Subgroups — C₂×C₇×#3333 #13238369

Order 1792.

2210 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#550	C₂	1	1	yes	maximal, characteristic
3	C₇×#3333	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#480	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#468	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#452	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#417	C₂	1	2	yes	maximal, complemented
13	C₂²×C₇×#41	C₂²	1	2	yes	—
14	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
16	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
17	C₇×#554	—	2	4	no	—
18	C₇×#554	C₂²	1	8	yes	complemented
19	C₇×#550	—	2	4	no	—
20	C₇×#550	C₂²	1	8	yes	—
21	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×#127	—	2	1	no	—
23	C₂×C₇×#127	C₂²	1	1	yes	characteristic
24	C₂×C₇×#125	—	2	1	no	—
25	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₇×#121	—	2	2	no	—
28	C₂×C₇×#121	C₂²	1	2	yes	—
29	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×D₄	—	2	1	no	—
32	C₂×C₄×C₇×D₄	—	2	1	no	—
33	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
34	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
35	C₇×#486	C₂²	1	8	yes	complemented
36	C₇×#480	C₂²	1	16	yes	complemented
37	C₇×#477	C₂²	1	8	yes	complemented
38	C₇×#468	C₂²	1	16	yes	complemented
39	C₇×#460	C₂²	1	8	yes	complemented
40	C₇×#452	C₂²	1	16	yes	complemented
41	C₄×C₇×Q₁₆	C₂²	1	8	yes	complemented
42	C₇×#436	C₂²	1	8	yes	complemented
43	C₂×C₇×#107	C₂²	1	1	yes	characteristic
44	C₂×C₇×#107	C₂²	1	2	yes	—
45	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₇×#417	C₂²	1	16	yes	complemented
47	C₂×C₇×#103	C₂²	1	1	yes	characteristic
48	C₂×C₇×#103	C₂²	1	1	yes	characteristic
49	C₂×C₇×#103	C₂²	1	2	yes	—
50	C₂×C₇×#102	C₂²	1	2	yes	—
51	C₂×C₇×#102	C₂²	1	2	yes	—
52	C₂×C₇×#102	C₂²	1	2	yes	—
53	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
55	C₂×#3333	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₇×#41	—	2	4	no	—
57	C₂×C₇×#41	—	2	8	no	—
58	C₂×C₇×#41	—	4	4	no	—
59	C₂×C₇×#41	C₂³	1	8	yes	—
60	C₂²×C₇×Q₈	—	2	1	no	—
61	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
63	C₂²×C₇×D₄	—	2	2	no	—
64	C₂²×C₇×D₄	—	2	2	no	—
65	C₂²×C₇×D₄	—	2	1	no	—
66	C₂²×C₇×D₄	—	2	1	no	—
67	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
68	C₂²×C₇×D₄	C₂³	1	2	yes	—
69	C₂³×C₄×C₇	—	2	2	no	—
70	C₂³×C₄×C₇	—	2	2	no	—
71	C₂³×C₄×C₇	C₂³	1	2	yes	—
72	C₂×C₇×Q₁₆	—	2	4	no	—
73	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
74	C₂×C₇×SD₁₆	—	4	4	no	—
75	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
76	C₂×C₇×M₁₆	—	4	4	no	—
77	C₂×C₇×M₁₆	C₂³	1	8	yes	—
78	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
79	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
80	C₂²×C₇×C₈	C₂³	1	2	yes	—
81	C₇×C₄⋊Q₈	—	2	2	no	—
82	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
83	C₇×#127	—	2	8	no	—
84	C₇×#127	—	2	2	no	—
85	C₇×#127	C₂³	1	4	yes	—
86	C₇×#125	—	2	8	no	—
87	C₇×C₂²⋊Q₈	—	2	4	no	—
88	C₇×C₂²⋊Q₈	—	2	16	no	—
89	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
90	C₇×#121	—	2	4	no	—
91	C₇×#121	—	2	16	no	—
92	C₇×#121	C₂³	1	8	yes	—
93	C₄×C₇×Q₈	—	2	2	no	—
94	C₄×C₇×Q₈	—	2	2	no	—
95	C₄×C₇×Q₈	C₂³	1	4	yes	—
96	C₄×C₇×Q₈	C₂³	1	4	yes	—
97	C₄×C₇×D₄	—	2	8	no	—
98	C₄×C₇×D₄	—	2	8	no	—
99	C₄×C₇×D₄	D₄	1	4	yes	—
100	C₄×C₇×D₄	C₂³	1	4	yes	—
101	C₂×C₇×C₄⋊C₄	—	2	2	no	—
102	C₂×C₇×C₄⋊C₄	—	2	4	no	—
103	C₂×C₇×C₄⋊C₄	—	2	1	no	—
104	C₂×C₇×C₄⋊C₄	—	2	1	no	—
105	C₂×C₇×C₄⋊C₄	—	2	1	no	—
106	C₂×C₇×C₄⋊C₄	—	2	2	no	—
107	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
108	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
109	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
112	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
114	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
115	C₂×C₄²×C₇	—	2	1	no	—
116	C₂×C₄²×C₇	—	2	1	no	—
117	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
118	C₇×#107	C₂³	1	8	yes	—
119	C₇×#107	C₂³	1	4	yes	—
120	C₇×C₄⋊C₈	C₂³	1	4	yes	—
121	C₇×#103	C₂³	1	4	yes	—
122	C₇×#103	C₂³	1	8	yes	—
123	C₇×#103	C₂³	1	4	yes	—
124	C₇×#102	C₂³	1	8	yes	—
125	C₇×#102	C₂³	1	8	yes	—
126	C₇×#102	C₂³	1	8	yes	—
127	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
128	C₄×C₇×C₈	C₂³	1	4	yes	—
129	C₂×#554	C₂×C₇	1	1	yes	characteristic, complemented
130	C₂×#550	C₂×C₇	1	1	yes	characteristic
131	#3333	C₂×C₇	1	16	yes	direct product factor
132	C₂×#486	C₂×C₇	1	1	yes	characteristic, complemented
133	C₂×#480	C₂×C₇	1	2	yes	complemented
134	C₂×#477	C₂×C₇	1	1	yes	characteristic, complemented
135	C₂×#468	C₂×C₇	1	2	yes	complemented
136	C₂×#460	C₂×C₇	1	1	yes	characteristic, complemented
137	C₂×#452	C₂×C₇	1	2	yes	complemented
138	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
139	C₂×#436	C₂×C₇	1	1	yes	characteristic, complemented
140	C₂×#417	C₂×C₇	1	2	yes	complemented
141	C₂⁴×C₇	—	2	2	no	—
142	C₂⁴×C₇	C₂×D₄	1	2	yes	—
143	C₇×#41	—	2	16	no	—
144	C₇×#41	—	4	8	no	—
145	C₇×#41	—	4	16	no	—
146	C₂×C₇×Q₈	—	2	1	no	—
147	C₂×C₇×Q₈	—	2	1	no	—
148	C₂×C₇×Q₈	—	2	4	no	—
149	C₂×C₇×Q₈	—	2	2	no	—
150	C₂×C₇×Q₈	—	2	2	no	—
151	C₂×C₇×Q₈	—	2	2	no	—
152	C₂×C₇×Q₈	—	8	2	no	—
153	C₂×C₇×Q₈	#41	1	4	yes	—
154	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
155	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
156	C₂×C₇×D₄	—	2	8	no	—
157	C₂×C₇×D₄	—	2	8	no	—
158	C₂×C₇×D₄	—	2	4	no	—
159	C₂×C₇×D₄	—	2	4	no	—
160	C₂×C₇×D₄	—	2	2	no	—
161	C₂×C₇×D₄	—	2	4	no	—
162	C₂×C₇×D₄	—	2	4	no	—
163	C₂×C₇×D₄	—	2	4	no	—
164	C₂×C₇×D₄	—	4	8	no	—
165	C₂×C₇×D₄	—	4	4	no	—
166	C₂×C₇×D₄	—	8	2	no	—
167	C₂×C₇×D₄	—	8	4	no	—
168	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
169	C₂×C₇×D₄	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₇	—	2	1	no	—
171	C₂²×C₄×C₇	—	2	4	no	—
172	C₂²×C₄×C₇	—	2	4	no	—
173	C₂²×C₄×C₇	—	2	1	no	—
174	C₂²×C₄×C₇	—	2	1	no	—
175	C₂²×C₄×C₇	—	2	1	no	—
176	C₂²×C₄×C₇	—	2	2	no	—
177	C₂²×C₄×C₇	—	2	8	no	—
178	C₂²×C₄×C₇	—	2	8	no	—
179	C₂²×C₄×C₇	—	4	8	no	—
180	C₂²×C₄×C₇	—	8	4	no	—
181	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
187	C₇×Q₁₆	—	2	8	no	—
188	C₇×SD₁₆	—	4	8	no	—
189	C₇×M₁₆	—	4	8	no	—
190	C₂×C₇×C₈	—	2	2	no	—
191	C₂×C₇×C₈	—	2	2	no	—
192	C₂×C₇×C₈	—	2	4	no	—
193	C₂×C₇×C₈	C₂⁴	1	2	yes	—
194	C₂×C₇×C₈	C₂⁴	1	2	yes	—
195	C₂×C₇×C₈	C₂⁴	1	4	yes	—
196	C₇×C₄⋊C₄	—	2	1	no	—
197	C₇×C₄⋊C₄	—	2	8	no	—
198	C₇×C₄⋊C₄	—	2	2	no	—
199	C₇×C₄⋊C₄	—	2	4	no	—
200	C₇×C₄⋊C₄	—	2	4	no	—
201	C₇×C₄⋊C₄	—	2	4	no	—
202	C₇×C₄⋊C₄	—	2	8	no	—
203	C₇×C₄⋊C₄	—	2	1	no	—
204	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
205	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
206	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
207	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
208	C₇×C₂²⋊C₄	—	2	8	no	—
209	C₇×C₂²⋊C₄	—	2	8	no	—
210	C₇×C₂²⋊C₄	C₂×D₄	1	8	yes	—
211	C₄²×C₇	—	2	4	no	—
212	C₄²×C₇	—	2	4	no	—
213	C₄²×C₇	C₂×D₄	1	2	yes	—
214	C₄²×C₇	C₂⁴	1	2	yes	—
215	C₂²×#41	C₂²×C₇	1	2	yes	—
216	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
217	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
218	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
219	#554	—	2	4	no	—
220	#554	C₂²×C₇	1	8	yes	complemented
221	#550	—	2	4	no	—
222	#550	C₂²×C₇	1	8	yes	—
223	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
224	C₂×#127	—	2	1	no	—
225	C₂×#127	C₂²×C₇	1	1	yes	characteristic
226	C₂×#125	—	2	1	no	—
227	C₂×C₂²⋊Q₈	—	2	2	no	—
228	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
229	C₂×#121	—	2	2	no	—
230	C₂×#121	C₂²×C₇	1	2	yes	—
231	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
232	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
233	C₂×C₄×D₄	—	2	1	no	—
234	C₂×C₄×D₄	—	2	1	no	—
235	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
236	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
237	#486	C₂²×C₇	1	8	yes	complemented
238	#480	C₂²×C₇	1	16	yes	complemented
239	#477	C₂²×C₇	1	8	yes	complemented
240	#468	C₂²×C₇	1	16	yes	complemented
241	#460	C₂²×C₇	1	8	yes	complemented
242	#452	C₂²×C₇	1	16	yes	complemented
243	C₄×Q₁₆	C₂²×C₇	1	8	yes	complemented
244	#436	C₂²×C₇	1	8	yes	complemented
245	C₂×#107	C₂²×C₇	1	2	yes	—
246	C₂×#107	C₂²×C₇	1	1	yes	characteristic
247	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
248	#417	C₂²×C₇	1	16	yes	complemented
249	C₂×#103	C₂²×C₇	1	2	yes	—
250	C₂×#103	C₂²×C₇	1	1	yes	characteristic
251	C₂×#103	C₂²×C₇	1	1	yes	characteristic
252	C₂×#102	C₂²×C₇	1	2	yes	—
253	C₂×#102	C₂²×C₇	1	2	yes	—
254	C₂×#102	C₂²×C₇	1	2	yes	—
255	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
256	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
257	C₂³×C₇	—	2	4	no	—
258	C₂³×C₇	—	2	4	no	—
259	C₂³×C₇	—	2	4	no	—
260	C₂³×C₇	—	2	4	no	—
261	C₂³×C₇	—	4	4	no	—
262	C₂³×C₇	—	4	4	no	—
263	C₂³×C₇	—	4	4	no	—
264	C₂³×C₇	C₂²×D₄	1	4	yes	—
265	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
266	C₇×Q₈	—	2	4	no	—
267	C₇×Q₈	—	2	2	no	—
268	C₇×Q₈	—	2	2	no	—
269	C₇×Q₈	—	2	2	no	—
270	C₇×Q₈	—	4	2	no	—
271	C₇×Q₈	—	8	4	no	—
272	C₇×Q₈	C₂×#41	1	4	yes	—
273	C₇×D₄	—	2	8	no	—
274	C₇×D₄	—	2	4	no	—
275	C₇×D₄	—	2	4	no	—
276	C₇×D₄	—	4	16	no	—
277	C₇×D₄	—	4	4	no	—
278	C₇×D₄	—	4	8	no	—
279	C₇×D₄	—	8	4	no	—
280	C₇×D₄	—	8	8	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	4	no	—
283	C₂×C₄×C₇	—	2	2	no	—
284	C₂×C₄×C₇	—	2	2	no	—
285	C₂×C₄×C₇	—	2	8	no	—
286	C₂×C₄×C₇	—	2	4	no	—
287	C₂×C₄×C₇	—	2	4	no	—
288	C₂×C₄×C₇	—	2	2	no	—
289	C₂×C₄×C₇	—	2	2	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	2	4	no	—
292	C₂×C₄×C₇	—	4	16	no	—
293	C₂×C₄×C₇	—	4	2	no	—
294	C₂×C₄×C₇	—	4	2	no	—
295	C₂×C₄×C₇	—	4	4	no	—
296	C₂×C₄×C₇	—	4	2	no	—
297	C₂×C₄×C₇	—	8	8	no	—
298	C₂×C₄×C₇	#137	1	4	yes	—
299	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
301	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
303	C₂×C₄×C₇	C₂×#41	1	2	yes	—
304	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
305	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
306	C₇×C₈	—	2	2	no	—
307	C₇×C₈	—	2	2	no	—
308	C₇×C₈	—	2	4	no	—
309	C₂×#41	—	2	4	no	—
310	C₂×#41	—	2	8	no	—
311	C₂×#41	—	4	4	no	—
312	C₂×#41	C₂³×C₇	1	8	yes	—
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
315	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
316	C₂²×D₄	—	2	2	no	—
317	C₂²×D₄	—	2	2	no	—
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
321	C₂²×D₄	C₂³×C₇	1	2	yes	—
322	C₂³×C₄	—	2	2	no	—
323	C₂³×C₄	—	2	2	no	—
324	C₂³×C₄	C₂³×C₇	1	2	yes	—
325	C₂×Q₁₆	—	2	4	no	—
326	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
327	C₂×SD₁₆	—	4	4	no	—
328	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
329	C₂×M₁₆	—	4	4	no	—
330	C₂×M₁₆	C₂³×C₇	1	8	yes	—
331	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₇	1	2	yes	—
333	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
334	C₄⋊Q₈	—	2	2	no	—
335	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
336	#127	—	2	8	no	—
337	#127	—	2	2	no	—
338	#127	C₂³×C₇	1	4	yes	—
339	#125	—	2	8	no	—
340	C₂²⋊Q₈	—	2	4	no	—
341	C₂²⋊Q₈	—	2	16	no	—
342	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
343	#121	—	2	4	no	—
344	#121	—	2	16	no	—
345	#121	C₂³×C₇	1	8	yes	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	C₂³×C₇	1	4	yes	—
349	C₄×Q₈	C₂³×C₇	1	4	yes	—
350	C₄×D₄	—	2	8	no	—
351	C₄×D₄	—	2	8	no	—
352	C₄×D₄	C₇×D₄	1	4	yes	—
353	C₄×D₄	C₂³×C₇	1	4	yes	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	—	2	4	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	C₂×C₄⋊C₄	—	2	2	no	—
360	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
361	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
363	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
364	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	—	2	2	no	—
367	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
368	C₂×C₄²	—	2	1	no	—
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
371	#107	C₂³×C₇	1	4	yes	—
372	#107	C₂³×C₇	1	8	yes	—
373	C₄⋊C₈	C₂³×C₇	1	4	yes	—
374	#103	C₂³×C₇	1	4	yes	—
375	#103	C₂³×C₇	1	8	yes	—
376	#103	C₂³×C₇	1	4	yes	—
377	#102	C₂³×C₇	1	8	yes	—
378	#102	C₂³×C₇	1	8	yes	—
379	#102	C₂³×C₇	1	8	yes	—
380	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
381	C₄×C₈	C₂³×C₇	1	4	yes	—
382	C₂²×C₇	—	2	4	no	—
383	C₂²×C₇	—	2	4	no	—
384	C₂²×C₇	—	2	4	no	—
385	C₂²×C₇	—	4	8	no	—
386	C₂²×C₇	—	4	4	no	—
387	C₂²×C₇	—	4	4	no	—
388	C₂²×C₇	—	4	4	no	—
389	C₂²×C₇	—	8	8	no	—
390	C₂²×C₇	#546	1	2	yes	—
391	C₂²×C₇	C₂×#137	1	2	yes	—
392	C₂²×C₇	#576	1	2	yes	—
393	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
394	C₄×C₇	—	2	4	no	—
395	C₄×C₇	—	2	4	no	—
396	C₄×C₇	—	2	2	no	—
397	C₄×C₇	—	4	2	no	—
398	C₄×C₇	—	4	2	no	—
399	C₄×C₇	—	4	4	no	—
400	C₄×C₇	—	4	2	no	—
401	C₄×C₇	#546	1	2	yes	—
402	C₄×C₇	C₂×#137	1	4	yes	—
403	C₄×C₇	C₂²×#41	1	1	yes	characteristic
404	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
405	C₂⁴	—	2	2	no	—
406	C₂⁴	C₂×C₇×D₄	1	2	yes	—
407	#41	—	2	16	no	—
408	#41	—	4	8	no	—
409	#41	—	4	16	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	8	2	no	—
417	C₂×Q₈	C₇×#41	1	4	yes	—
418	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
419	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	4	8	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	8	2	no	—
431	C₂×D₄	—	8	4	no	—
432	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
433	C₂×D₄	C₂⁴×C₇	1	4	yes	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	8	no	—
442	C₂²×C₄	—	2	8	no	—
443	C₂²×C₄	—	4	8	no	—
444	C₂²×C₄	—	8	4	no	—
445	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
446	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
447	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
448	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
451	Q₁₆	—	2	8	no	—
452	SD₁₆	—	4	8	no	—
453	M₁₆	—	4	8	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	4	no	—
457	C₂×C₈	C₂⁴×C₇	1	2	yes	—
458	C₂×C₈	C₂⁴×C₇	1	2	yes	—
459	C₂×C₈	C₂⁴×C₇	1	4	yes	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
469	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
470	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
471	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	C₂×C₇×D₄	1	8	yes	—
475	C₄²	—	2	4	no	—
476	C₄²	—	2	4	no	—
477	C₄²	C₂×C₇×D₄	1	2	yes	—
478	C₄²	C₂⁴×C₇	1	2	yes	—
479	C₂×C₇	—	4	4	no	—
480	C₂×C₇	—	8	4	no	—
481	C₂×C₇	#3333	1	4	yes	direct product factor
482	C₂×C₇	C₂×#546	1	1	yes	characteristic
483	C₂×C₇	C₂²×#137	1	1	yes	characteristic
484	C₂×C₇	C₂×#576	1	1	yes	characteristic
485	C₂³	—	2	4	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	4	no	—
492	C₂³	C₂²×C₇×D₄	1	4	yes	—
493	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
494	Q₈	—	2	2	no	—
495	Q₈	—	2	4	no	—
496	Q₈	—	2	2	no	—
497	Q₈	—	2	2	no	—
498	Q₈	—	4	2	no	—
499	Q₈	—	8	4	no	—
500	Q₈	C₂×C₇×#41	1	4	yes	—
501	D₄	—	2	4	no	—
502	D₄	—	2	4	no	—
503	D₄	—	2	8	no	—
504	D₄	—	4	16	no	—
505	D₄	—	4	4	no	—
506	D₄	—	4	8	no	—
507	D₄	—	8	4	no	—
508	D₄	—	8	8	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	4	16	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	8	8	no	—
526	C₂×C₄	C₇×#137	1	4	yes	—
527	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
528	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
529	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
530	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
531	C₂×C₄	C₂×C₇×#41	1	2	yes	—
532	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
533	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₈	—	2	4	no	—
537	C₇	C₂×#3333	1	1	yes	characteristic, direct product factor, Sylow
538	C₂²	—	2	4	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	4	8	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	8	8	no	—
546	C₂²	C₇×#546	1	2	yes	—
547	C₂²	C₂×C₇×#137	1	2	yes	—
548	C₂²	C₇×#576	1	2	yes	—
549	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
550	C₄	—	2	2	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	4	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	2	no	—
556	C₄	—	4	2	no	—
557	C₄	C₇×#546	1	2	yes	—
558	C₄	C₂×C₇×#137	1	4	yes	—
559	C₄	C₂²×C₇×#41	1	1	yes	characteristic
560	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
561	C₂	—	4	4	no	—
562	C₂	—	8	4	no	—
563	C₂	C₇×#3333	1	4	yes	direct product factor
564	C₂	C₂×C₇×#546	1	1	yes	characteristic
565	C₂	C₂²×C₇×#137	1	1	yes	characteristic
566	C₂	C₂×C₇×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#554	C₂	semidirect
3. C₇×#3333	C₂	direct (N × H)
4. C₂×C₇×#486	C₂	semidirect
5. C₂×C₇×#480	C₂	semidirect
6. C₂×C₇×#477	C₂	semidirect
7. C₂×C₇×#468	C₂	semidirect
8. C₂×C₇×#460	C₂	semidirect
9. C₂×C₇×#452	C₂	semidirect
10. C₂×C₄×C₇×Q₁₆	C₂	semidirect
11. C₂×C₇×#436	C₂	semidirect
12. C₂×C₇×#417	C₂	semidirect
18. C₇×#554	C₂²	semidirect
35. C₇×#486	C₂²	semidirect
36. C₇×#480	C₂²	semidirect
37. C₇×#477	C₂²	semidirect
38. C₇×#468	C₂²	semidirect
39. C₇×#460	C₂²	semidirect
40. C₇×#452	C₂²	semidirect
41. C₄×C₇×Q₁₆	C₂²	semidirect
42. C₇×#436	C₂²	semidirect
46. C₇×#417	C₂²	semidirect
55. C₂×#3333	C₇	direct (N × H)
129. C₂×#554	C₂×C₇	semidirect
131. #3333	C₂×C₇	direct (N × H)
132. C₂×#486	C₂×C₇	semidirect
133. C₂×#480	C₂×C₇	semidirect
134. C₂×#477	C₂×C₇	semidirect
135. C₂×#468	C₂×C₇	semidirect
136. C₂×#460	C₂×C₇	semidirect
137. C₂×#452	C₂×C₇	semidirect
138. C₂×C₄×Q₁₆	C₂×C₇	semidirect
139. C₂×#436	C₂×C₇	semidirect
140. C₂×#417	C₂×C₇	semidirect
220. #554	C₂²×C₇	semidirect
237. #486	C₂²×C₇	semidirect
238. #480	C₂²×C₇	semidirect
239. #477	C₂²×C₇	semidirect
240. #468	C₂²×C₇	semidirect
241. #460	C₂²×C₇	semidirect
242. #452	C₂²×C₇	semidirect
243. C₄×Q₁₆	C₂²×C₇	semidirect
244. #436	C₂²×C₇	semidirect
248. #417	C₂²×C₇	semidirect
481. C₂×C₇	#3333	direct (N × H)
537. C₇	C₂×#3333	direct (N × H)
563. C₂	C₇×#3333	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×#554	2
C₂×C₇×#550	2
C₇×#3333	2
C₂×C₇×#486	2
C₂×C₇×#480	2
C₂×C₇×#477	2
C₂×C₇×#468	2
C₂×C₇×#460	2
C₂×C₇×#452	2
C₂×C₄×C₇×Q₁₆	2
C₂×C₇×#436	2
C₂×C₇×#417	2
C₂×#3333	7

Subgroups — C2×C7×#3333 #13238369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3333 #13238369