Subgroups — C₇×#23215 #13200431

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#486	C₂	1	1	yes	maximal, characteristic
2	C₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2366	C₂	1	1	yes	maximal, characteristic
4	C₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1982	C₂	1	1	yes	maximal, characteristic
6	C₇×#1934	C₂	1	2	yes	maximal, complemented
7	C₇×#1926	C₂	1	2	yes	maximal, complemented
8	C₇×#1864	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#102	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×D₄	—	2	1	no	—
15	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
16	C₇×#486	C₄	1	8	yes	complemented
17	C₂×C₇×#106	C₂²	1	1	yes	characteristic
18	C₂×C₇×#106	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×#102	C₂²	1	2	yes	—
21	C₂×C₇×#102	C₂²	1	2	yes	—
22	C₂×C₇×#102	C₂²	1	2	yes	—
23	C₂×C₇×#102	C₂²	1	2	yes	—
24	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#391	—	2	2	no	—
27	C₇×#390	—	2	1	no	—
28	C₇×#390	C₂²	1	2	yes	—
29	C₇×#389	—	2	1	no	—
30	C₇×#389	C₂²	1	2	yes	—
31	C₇×#386	—	2	2	no	—
32	C₄×C₇×C₄⋊C₄	—	2	1	no	—
33	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
36	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
41	C₇×#336	C₂²	1	2	yes	—
42	#23215	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×Q₈	—	2	1	no	—
44	C₂²×C₇×D₄	—	2	1	no	—
45	C₂²×C₇×D₄	—	2	1	no	—
46	C₂²×C₇×D₄	C₂³	1	2	yes	—
47	C₂³×C₄×C₇	—	2	2	no	—
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	C₇×C₄⋊Q₈	—	2	2	no	—
52	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
53	C₇×#127	—	2	2	no	—
54	C₇×#127	C₂×C₄	1	4	yes	—
55	C₄×C₇×D₄	—	2	4	no	—
56	C₄×C₇×D₄	—	4	2	no	—
57	C₄×C₇×D₄	—	4	4	no	—
58	C₂×C₇×C₄⋊C₄	—	2	1	no	—
59	C₂×C₇×C₄⋊C₄	—	2	2	no	—
60	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
63	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
65	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
68	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
69	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
70	C₂×C₄²×C₇	—	2	1	no	—
71	C₂×C₄²×C₇	—	2	1	no	—
72	C₂×C₄²×C₇	—	2	2	no	—
73	C₂×C₄²×C₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
75	C₇×#106	—	2	2	no	—
76	C₇×#106	—	2	2	no	—
77	C₇×C₄⋊C₈	—	2	2	no	—
78	C₇×#102	—	2	4	no	—
79	C₇×#102	—	4	2	no	—
80	C₇×#102	C₂×C₄	1	8	yes	complemented
81	C₇×#102	C₂×C₄	1	8	yes	complemented
82	C₄×C₇×C₈	—	4	2	no	—
83	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
84	C₇×#95	—	2	2	no	—
85	C₇×#95	—	2	1	no	—
86	C₇×#95	—	2	2	no	—
87	C₇×#95	—	2	1	no	—
88	C₇×#95	D₄	1	2	yes	—
89	C₂×#486	C₂×C₇	1	1	yes	characteristic
90	#2369	C₂×C₇	1	1	yes	characteristic, complemented
91	#2366	C₂×C₇	1	1	yes	characteristic
92	#1987	C₂×C₇	1	1	yes	characteristic, complemented
93	#1982	C₂×C₇	1	1	yes	characteristic
94	#1934	C₂×C₇	1	2	yes	complemented
95	#1926	C₂×C₇	1	2	yes	complemented
96	#1864	C₂×C₇	1	2	yes	complemented
97	C₄×#106	C₂×C₇	1	1	yes	characteristic, complemented
98	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
99	C₄×#102	C₂×C₇	1	2	yes	complemented
100	C₂⁴×C₇	—	2	2	no	—
101	C₂×C₇×Q₈	—	2	4	no	—
102	C₂×C₇×Q₈	—	8	1	no	—
103	C₂×C₇×Q₈	—	8	1	no	—
104	C₂×C₇×D₄	—	2	4	no	—
105	C₂×C₇×D₄	—	2	4	no	—
106	C₂×C₇×D₄	—	2	2	no	—
107	C₂×C₇×D₄	—	2	2	no	—
108	C₂×C₇×D₄	—	2	2	no	—
109	C₂×C₇×D₄	—	2	2	no	—
110	C₂×C₇×D₄	—	2	2	no	—
111	C₂×C₇×D₄	—	4	2	no	—
112	C₂×C₇×D₄	—	4	2	no	—
113	C₂×C₇×D₄	—	8	1	no	—
114	C₂×C₇×D₄	—	8	1	no	—
115	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
116	C₂²×C₄×C₇	—	2	1	no	—
117	C₂²×C₄×C₇	—	2	2	no	—
118	C₂²×C₄×C₇	—	2	1	no	—
119	C₂²×C₄×C₇	—	2	1	no	—
120	C₂²×C₄×C₇	—	2	1	no	—
121	C₂²×C₄×C₇	—	4	4	no	—
122	C₂²×C₄×C₇	—	4	4	no	—
123	C₂²×C₄×C₇	—	8	2	no	—
124	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
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	2	no	—
134	C₂×C₇×C₈	—	2	2	no	—
135	C₂×C₇×C₈	—	2	2	no	—
136	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
138	C₇×C₄⋊C₄	—	2	4	no	—
139	C₇×C₄⋊C₄	—	2	2	no	—
140	C₇×C₄⋊C₄	—	2	2	no	—
141	C₇×C₄⋊C₄	—	2	1	no	—
142	C₇×C₄⋊C₄	—	2	1	no	—
143	C₇×C₄⋊C₄	—	2	1	no	—
144	C₇×C₄⋊C₄	—	2	1	no	—
145	C₇×C₄⋊C₄	—	2	1	no	—
146	C₇×C₄⋊C₄	—	2	1	no	—
147	C₇×C₄⋊C₄	—	8	2	no	—
148	C₇×C₄⋊C₄	SD₁₆	1	4	yes	—
149	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
150	C₇×C₄⋊C₄	#41	1	2	yes	—
151	C₇×C₄⋊C₄	#41	1	2	yes	—
152	C₇×C₂²⋊C₄	—	4	4	no	—
153	C₇×C₂²⋊C₄	—	4	4	no	—
154	C₇×C₂²⋊C₄	—	8	2	no	—
155	C₄²×C₇	—	2	2	no	—
156	C₄²×C₇	—	2	2	no	—
157	C₄²×C₇	—	4	2	no	—
158	C₄²×C₇	—	4	2	no	—
159	C₄²×C₇	—	8	2	no	—
160	C₄²×C₇	C₂²×C₄	1	2	yes	—
161	C₄²×C₇	#41	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
163	C₂×#127	C₂²×C₇	1	1	yes	characteristic
164	C₂×C₄×D₄	—	2	1	no	—
165	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
166	#486	C₄×C₇	1	8	yes	complemented
167	C₂×#106	C₂²×C₇	1	1	yes	characteristic
168	C₂×#106	C₂²×C₇	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
170	C₂×#102	C₂²×C₇	1	2	yes	—
171	C₂×#102	C₂²×C₇	1	2	yes	—
172	C₂×#102	C₂²×C₇	1	2	yes	—
173	C₂×#102	C₂²×C₇	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
176	#391	—	2	2	no	—
177	#390	—	2	1	no	—
178	#390	C₂²×C₇	1	2	yes	—
179	#389	—	2	1	no	—
180	#389	C₂²×C₇	1	2	yes	—
181	#386	—	2	2	no	—
182	C₄×C₄⋊C₄	—	2	1	no	—
183	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
184	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
185	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
186	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
187	#340	C₂²×C₇	1	2	yes	—
188	#340	C₂²×C₇	1	2	yes	—
189	#340	C₂²×C₇	1	1	yes	characteristic
190	#340	C₂²×C₇	1	1	yes	characteristic
191	#336	C₂²×C₇	1	2	yes	—
192	C₂³×C₇	—	2	4	no	—
193	C₂³×C₇	—	4	2	no	—
194	C₂³×C₇	—	4	2	no	—
195	C₂³×C₇	—	4	2	no	—
196	C₂³×C₇	—	4	2	no	—
197	C₂³×C₇	—	4	2	no	—
198	C₂³×C₇	—	4	2	no	—
199	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
200	C₇×Q₈	—	8	4	no	—
201	C₇×D₄	—	2	4	no	—
202	C₇×D₄	—	2	4	no	—
203	C₇×D₄	—	4	2	no	—
204	C₇×D₄	—	4	2	no	—
205	C₇×D₄	—	4	8	no	—
206	C₇×D₄	—	8	4	no	—
207	C₂×C₄×C₇	—	2	2	no	—
208	C₂×C₄×C₇	—	2	2	no	—
209	C₂×C₄×C₇	—	2	1	no	—
210	C₂×C₄×C₇	—	2	1	no	—
211	C₂×C₄×C₇	—	2	4	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	—	2	2	no	—
216	C₂×C₄×C₇	—	2	2	no	—
217	C₂×C₄×C₇	—	4	1	no	—
218	C₂×C₄×C₇	—	4	1	no	—
219	C₂×C₄×C₇	—	4	1	no	—
220	C₂×C₄×C₇	—	4	2	no	—
221	C₂×C₄×C₇	—	4	2	no	—
222	C₂×C₄×C₇	—	4	1	no	—
223	C₂×C₄×C₇	—	4	1	no	—
224	C₂×C₄×C₇	—	4	1	no	—
225	C₂×C₄×C₇	—	4	1	no	—
226	C₂×C₄×C₇	—	4	1	no	—
227	C₂×C₄×C₇	—	4	1	no	—
228	C₂×C₄×C₇	—	4	1	no	—
229	C₂×C₄×C₇	—	8	8	no	—
230	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
231	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
232	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
233	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
234	C₂×C₄×C₇	#135	1	2	yes	—
235	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
240	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
241	C₇×C₈	—	2	2	no	—
242	C₇×C₈	—	2	2	no	—
243	C₇×C₈	—	4	2	no	—
244	C₂²×Q₈	—	2	1	no	—
245	C₂²×D₄	—	2	1	no	—
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	C₂³×C₇	1	2	yes	—
248	C₂³×C₄	—	2	2	no	—
249	C₂²×C₈	C₂³×C₇	1	2	yes	—
250	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
252	C₄⋊Q₈	—	2	2	no	—
253	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
254	#127	—	2	2	no	—
255	#127	C₂×C₄×C₇	1	4	yes	—
256	C₄×D₄	—	2	4	no	—
257	C₄×D₄	—	4	2	no	—
258	C₄×D₄	—	4	4	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
262	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
263	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
264	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
266	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
268	C₂×C₂²⋊C₄	—	2	2	no	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₄²	—	2	1	no	—
272	C₂×C₄²	—	2	1	no	—
273	C₂×C₄²	—	2	2	no	—
274	C₂×C₄²	C₂³×C₇	1	2	yes	—
275	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
276	#106	—	2	2	no	—
277	#106	—	2	2	no	—
278	C₄⋊C₈	—	2	2	no	—
279	#102	—	2	4	no	—
280	#102	—	4	2	no	—
281	#102	C₂×C₄×C₇	1	8	yes	complemented
282	#102	C₂×C₄×C₇	1	8	yes	complemented
283	C₄×C₈	—	4	2	no	—
284	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
285	#95	—	2	2	no	—
286	#95	—	2	1	no	—
287	#95	—	2	2	no	—
288	#95	—	2	1	no	—
289	#95	C₇×D₄	1	2	yes	—
290	C₂²×C₇	—	4	4	no	—
291	C₂²×C₇	—	4	4	no	—
292	C₂²×C₇	—	4	4	no	—
293	C₂²×C₇	—	8	2	no	—
294	C₂²×C₇	—	8	2	no	—
295	C₂²×C₇	—	8	2	no	—
296	C₂²×C₇	—	8	2	no	—
297	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
298	C₂²×C₇	#546	1	1	yes	characteristic
299	C₂²×C₇	#548	1	1	yes	characteristic
300	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
301	C₂²×C₇	C₂×#135	1	1	yes	characteristic
302	C₂²×C₇	#576	1	1	yes	characteristic
303	C₂²×C₇	#576	1	1	yes	characteristic
304	C₄×C₇	—	2	2	no	—
305	C₄×C₇	—	2	4	no	—
306	C₄×C₇	—	4	2	no	—
307	C₄×C₇	—	4	4	no	—
308	C₄×C₇	—	4	2	no	—
309	C₄×C₇	—	4	2	no	—
310	C₄×C₇	—	8	1	no	—
311	C₄×C₇	—	8	1	no	—
312	C₄×C₇	C₄×SD₁₆	1	4	yes	—
313	C₄×C₇	C₄×#41	1	1	yes	characteristic
314	C₄×C₇	#520	1	1	yes	characteristic
315	C₄×C₇	#556	1	1	yes	characteristic
316	C₄×C₇	#559	1	1	yes	characteristic
317	C₂⁴	—	2	2	no	—
318	C₂×Q₈	—	2	4	no	—
319	C₂×Q₈	—	8	1	no	—
320	C₂×Q₈	—	8	1	no	—
321	C₂×D₄	—	2	4	no	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	8	1	no	—
331	C₂×D₄	—	8	1	no	—
332	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	4	4	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	—	8	2	no	—
341	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
342	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
343	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
344	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
345	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
354	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	8	2	no	—
365	C₄⋊C₄	C₇×SD₁₆	1	4	yes	—
366	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
367	C₄⋊C₄	C₇×#41	1	2	yes	—
368	C₄⋊C₄	C₇×#41	1	2	yes	—
369	C₂²⋊C₄	—	4	4	no	—
370	C₂²⋊C₄	—	4	4	no	—
371	C₂²⋊C₄	—	8	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	4	2	no	—
375	C₄²	—	4	2	no	—
376	C₄²	—	8	2	no	—
377	C₄²	C₂²×C₄×C₇	1	2	yes	—
378	C₄²	C₇×#41	1	2	yes	—
379	C₂×C₇	—	8	4	no	—
380	C₂×C₇	#2367	1	1	yes	characteristic
381	C₂×C₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
382	C₂×C₇	#2948	1	1	yes	characteristic
383	C₂×C₇	#3295	1	1	yes	characteristic
384	C₂×C₇	#3305	1	1	yes	characteristic
385	C₂×C₇	#3361	1	1	yes	characteristic
386	C₂×C₇	#3362	1	1	yes	characteristic
387	C₂³	—	2	4	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
395	Q₈	—	8	4	no	—
396	D₄	—	2	4	no	—
397	D₄	—	2	4	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	8	no	—
401	D₄	—	8	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	8	8	no	—
425	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
426	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
427	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
428	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
429	C₂×C₄	C₇×#135	1	2	yes	—
430	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
431	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
432	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
433	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
434	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
435	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
436	C₈	—	2	2	no	—
437	C₈	—	2	2	no	—
438	C₈	—	4	2	no	—
439	C₇	#23215	1	1	yes	characteristic, direct product factor, Sylow
440	C₂²	—	4	4	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
448	C₂²	C₇×#546	1	1	yes	characteristic
449	C₂²	C₇×#548	1	1	yes	characteristic
450	C₂²	C₂²×C₇×SD₁₆	1	1	yes	characteristic
451	C₂²	C₂×C₇×#135	1	1	yes	characteristic
452	C₂²	C₇×#576	1	1	yes	characteristic
453	C₂²	C₇×#576	1	1	yes	characteristic
454	C₄	—	2	2	no	—
455	C₄	—	2	4	no	—
456	C₄	—	4	4	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	8	1	no	—
461	C₄	—	8	1	no	—
462	C₄	C₄×C₇×SD₁₆	1	4	yes	—
463	C₄	C₄×C₇×#41	1	1	yes	characteristic
464	C₄	C₇×#520	1	1	yes	characteristic
465	C₄	C₇×#556	1	1	yes	characteristic
466	C₄	C₇×#559	1	1	yes	characteristic
467	C₂	—	8	4	no	—
468	C₂	C₇×#2367	1	1	yes	characteristic
469	C₂	C₂×C₄×C₇×SD₁₆	1	1	yes	characteristic
470	C₂	C₇×#2948	1	1	yes	characteristic
471	C₂	C₇×#3295	1	1	yes	characteristic
472	C₂	C₇×#3305	1	1	yes	characteristic
473	C₂	C₇×#3361	1	1	yes	characteristic
474	C₂	C₇×#3362	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	C₂×C₇×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2369	C₂	semidirect
4. C₇×#1987	C₂	semidirect
6. C₇×#1934	C₂	semidirect
7. C₇×#1926	C₂	semidirect
8. C₇×#1864	C₂	semidirect
9. C₄×C₇×#106	C₂	semidirect
10. C₇×C₈×C₄⋊C₄	C₂	semidirect
11. C₄×C₇×#102	C₂	semidirect
16. C₇×#486	C₄	semidirect
42. #23215	C₇	direct (N × H)
52. C₇×C₄⋊Q₈	C₂×C₄	semidirect
80. C₇×#102	C₂×C₄	semidirect
81. C₇×#102	C₂×C₄	semidirect
83. C₄×C₇×C₈	C₂×C₄	semidirect
90. #2369	C₂×C₇	semidirect
92. #1987	C₂×C₇	semidirect
94. #1934	C₂×C₇	semidirect
95. #1926	C₂×C₇	semidirect
96. #1864	C₂×C₇	semidirect
97. C₄×#106	C₂×C₇	semidirect
98. C₈×C₄⋊C₄	C₂×C₇	semidirect
99. C₄×#102	C₂×C₇	semidirect
166. #486	C₄×C₇	semidirect
253. C₄⋊Q₈	C₂×C₄×C₇	semidirect
281. #102	C₂×C₄×C₇	semidirect
282. #102	C₂×C₄×C₇	semidirect
284. C₄×C₈	C₂×C₄×C₇	semidirect
439. C₇	#23215	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×#486	2
C₇×#2369	2
C₇×#2366	2
C₇×#1987	2
C₇×#1982	2
C₇×#1934	2
C₇×#1926	2
C₇×#1864	2
C₄×C₇×#106	2
C₇×C₈×C₄⋊C₄	2
C₄×C₇×#102	2
#23215	7

Subgroups — C7×#23215 #13200431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23215 #13200431