Subgroups — C₂×C₇×#3248 #13238113

Order 1792.

1930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#529	C₂	1	1	yes	maximal, characteristic
3	C₇×#3248	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#479	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#467	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#465	C₂	1	2	yes	maximal
9	C₂×C₇×#464	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
14	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
15	C₂³×C₇×C₈	C₂²	1	2	yes	—
16	C₇×#533	—	2	4	no	—
17	C₇×#533	C₂²	1	8	yes	complemented
18	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₇×#529	—	2	4	no	—
21	C₇×#529	C₂²	1	8	yes	—
22	C₂×C₇×#126	—	2	1	no	—
23	C₂×C₇×#125	—	2	1	no	—
24	C₂×C₇×#125	C₂²	1	1	yes	characteristic
25	C₂×C₇×#124	C₂²	1	1	yes	characteristic
26	C₂×C₇×#123	—	2	1	no	—
27	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
28	C₂×C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
30	C₂×C₇×#121	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×Q₈	—	2	1	no	—
32	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
34	C₂×C₄×C₇×D₄	—	2	1	no	—
35	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
36	C₂×C₇×#117	—	2	1	no	—
37	C₂×C₇×#117	—	2	1	no	—
38	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
39	C₇×#479	C₂²	1	16	yes	complemented
40	C₇×#475	C₂²	1	8	yes	complemented
41	C₇×#474	C₂²	1	8	yes	complemented
42	C₇×#467	C₂²	1	16	yes	complemented
43	C₇×#465	C₂²	1	16	yes	—
44	C₇×#464	C₂²	1	16	yes	complemented
45	C₇×#462	C₂²	1	8	yes	complemented
46	C₇×#461	C₂²	1	8	yes	complemented
47	C₂×C₇×#107	C₂²	1	2	yes	—
48	C₂×C₇×#106	C₂²	1	2	yes	—
49	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
52	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	complemented
53	C₂×C₇×#103	C₂²	1	2	yes	—
54	C₂×C₇×#103	C₂²	1	2	yes	—
55	C₂×C₇×#103	C₂²	1	2	yes	—
56	C₂×C₇×#102	C₂²	1	2	yes	—
57	C₂×#3248	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂²×C₇×Q₈	—	2	1	no	—
59	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
60	C₂²×C₇×Q₈	C₂³	1	2	yes	—
61	C₂²×C₇×D₄	—	2	1	no	—
62	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₇	—	2	1	no	—
64	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
67	C₂×C₇×Q₁₆	—	4	4	no	—
68	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
69	C₂×C₇×SD₁₆	—	4	4	no	—
70	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
71	C₂²×C₇×C₈	—	4	4	no	—
72	C₂²×C₇×C₈	C₂³	1	2	yes	—
73	C₂²×C₇×C₈	C₂³	1	2	yes	—
74	C₂²×C₇×C₈	C₂³	1	8	yes	—
75	C₇×C₄⋊Q₈	—	2	2	no	—
76	C₇×C₄⋊Q₈	—	2	2	no	—
77	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
78	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₇×#126	—	2	8	no	—
80	C₇×#125	—	2	8	no	—
81	C₇×#125	—	2	2	no	—
82	C₇×#125	C₂³	1	4	yes	—
83	C₇×#124	—	2	2	no	—
84	C₇×#124	C₂³	1	4	yes	—
85	C₇×#123	—	2	8	no	—
86	C₇×C₂²⋊Q₈	—	2	8	no	—
87	C₇×C₂²⋊Q₈	—	2	2	no	—
88	C₇×C₂²⋊Q₈	—	2	4	no	—
89	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
90	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
91	C₇×#121	—	2	2	no	—
92	C₇×#121	C₂³	1	4	yes	—
93	C₄×C₇×Q₈	—	2	8	no	—
94	C₄×C₇×Q₈	—	2	2	no	—
95	C₄×C₇×Q₈	—	2	4	no	—
96	C₄×C₇×Q₈	C₂³	1	4	yes	—
97	C₄×C₇×Q₈	C₂³	1	8	yes	—
98	C₄×C₇×D₄	—	2	8	no	—
99	C₄×C₇×D₄	—	2	2	no	—
100	C₄×C₇×D₄	C₂³	1	4	yes	—
101	C₇×#117	—	2	8	no	—
102	C₇×#117	—	2	8	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	1	no	—
107	C₂×C₇×C₄⋊C₄	—	2	1	no	—
108	C₂×C₇×C₄⋊C₄	—	2	1	no	—
109	C₂×C₇×C₄⋊C₄	—	2	2	no	—
110	C₂×C₇×C₄⋊C₄	—	2	2	no	—
111	C₂×C₇×C₄⋊C₄	—	2	1	no	—
112	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
115	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
116	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
117	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
118	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
119	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
120	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₄²×C₇	—	2	1	no	—
122	C₂×C₄²×C₇	—	2	2	no	—
123	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₇	D₄	1	4	yes	—
126	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₇	C₂³	1	4	yes	—
129	C₇×#107	C₂³	1	8	yes	—
130	C₇×#106	C₂³	1	8	yes	—
131	C₇×C₄⋊C₈	C₂³	1	4	yes	—
132	C₇×C₄⋊C₈	C₂³	1	4	yes	—
133	C₇×C₄⋊C₈	C₂³	1	8	yes	—
134	C₇×#103	C₂³	1	8	yes	—
135	C₇×#103	C₂³	1	8	yes	—
136	C₇×#103	C₂³	1	8	yes	—
137	C₇×#102	C₂³	1	8	yes	—
138	C₂×#533	C₂×C₇	1	1	yes	characteristic, complemented
139	C₂×#529	C₂×C₇	1	1	yes	characteristic
140	#3248	C₂×C₇	1	16	yes	direct product factor
141	C₂×#479	C₂×C₇	1	2	yes	complemented
142	C₂×#475	C₂×C₇	1	1	yes	characteristic, complemented
143	C₂×#474	C₂×C₇	1	1	yes	characteristic, complemented
144	C₂×#467	C₂×C₇	1	2	yes	complemented
145	C₂×#465	C₂×C₇	1	2	yes	—
146	C₂×#464	C₂×C₇	1	2	yes	complemented
147	C₂×#462	C₂×C₇	1	1	yes	characteristic, complemented
148	C₂×#461	C₂×C₇	1	1	yes	characteristic, complemented
149	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
150	C₂⁴×C₇	—	2	1	no	—
151	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
152	C₂×C₇×Q₈	—	2	1	no	—
153	C₂×C₇×Q₈	—	2	2	no	—
154	C₂×C₇×Q₈	—	2	4	no	—
155	C₂×C₇×Q₈	—	2	4	no	—
156	C₂×C₇×Q₈	—	2	2	no	—
157	C₂×C₇×Q₈	—	2	2	no	—
158	C₂×C₇×Q₈	—	2	4	no	—
159	C₂×C₇×Q₈	—	4	4	no	—
160	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
161	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
162	C₂×C₇×D₄	—	2	4	no	—
163	C₂×C₇×D₄	—	2	1	no	—
164	C₂×C₇×D₄	—	2	2	no	—
165	C₂×C₇×D₄	—	2	2	no	—
166	C₂×C₇×D₄	—	4	4	no	—
167	C₂×C₇×D₄	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₇	—	2	4	no	—
169	C₂²×C₄×C₇	—	2	4	no	—
170	C₂²×C₄×C₇	—	2	2	no	—
171	C₂²×C₄×C₇	—	2	2	no	—
172	C₂²×C₄×C₇	—	2	1	no	—
173	C₂²×C₄×C₇	—	2	2	no	—
174	C₂²×C₄×C₇	—	2	1	no	—
175	C₂²×C₄×C₇	—	2	2	no	—
176	C₂²×C₄×C₇	—	2	1	no	—
177	C₂²×C₄×C₇	—	2	4	no	—
178	C₂²×C₄×C₇	—	4	4	no	—
179	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
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	1	yes	characteristic
184	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
189	C₇×Q₁₆	—	4	8	no	—
190	C₇×SD₁₆	—	4	8	no	—
191	C₂×C₇×C₈	—	2	4	no	—
192	C₂×C₇×C₈	—	2	4	no	—
193	C₂×C₇×C₈	—	4	8	no	—
194	C₂×C₇×C₈	C₂⁴	1	4	yes	—
195	C₂×C₇×C₈	C₂⁴	1	4	yes	—
196	C₇×C₄⋊C₄	—	2	4	no	—
197	C₇×C₄⋊C₄	—	2	4	no	—
198	C₇×C₄⋊C₄	—	2	4	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	8	no	—
204	C₇×C₄⋊C₄	—	2	4	no	—
205	C₇×C₄⋊C₄	—	2	1	no	—
206	C₇×C₄⋊C₄	—	2	1	no	—
207	C₇×C₄⋊C₄	—	2	2	no	—
208	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
209	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
210	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
211	C₇×C₂²⋊C₄	—	2	8	no	—
212	C₇×C₂²⋊C₄	—	2	4	no	—
213	C₇×C₂²⋊C₄	—	2	4	no	—
214	C₇×C₂²⋊C₄	—	2	4	no	—
215	C₇×C₂²⋊C₄	—	2	4	no	—
216	C₇×C₂²⋊C₄	—	2	4	no	—
217	C₄²×C₇	—	2	4	no	—
218	C₄²×C₇	—	2	8	no	—
219	C₄²×C₇	C₂×D₄	1	4	yes	—
220	C₄²×C₇	C₂×D₄	1	2	yes	—
221	C₄²×C₇	C₂×D₄	1	2	yes	—
222	C₄²×C₇	C₂×D₄	1	4	yes	—
223	C₄²×C₇	C₂⁴	1	2	yes	—
224	C₄²×C₇	C₂⁴	1	2	yes	—
225	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
226	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
227	C₂³×C₈	C₂²×C₇	1	2	yes	—
228	#533	—	2	4	no	—
229	#533	C₂²×C₇	1	8	yes	complemented
230	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
231	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
232	#529	—	2	4	no	—
233	#529	C₂²×C₇	1	8	yes	—
234	C₂×#126	—	2	1	no	—
235	C₂×#125	—	2	1	no	—
236	C₂×#125	C₂²×C₇	1	1	yes	characteristic
237	C₂×#124	C₂²×C₇	1	1	yes	characteristic
238	C₂×#123	—	2	1	no	—
239	C₂×C₂²⋊Q₈	—	2	1	no	—
240	C₂×C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
241	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
242	C₂×#121	C₂²×C₇	1	1	yes	characteristic
243	C₂×C₄×Q₈	—	2	1	no	—
244	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
245	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
246	C₂×C₄×D₄	—	2	1	no	—
247	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
248	C₂×#117	—	2	1	no	—
249	C₂×#117	—	2	1	no	—
250	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
251	#479	C₂²×C₇	1	16	yes	complemented
252	#475	C₂²×C₇	1	8	yes	complemented
253	#474	C₂²×C₇	1	8	yes	complemented
254	#467	C₂²×C₇	1	16	yes	complemented
255	#465	C₂²×C₇	1	16	yes	—
256	#464	C₂²×C₇	1	16	yes	complemented
257	#462	C₂²×C₇	1	8	yes	complemented
258	#461	C₂²×C₇	1	8	yes	complemented
259	C₂×#107	C₂²×C₇	1	2	yes	—
260	C₂×#106	C₂²×C₇	1	2	yes	—
261	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
263	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
264	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	complemented
265	C₂×#103	C₂²×C₇	1	2	yes	—
266	C₂×#103	C₂²×C₇	1	2	yes	—
267	C₂×#103	C₂²×C₇	1	2	yes	—
268	C₂×#102	C₂²×C₇	1	2	yes	—
269	C₂³×C₇	—	2	2	no	—
270	C₂³×C₇	—	2	2	no	—
271	C₂³×C₇	—	2	2	no	—
272	C₂³×C₇	—	4	2	no	—
273	C₂³×C₇	—	4	2	no	—
274	C₂³×C₇	—	4	2	no	—
275	C₂³×C₇	#137	1	4	yes	—
276	C₂³×C₇	C₂²×D₄	1	2	yes	—
277	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
278	C₇×Q₈	—	2	4	no	—
279	C₇×Q₈	—	2	2	no	—
280	C₇×Q₈	—	2	2	no	—
281	C₇×Q₈	—	2	4	no	—
282	C₇×Q₈	—	4	2	no	—
283	C₇×Q₈	—	4	4	no	—
284	C₇×Q₈	—	4	8	no	—
285	C₇×D₄	—	2	2	no	—
286	C₇×D₄	—	2	2	no	—
287	C₇×D₄	—	4	8	no	—
288	C₇×D₄	—	4	2	no	—
289	C₂×C₄×C₇	—	2	2	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	2	8	no	—
292	C₂×C₄×C₇	—	2	2	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	4	no	—
295	C₂×C₄×C₇	—	2	8	no	—
296	C₂×C₄×C₇	—	2	2	no	—
297	C₂×C₄×C₇	—	2	2	no	—
298	C₂×C₄×C₇	—	2	2	no	—
299	C₂×C₄×C₇	—	2	4	no	—
300	C₂×C₄×C₇	—	2	2	no	—
301	C₂×C₄×C₇	—	2	4	no	—
302	C₂×C₄×C₇	—	2	2	no	—
303	C₂×C₄×C₇	—	4	2	no	—
304	C₂×C₄×C₇	—	4	4	no	—
305	C₂×C₄×C₇	—	4	2	no	—
306	C₂×C₄×C₇	—	4	4	no	—
307	C₂×C₄×C₇	—	4	2	no	—
308	C₂×C₄×C₇	—	4	8	no	—
309	C₂×C₄×C₇	#135	1	4	yes	—
310	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
311	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
312	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
313	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
314	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
316	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
317	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
318	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
319	C₇×C₈	—	2	4	no	—
320	C₇×C₈	—	2	4	no	—
321	C₂²×Q₈	—	2	1	no	—
322	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
323	C₂²×Q₈	C₂³×C₇	1	2	yes	—
324	C₂²×D₄	—	2	1	no	—
325	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
326	C₂³×C₄	—	2	1	no	—
327	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
328	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
330	C₂×Q₁₆	—	4	4	no	—
331	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
332	C₂×SD₁₆	—	4	4	no	—
333	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
334	C₂²×C₈	—	4	4	no	—
335	C₂²×C₈	C₂³×C₇	1	2	yes	—
336	C₂²×C₈	C₂³×C₇	1	2	yes	—
337	C₂²×C₈	C₂³×C₇	1	8	yes	—
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
341	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
342	#126	—	2	8	no	—
343	#125	—	2	8	no	—
344	#125	—	2	2	no	—
345	#125	C₂³×C₇	1	4	yes	—
346	#124	—	2	2	no	—
347	#124	C₂³×C₇	1	4	yes	—
348	#123	—	2	8	no	—
349	C₂²⋊Q₈	—	2	8	no	—
350	C₂²⋊Q₈	—	2	2	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
353	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
354	#121	—	2	2	no	—
355	#121	C₂³×C₇	1	4	yes	—
356	C₄×Q₈	—	2	8	no	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	—	2	4	no	—
359	C₄×Q₈	C₂³×C₇	1	4	yes	—
360	C₄×Q₈	C₂³×C₇	1	8	yes	—
361	C₄×D₄	—	2	8	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	C₂³×C₇	1	4	yes	—
364	#117	—	2	8	no	—
365	#117	—	2	8	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	2	no	—
373	C₂×C₄⋊C₄	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
378	C₂×C₂²⋊C₄	—	2	2	no	—
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	—	2	2	no	—
386	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
387	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
388	C₂×C₄²	C₇×D₄	1	4	yes	—
389	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₇	1	4	yes	—
392	#107	C₂³×C₇	1	8	yes	—
393	#106	C₂³×C₇	1	8	yes	—
394	C₄⋊C₈	C₂³×C₇	1	4	yes	—
395	C₄⋊C₈	C₂³×C₇	1	4	yes	—
396	C₄⋊C₈	C₂³×C₇	1	8	yes	—
397	#103	C₂³×C₇	1	8	yes	—
398	#103	C₂³×C₇	1	8	yes	—
399	#103	C₂³×C₇	1	8	yes	—
400	#102	C₂³×C₇	1	8	yes	—
401	C₂²×C₇	—	2	2	no	—
402	C₂²×C₇	—	2	8	no	—
403	C₂²×C₇	—	2	2	no	—
404	C₂²×C₇	—	4	2	no	—
405	C₂²×C₇	—	4	2	no	—
406	C₂²×C₇	—	4	2	no	—
407	C₂²×C₇	—	8	4	no	—
408	C₂²×C₇	#537	1	2	yes	—
409	C₂²×C₇	C₂×#135	1	2	yes	—
410	C₂²×C₇	C₂×#137	1	4	yes	—
411	C₂²×C₇	C₂×#137	1	2	yes	—
412	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
413	C₄×C₇	—	2	2	no	—
414	C₄×C₇	—	2	2	no	—
415	C₄×C₇	—	2	2	no	—
416	C₄×C₇	—	2	2	no	—
417	C₄×C₇	—	4	2	no	—
418	C₄×C₇	—	4	4	no	—
419	C₄×C₇	—	4	2	no	—
420	C₄×C₇	—	4	4	no	—
421	C₄×C₇	—	4	2	no	—
422	C₄×C₇	#537	1	2	yes	—
423	C₄×C₇	C₂×#135	1	4	yes	—
424	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
425	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
426	C₂⁴	—	2	1	no	—
427	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	2	4	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	4	4	no	—
436	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
437	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	4	4	no	—
443	C₂×D₄	C₂⁴×C₇	1	2	yes	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
456	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
457	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
458	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
459	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
460	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
461	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
462	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
464	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
465	Q₁₆	—	4	8	no	—
466	SD₁₆	—	4	8	no	—
467	C₂×C₈	—	2	4	no	—
468	C₂×C₈	—	2	4	no	—
469	C₂×C₈	—	4	8	no	—
470	C₂×C₈	C₂⁴×C₇	1	4	yes	—
471	C₂×C₈	C₂⁴×C₇	1	4	yes	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	8	no	—
479	C₄⋊C₄	—	2	8	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
485	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
486	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₄²	—	2	4	no	—
494	C₄²	—	2	8	no	—
495	C₄²	C₂×C₇×D₄	1	4	yes	—
496	C₄²	C₂×C₇×D₄	1	2	yes	—
497	C₄²	C₂×C₇×D₄	1	2	yes	—
498	C₄²	C₂×C₇×D₄	1	4	yes	—
499	C₄²	C₂⁴×C₇	1	2	yes	—
500	C₄²	C₂⁴×C₇	1	2	yes	—
501	C₂×C₇	—	2	4	no	—
502	C₂×C₇	—	8	2	no	—
503	C₂×C₇	#3248	1	4	yes	direct product factor
504	C₂×C₇	C₂×#537	1	1	yes	characteristic
505	C₂×C₇	C₂²×#135	1	1	yes	characteristic
506	C₂×C₇	C₂²×#137	1	1	yes	characteristic
507	C₂³	—	2	2	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	C₇×#137	1	4	yes	—
514	C₂³	C₂²×C₇×D₄	1	2	yes	—
515	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
516	Q₈	—	2	4	no	—
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	2	4	no	—
520	Q₈	—	4	2	no	—
521	Q₈	—	4	4	no	—
522	Q₈	—	4	8	no	—
523	D₄	—	2	2	no	—
524	D₄	—	2	2	no	—
525	D₄	—	4	8	no	—
526	D₄	—	4	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	4	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	8	no	—
547	C₂×C₄	C₇×#135	1	4	yes	—
548	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
549	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
550	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
551	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
552	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
553	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
554	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
555	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
556	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
557	C₈	—	2	4	no	—
558	C₈	—	2	4	no	—
559	C₇	C₂×#3248	1	1	yes	characteristic, direct product factor, Sylow
560	C₂²	—	2	2	no	—
561	C₂²	—	2	8	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	8	4	no	—
567	C₂²	C₇×#537	1	2	yes	—
568	C₂²	C₂×C₇×#135	1	2	yes	—
569	C₂²	C₂×C₇×#137	1	4	yes	—
570	C₂²	C₂×C₇×#137	1	2	yes	—
571	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	4	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	2	no	—
580	C₄	—	4	4	no	—
581	C₄	C₇×#537	1	2	yes	—
582	C₄	C₂×C₇×#135	1	4	yes	—
583	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
584	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
585	C₂	—	2	4	no	—
586	C₂	—	8	2	no	—
587	C₂	C₇×#3248	1	4	yes	direct product factor
588	C₂	C₂×C₇×#537	1	1	yes	characteristic
589	C₂	C₂²×C₇×#135	1	1	yes	characteristic
590	C₂	C₂²×C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#533	C₂	semidirect
3. C₇×#3248	C₂	direct (N × H)
4. C₂×C₇×#479	C₂	semidirect
5. C₂×C₇×#475	C₂	semidirect
6. C₂×C₇×#474	C₂	semidirect
7. C₂×C₇×#467	C₂	semidirect
9. C₂×C₇×#464	C₂	semidirect
10. C₂×C₇×#462	C₂	semidirect
11. C₂×C₇×#461	C₂	semidirect
12. C₂²×C₇×C₄⋊C₈	C₂	semidirect
17. C₇×#533	C₂²	semidirect
39. C₇×#479	C₂²	semidirect
40. C₇×#475	C₂²	semidirect
41. C₇×#474	C₂²	semidirect
42. C₇×#467	C₂²	semidirect
44. C₇×#464	C₂²	semidirect
45. C₇×#462	C₂²	semidirect
46. C₇×#461	C₂²	semidirect
52. C₂×C₇×C₄⋊C₈	C₂²	semidirect
57. C₂×#3248	C₇	direct (N × H)
138. C₂×#533	C₂×C₇	semidirect
140. #3248	C₂×C₇	direct (N × H)
141. C₂×#479	C₂×C₇	semidirect
142. C₂×#475	C₂×C₇	semidirect
143. C₂×#474	C₂×C₇	semidirect
144. C₂×#467	C₂×C₇	semidirect
146. C₂×#464	C₂×C₇	semidirect
147. C₂×#462	C₂×C₇	semidirect
148. C₂×#461	C₂×C₇	semidirect
149. C₂²×C₄⋊C₈	C₂×C₇	semidirect
229. #533	C₂²×C₇	semidirect
251. #479	C₂²×C₇	semidirect
252. #475	C₂²×C₇	semidirect
253. #474	C₂²×C₇	semidirect
254. #467	C₂²×C₇	semidirect
256. #464	C₂²×C₇	semidirect
257. #462	C₂²×C₇	semidirect
258. #461	C₂²×C₇	semidirect
264. C₂×C₄⋊C₈	C₂²×C₇	semidirect
503. C₂×C₇	#3248	direct (N × H)
559. C₇	C₂×#3248	direct (N × H)
587. C₂	C₇×#3248	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×#533	2
C₂×C₇×#529	2
C₇×#3248	2
C₂×C₇×#479	2
C₂×C₇×#475	2
C₂×C₇×#474	2
C₂×C₇×#467	2
C₂×C₇×#465	2
C₂×C₇×#464	2
C₂×C₇×#462	2
C₂×C₇×#461	2
C₂²×C₇×C₄⋊C₈	2
C₂×#3248	7

Subgroups — C2×C7×#3248 #13238113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3248 #13238113