Subgroups — C₇×#23021 #13200237

Order 1792.

874 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	C₇×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2933	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#430	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1974	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1929	C₂	1	2	yes	maximal, complemented
8	C₇×#1871	C₂	1	2	yes	maximal, complemented
9	C₇×#1866	C₂	1	2	yes	maximal, complemented
10	C₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	1	yes	characteristic
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
15	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
16	C₇×#525	C₂²	1	2	yes	—
17	C₇×#470	C₄	1	2	yes	complemented
18	C₇×#469	C₄	1	2	yes	complemented
19	C₇×#468	C₄	1	2	yes	complemented
20	C₇×#468	C₄	1	2	yes	complemented
21	C₇×#444	—	2	4	no	—
22	C₇×#436	—	2	2	no	—
23	C₇×#435	—	2	2	no	—
24	C₇×#435	—	2	2	no	—
25	C₇×C₈×D₄	—	2	2	no	—
26	C₇×#430	—	2	4	no	—
27	C₂×C₇×#108	C₂²	1	1	yes	characteristic
28	C₂×C₇×#108	C₂²	1	2	yes	—
29	C₂×C₇×#108	C₂²	1	1	yes	characteristic
30	C₇×#427	C₂²	1	1	yes	characteristic
31	C₇×#424	C₂²	1	2	yes	—
32	C₂×C₇×#104	C₂²	1	2	yes	—
33	C₂×C₇×#104	C₂²	1	2	yes	—
34	C₇×#416	C₂²	1	2	yes	—
35	C₇×#408	C₂²	1	2	yes	—
36	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₇×#405	C₂²	1	2	yes	—
38	C₂×C₇×#97	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
40	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
41	C₇×#339	C₂²	1	2	yes	—
42	C₇×#339	C₂²	1	2	yes	—
43	C₇×#336	C₂²	1	1	yes	characteristic
44	C₇×#336	C₂²	1	1	yes	characteristic
45	#23021	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₇×#41	C₂³	1	2	yes	—
47	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×#135	—	2	2	no	—
49	C₇×#135	—	2	2	no	—
50	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
52	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
53	C₇×#131	—	2	4	no	—
54	C₇×#131	—	4	2	no	—
55	C₂×C₇×M₁₆	—	2	2	no	—
56	C₂×C₇×M₁₆	—	2	2	no	—
57	C₂×C₇×M₁₆	—	2	2	no	—
58	C₂×C₇×M₁₆	—	2	2	no	—
59	C₂×C₇×M₁₆	—	2	4	no	—
60	C₂×C₇×M₁₆	—	4	2	no	—
61	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
62	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
63	C₂×C₇×M₁₆	C₂³	1	2	yes	—
64	C₂×C₇×M₁₆	C₂³	1	2	yes	—
65	C₂²×C₇×C₈	—	2	2	no	—
66	C₂²×C₇×C₈	—	2	2	no	—
67	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
71	C₇×#123	—	2	2	no	—
72	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
73	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
74	C₇×#121	C₂×C₄	1	2	yes	—
75	C₇×#121	C₂×C₄	1	2	yes	—
76	C₇×#120	—	2	2	no	—
77	C₄×C₇×D₄	—	2	2	no	—
78	C₄×C₇×D₄	—	2	2	no	—
79	C₇×#117	C₂³	1	2	yes	—
80	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
81	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
83	C₇×#108	—	2	4	no	—
84	C₇×#108	—	2	2	no	—
85	C₇×#108	D₄	1	4	yes	complemented
86	C₇×#107	C₂×C₄	1	1	yes	characteristic
87	C₇×#107	C₂×C₄	1	1	yes	characteristic
88	C₇×#106	C₂×C₄	1	2	yes	—
89	C₇×C₄⋊C₈	—	2	2	no	—
90	C₇×C₄⋊C₈	—	2	2	no	—
91	C₇×#104	—	2	4	no	—
92	C₇×#104	—	4	2	no	—
93	C₇×#103	C₂×C₄	1	2	yes	—
94	C₇×#103	C₂×C₄	1	2	yes	—
95	C₇×#102	C₂×C₄	1	2	yes	—
96	C₇×#102	C₂×C₄	1	2	yes	—
97	C₇×C₂²⋊C₈	—	2	2	no	—
98	C₇×C₂²⋊C₈	—	2	2	no	—
99	C₇×C₂²⋊C₈	D₄	1	2	yes	—
100	C₇×C₂²⋊C₈	D₄	1	2	yes	—
101	C₇×#97	—	2	2	no	—
102	C₇×#97	—	2	2	no	—
103	C₄×C₇×C₈	—	2	2	no	—
104	C₄×C₇×C₈	—	2	2	no	—
105	C₇×#95	—	2	1	no	—
106	#3054	C₂×C₇	1	1	yes	characteristic
107	C₂×#444	C₂×C₇	1	1	yes	characteristic, complemented
108	#2933	C₂×C₇	1	2	yes	complemented
109	C₂×#430	C₂×C₇	1	1	yes	characteristic, complemented
110	#1974	C₂×C₇	1	1	yes	characteristic, complemented
111	#1946	C₂×C₇	1	1	yes	characteristic, complemented
112	#1929	C₂×C₇	1	2	yes	complemented
113	#1871	C₂×C₇	1	2	yes	complemented
114	#1866	C₂×C₇	1	2	yes	complemented
115	#1772	C₂×C₇	1	1	yes	characteristic, complemented
116	#1753	C₂×C₇	1	1	yes	characteristic, complemented
117	C₂⁴×C₇	#41	1	1	yes	characteristic
118	C₇×#41	—	2	2	no	—
119	C₇×#41	—	2	2	no	—
120	C₇×#41	—	2	2	no	—
121	C₇×#41	—	2	2	no	—
122	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
123	C₂×C₇×D₄	—	2	2	no	—
124	C₂×C₇×D₄	—	2	2	no	—
125	C₂×C₇×D₄	—	2	2	no	—
126	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₇	—	2	1	no	—
128	C₂²×C₄×C₇	—	2	1	no	—
129	C₂²×C₄×C₇	—	2	1	no	—
130	C₂²×C₄×C₇	—	2	1	no	—
131	C₂²×C₄×C₇	—	2	1	no	—
132	C₂²×C₄×C₇	—	2	2	no	—
133	C₂²×C₄×C₇	—	2	1	no	—
134	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₇	#41	1	1	yes	characteristic
137	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
138	C₇×Q₁₆	—	2	2	no	—
139	C₇×SD₁₆	—	2	4	no	—
140	C₇×D₈	—	2	2	no	—
141	C₇×M₁₆	—	2	2	no	—
142	C₇×M₁₆	—	2	4	no	—
143	C₇×M₁₆	—	2	4	no	—
144	C₇×M₁₆	—	2	2	no	—
145	C₇×M₁₆	—	2	2	no	—
146	C₇×M₁₆	—	4	4	no	—
147	C₇×M₁₆	—	4	4	no	—
148	C₇×M₁₆	C₂×D₄	1	4	yes	—
149	C₂×C₇×C₈	—	2	1	no	—
150	C₂×C₇×C₈	—	2	1	no	—
151	C₂×C₇×C₈	—	2	2	no	—
152	C₂×C₇×C₈	—	2	1	no	—
153	C₂×C₇×C₈	—	2	1	no	—
154	C₂×C₇×C₈	—	2	1	no	—
155	C₂×C₇×C₈	—	2	1	no	—
156	C₂×C₇×C₈	—	2	2	no	—
157	C₂×C₇×C₈	—	4	4	no	—
158	C₂×C₇×C₈	—	4	4	no	—
159	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
160	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
163	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
164	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
165	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
166	C₂×C₇×C₈	#41	1	2	yes	—
167	C₇×C₄⋊C₄	—	2	2	no	—
168	C₇×C₄⋊C₄	—	2	2	no	—
169	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
170	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
171	C₇×C₂²⋊C₄	—	2	2	no	—
172	C₇×C₂²⋊C₄	—	2	2	no	—
173	C₇×C₂²⋊C₄	—	2	2	no	—
174	C₇×C₂²⋊C₄	—	2	2	no	—
175	C₇×C₂²⋊C₄	—	2	2	no	—
176	C₇×C₂²⋊C₄	—	4	2	no	—
177	C₄²×C₇	—	2	2	no	—
178	C₄²×C₇	—	2	1	no	—
179	C₄²×C₇	—	2	1	no	—
180	C₄²×C₇	—	2	1	no	—
181	C₄²×C₇	—	2	1	no	—
182	C₂×#135	C₂²×C₇	1	1	yes	characteristic
183	C₂×#131	C₂²×C₇	1	2	yes	—
184	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
185	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
186	#525	C₂²×C₇	1	2	yes	—
187	#470	C₄×C₇	1	2	yes	complemented
188	#469	C₄×C₇	1	2	yes	complemented
189	#468	C₄×C₇	1	2	yes	complemented
190	#468	C₄×C₇	1	2	yes	complemented
191	#444	—	2	4	no	—
192	#436	—	2	2	no	—
193	#435	—	2	2	no	—
194	#435	—	2	2	no	—
195	C₈×D₄	—	2	2	no	—
196	#430	—	2	4	no	—
197	C₂×#108	C₂²×C₇	1	2	yes	—
198	C₂×#108	C₂²×C₇	1	1	yes	characteristic
199	C₂×#108	C₂²×C₇	1	1	yes	characteristic
200	#427	C₂²×C₇	1	1	yes	characteristic
201	#424	C₂²×C₇	1	2	yes	—
202	C₂×#104	C₂²×C₇	1	2	yes	—
203	C₂×#104	C₂²×C₇	1	2	yes	—
204	#416	C₂²×C₇	1	2	yes	—
205	#408	C₂²×C₇	1	2	yes	—
206	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
207	#405	C₂²×C₇	1	2	yes	—
208	C₂×#97	C₂²×C₇	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
210	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
211	#339	C₂²×C₇	1	2	yes	—
212	#339	C₂²×C₇	1	2	yes	—
213	#336	C₂²×C₇	1	1	yes	characteristic
214	#336	C₂²×C₇	1	1	yes	characteristic
215	C₂³×C₇	—	2	1	no	—
216	C₂³×C₇	—	2	1	no	—
217	C₂³×C₇	—	2	2	no	—
218	C₂³×C₇	—	4	1	no	—
219	C₂³×C₇	—	4	1	no	—
220	C₂³×C₇	C₄×D₄	1	2	yes	—
221	C₂³×C₇	C₂×#41	1	1	yes	characteristic
222	C₇×Q₈	—	2	2	no	—
223	C₇×Q₈	—	2	2	no	—
224	C₇×D₄	—	2	2	no	—
225	C₇×D₄	—	2	2	no	—
226	C₇×D₄	—	4	4	no	—
227	C₇×D₄	—	4	2	no	—
228	C₇×D₄	—	4	2	no	—
229	C₇×D₄	—	8	2	no	—
230	C₂×C₄×C₇	—	2	1	no	—
231	C₂×C₄×C₇	—	2	1	no	—
232	C₂×C₄×C₇	—	2	2	no	—
233	C₂×C₄×C₇	—	2	2	no	—
234	C₂×C₄×C₇	—	2	2	no	—
235	C₂×C₄×C₇	—	2	2	no	—
236	C₂×C₄×C₇	—	2	2	no	—
237	C₂×C₄×C₇	—	2	2	no	—
238	C₂×C₄×C₇	—	2	2	no	—
239	C₂×C₄×C₇	—	2	2	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	1	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	2	no	—
244	C₂×C₄×C₇	—	4	1	no	—
245	C₂×C₄×C₇	—	4	1	no	—
246	C₂×C₄×C₇	—	4	1	no	—
247	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
248	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
252	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
253	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
254	C₇×C₈	—	2	2	no	—
255	C₇×C₈	—	2	2	no	—
256	C₇×C₈	—	2	4	no	—
257	C₇×C₈	—	2	1	no	—
258	C₇×C₈	—	2	1	no	—
259	C₇×C₈	—	4	2	no	—
260	C₇×C₈	C₄×D₄	1	2	yes	—
261	C₇×C₈	C₄×D₄	1	2	yes	—
262	C₂×#41	C₂³×C₇	1	2	yes	—
263	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
264	#135	—	2	2	no	—
265	#135	—	2	2	no	—
266	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
267	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
268	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
269	#131	—	2	4	no	—
270	#131	—	4	2	no	—
271	C₂×M₁₆	—	2	2	no	—
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	2	2	no	—
275	C₂×M₁₆	—	2	4	no	—
276	C₂×M₁₆	—	4	2	no	—
277	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
278	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
279	C₂×M₁₆	C₂³×C₇	1	2	yes	—
280	C₂×M₁₆	C₂³×C₇	1	2	yes	—
281	C₂²×C₈	—	2	2	no	—
282	C₂²×C₈	—	2	2	no	—
283	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
287	#123	—	2	2	no	—
288	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
289	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
290	#121	C₂×C₄×C₇	1	2	yes	—
291	#121	C₂×C₄×C₇	1	2	yes	—
292	#120	—	2	2	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	#117	C₂³×C₇	1	2	yes	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
299	#108	—	2	4	no	—
300	#108	—	2	2	no	—
301	#108	C₇×D₄	1	4	yes	complemented
302	#107	C₂×C₄×C₇	1	1	yes	characteristic
303	#107	C₂×C₄×C₇	1	1	yes	characteristic
304	#106	C₂×C₄×C₇	1	2	yes	—
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	2	2	no	—
307	#104	—	2	4	no	—
308	#104	—	4	2	no	—
309	#103	C₂×C₄×C₇	1	2	yes	—
310	#103	C₂×C₄×C₇	1	2	yes	—
311	#102	C₂×C₄×C₇	1	2	yes	—
312	#102	C₂×C₄×C₇	1	2	yes	—
313	C₂²⋊C₈	—	2	2	no	—
314	C₂²⋊C₈	—	2	2	no	—
315	C₂²⋊C₈	C₇×D₄	1	2	yes	—
316	C₂²⋊C₈	C₇×D₄	1	2	yes	—
317	#97	—	2	2	no	—
318	#97	—	2	2	no	—
319	C₄×C₈	—	2	2	no	—
320	C₄×C₈	—	2	2	no	—
321	#95	—	2	1	no	—
322	C₂²×C₇	—	2	1	no	—
323	C₂²×C₇	—	2	1	no	—
324	C₂²×C₇	—	2	2	no	—
325	C₂²×C₇	—	2	2	no	—
326	C₂²×C₇	—	2	2	no	—
327	C₂²×C₇	—	4	2	no	—
328	C₂²×C₇	—	4	2	no	—
329	C₂²×C₇	—	4	2	no	—
330	C₂²×C₇	—	8	1	no	—
331	C₂²×C₇	—	8	1	no	—
332	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
333	C₂²×C₇	#548	1	1	yes	characteristic
334	C₂²×C₇	#553	1	1	yes	characteristic
335	C₄×C₇	—	2	2	no	—
336	C₄×C₇	—	4	2	no	—
337	C₄×C₇	—	4	2	no	—
338	C₄×C₇	—	4	2	no	—
339	C₄×C₇	—	4	2	no	—
340	C₄×C₇	—	8	1	no	—
341	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
342	C₄×C₇	#520	1	1	yes	characteristic
343	C₄×C₇	D₄²	1	1	yes	characteristic
344	C₄×C₇	#547	1	1	yes	characteristic
345	C₂⁴	C₇×#41	1	1	yes	characteristic
346	#41	—	2	2	no	—
347	#41	—	2	2	no	—
348	#41	—	2	2	no	—
349	#41	—	2	2	no	—
350	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
363	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
364	C₂²×C₄	C₇×#41	1	1	yes	characteristic
365	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
366	Q₁₆	—	2	2	no	—
367	SD₁₆	—	2	4	no	—
368	D₈	—	2	2	no	—
369	M₁₆	—	2	2	no	—
370	M₁₆	—	2	4	no	—
371	M₁₆	—	2	4	no	—
372	M₁₆	—	2	2	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	4	4	no	—
375	M₁₆	—	4	4	no	—
376	M₁₆	C₂×C₇×D₄	1	4	yes	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	4	4	no	—
386	C₂×C₈	—	4	4	no	—
387	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
388	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
389	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
390	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
391	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
392	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
393	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
394	C₂×C₈	C₇×#41	1	2	yes	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
398	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	2	no	—
401	C₂²⋊C₄	—	2	2	no	—
402	C₂²⋊C₄	—	2	2	no	—
403	C₂²⋊C₄	—	2	2	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₂×C₇	—	2	1	no	—
411	C₂×C₇	—	2	1	no	—
412	C₂×C₇	—	4	2	no	—
413	C₂×C₇	—	8	2	no	—
414	C₂×C₇	#2361	1	1	yes	characteristic
415	C₂×C₇	#2955	1	1	yes	characteristic
416	C₂×C₇	#2956	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	4	1	no	—
421	C₂³	—	4	1	no	—
422	C₂³	C₄×C₇×D₄	1	2	yes	—
423	C₂³	C₂×C₇×#41	1	1	yes	characteristic
424	Q₈	—	2	2	no	—
425	Q₈	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	2	no	—
428	D₄	—	4	4	no	—
429	D₄	—	4	2	no	—
430	D₄	—	4	2	no	—
431	D₄	—	8	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	C₄×C₇×D₄	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×D₄	1	1	yes	characteristic
453	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
454	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
455	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
456	C₈	—	2	2	no	—
457	C₈	—	2	2	no	—
458	C₈	—	2	4	no	—
459	C₈	—	2	1	no	—
460	C₈	—	2	1	no	—
461	C₈	—	4	2	no	—
462	C₈	C₄×C₇×D₄	1	2	yes	—
463	C₈	C₄×C₇×D₄	1	2	yes	—
464	C₇	#23021	1	1	yes	characteristic, direct product factor, Sylow
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	8	1	no	—
474	C₂²	—	8	1	no	—
475	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
476	C₂²	C₇×#548	1	1	yes	characteristic
477	C₂²	C₇×#553	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	8	1	no	—
484	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
485	C₄	C₇×#520	1	1	yes	characteristic
486	C₄	C₇×D₄²	1	1	yes	characteristic
487	C₄	C₇×#547	1	1	yes	characteristic
488	C₂	—	2	1	no	—
489	C₂	—	2	1	no	—
490	C₂	—	4	2	no	—
491	C₂	—	8	2	no	—
492	C₂	C₇×#2361	1	1	yes	characteristic
493	C₂	C₇×#2955	1	1	yes	characteristic
494	C₂	C₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3054	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#444	C₂	semidirect
3. C₇×#2933	C₂	semidirect
4. C₂×C₇×#430	C₂	semidirect
5. C₇×#1974	C₂	semidirect
6. C₇×#1946	C₂	semidirect
7. C₇×#1929	C₂	semidirect
8. C₇×#1871	C₂	semidirect
9. C₇×#1866	C₂	semidirect
10. C₇×#1772	C₂	semidirect
11. C₇×#1753	C₂	semidirect
17. C₇×#470	C₄	semidirect
18. C₇×#469	C₄	semidirect
19. C₇×#468	C₄	semidirect
20. C₇×#468	C₄	semidirect
45. #23021	C₇	direct (N × H)
85. C₇×#108	D₄	semidirect
107. C₂×#444	C₂×C₇	semidirect
108. #2933	C₂×C₇	semidirect
109. C₂×#430	C₂×C₇	semidirect
110. #1974	C₂×C₇	semidirect
111. #1946	C₂×C₇	semidirect
112. #1929	C₂×C₇	semidirect
113. #1871	C₂×C₇	semidirect
114. #1866	C₂×C₇	semidirect
115. #1772	C₂×C₇	semidirect
116. #1753	C₂×C₇	semidirect
187. #470	C₄×C₇	semidirect
188. #469	C₄×C₇	semidirect
189. #468	C₄×C₇	semidirect
190. #468	C₄×C₇	semidirect
301. #108	C₇×D₄	semidirect
464. C₇	#23021	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3054	2
C₂×C₇×#444	2
C₇×#2933	2
C₂×C₇×#430	2
C₇×#1974	2
C₇×#1946	2
C₇×#1929	2
C₇×#1871	2
C₇×#1866	2
C₇×#1772	2
C₇×#1753	2
#23021	7

Subgroups — C7×#23021 #13200237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23021 #13200237