2929	1	1	yes	characteristic
362	C₂×C₇	#2932	1	1	yes	characteristic
363	C₂×C₇	#2985	1	2	yes	—
364	C₂×C₇	#2993	1	2	yes	—
365	C₂³	—	2	1	no	—
366	C₂³	—	2	1	no	—
367	C₂³	—	2	1	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	2	no	—
370	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
371	Q₈	—	2	8	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	1	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	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
397	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
398	C₂×C₄	C₇×#131	1	4	yes	—
399	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
400	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
401	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
402	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
403	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
404	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
405	C₈	—	2	2	no	—
406	C₈	—	2	2	no	—
407	C₈	—	4	1	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	1	no	—
410	C₈	—	4	2	no	—
411	C₇	#22101	1	1	yes	characteristic, direct product factor, Sylow
412	C₂²	—	4	1	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	1	no	—
417	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
418	C₂²	C₇×#519	1	1	yes	characteristic
419	C₂²	C₇×#520	1	1	yes	characteristic
420	C₂²	C₂×C₇×#131	1	2	yes	—
421	C₂²	C₇×#568	1	2	yes	—
422	C₄	—	2	2	no	—
423	C₄	—	2	4	no	—
424	C₄	—	2	2	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	1	no	—
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	1	no	—
431	C₄	C₇×#536	1	1	yes	characteristic
432	C₄	C₇×#537	1	1	yes	characteristic
433	C₄	C₇×#541	1	1	yes	characteristic
434	C₄	C₇×#542	1	1	yes	characteristic
435	C₂	—	8	1	no	—
436	C₂	C₇×#2323	1	1	yes	characteristic
437	C₂	C₇×#2929	1	1	yes	characteristic
438	C₂	C₇×#2932	1	1	yes	characteristic
439	C₂	C₇×#2985	1	2	yes	—
440	C₂	C₇×#2993	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₈×Q₈	C₂	semidirect
4. C₇×#1985	C₂	semidirect
5. C₇×#1940	C₂	semidirect
6. C₇×#1940	C₂	semidirect
9. C₇×#1916	C₂	semidirect
11. C₇×#1770	C₂	semidirect
43. #22101	C₇	direct (N × H)
90. C₂×C₈×Q₈	C₂×C₇	semidirect
93. #1985	C₂×C₇	semidirect
94. #1940	C₂×C₇	semidirect
95. #1940	C₂×C₇	semidirect
98. #1916	C₂×C₇	semidirect
100. #1770	C₂×C₇	semidirect
411. C₇	#22101	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×C₈×Q₈	2
C₂×C₇×#433	2
C₇×#2396	2
C₇×#1985	2
C₇×#1940	2
C₇×#1924	2
C₇×#1923	2
C₇×#1916	2
C₇×#1798	2
C₇×#1770	2
#22101	7

Subgroups — C7×#22101 #13199317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22101 #13199317