Subgroups — C₆₇×#33014 #30712265

Order 17152.

1474 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#518×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#482×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#447×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2624	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1862	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1825	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1786	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
15	#518×C₆₇	—	2	4	no	—
16	#518×C₆₇	—	2	4	no	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
24	#482×C₆₇	—	2	4	no	—
25	#447×C₆₇	—	2	4	no	—
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#399×C₆₇	—	2	1	no	—
36	#397×C₆₇	—	2	1	no	—
37	#394×C₆₇	—	2	2	no	—
38	#390×C₆₇	C₂²	1	2	yes	complemented
39	#389×C₆₇	C₂²	1	2	yes	complemented
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	complemented
42	#340×C₆₇	C₂²	1	2	yes	complemented
43	#336×C₆₇	C₂²	1	2	yes	complemented
44	#336×C₆₇	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	4	no	—
47	C₂²×D₄×C₆₇	—	2	4	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	D₄	1	4	yes	—
52	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	—	2	2	no	—
55	C₂³×C₄×C₆₇	—	2	2	no	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×D₈×C₆₇	—	2	4	no	—
58	C₂×D₈×C₆₇	—	2	4	no	—
59	C₂×D₈×C₆₇	—	2	4	no	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	#121×C₆₇	—	2	2	no	—
64	#121×C₆₇	—	2	2	no	—
65	C₄×D₄×C₆₇	—	2	8	no	—
66	C₄×D₄×C₆₇	—	2	8	no	—
67	C₄×D₄×C₆₇	—	2	4	no	—
68	C₄×D₄×C₆₇	—	2	4	no	—
69	C₄×D₄×C₆₇	—	2	4	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	—	2	2	no	—
74	#117×C₆₇	D₄	1	4	yes	—
75	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
93	#107×C₆₇	—	2	2	no	—
94	#106×C₆₇	—	2	2	no	—
95	#102×C₆₇	—	2	4	no	—
96	#102×C₆₇	—	2	4	no	—
97	#102×C₆₇	—	2	4	no	—
98	#102×C₆₇	—	2	2	no	—
99	#102×C₆₇	—	2	2	no	—
100	C₂²⋊C₈×C₆₇	—	2	2	no	—
101	C₂²⋊C₈×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	2	no	—
104	C₂⁴×C₆₇	—	2	8	no	—
105	C₂⁴×C₆₇	—	2	2	no	—
106	C₂⁴×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	—	2	1	no	—
108	C₂⁴×C₆₇	—	8	1	no	—
109	C₂⁴×C₆₇	—	8	2	no	—
110	C₂⁴×C₆₇	—	8	1	no	—
111	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×D₄×C₆₇	—	2	8	no	—
113	C₂×D₄×C₆₇	—	2	8	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	2	16	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	2	1	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	4	2	no	—
125	C₂×D₄×C₆₇	—	8	2	no	—
126	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
127	C₂×D₄×C₆₇	#41	1	4	yes	—
128	C₂×D₄×C₆₇	#41	1	4	yes	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	8	no	—
134	C₂²×C₄×C₆₇	—	2	8	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	8	2	no	—
138	C₂²×C₄×C₆₇	—	8	2	no	—
139	C₂²×C₄×C₆₇	—	8	2	no	—
140	C₂²×C₄×C₆₇	—	8	2	no	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	D₈×C₆₇	—	4	8	no	—
148	C₂×C₈×C₆₇	—	2	1	no	—
149	C₂×C₈×C₆₇	—	2	1	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	1	no	—
159	C₄⋊C₄×C₆₇	—	2	1	no	—
160	C₄⋊C₄×C₆₇	—	2	1	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	8	2	no	—
164	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₆₇	—	2	8	no	—
167	C₂²⋊C₄×C₆₇	—	2	8	no	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	2	4	no	—
170	C₂²⋊C₄×C₆₇	—	4	2	no	—
171	C₂²⋊C₄×C₆₇	—	8	2	no	—
172	C₂²⋊C₄×C₆₇	—	8	2	no	—
173	C₂²⋊C₄×C₆₇	—	8	1	no	—
174	C₄²×C₆₇	—	2	4	no	—
175	C₄²×C₆₇	—	2	2	no	—
176	C₄²×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	16	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	—	2	4	no	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	1	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	4	2	no	—
187	C₂³×C₆₇	—	4	2	no	—
188	C₂³×C₆₇	—	4	2	no	—
189	C₂³×C₆₇	—	4	1	no	—
190	C₂³×C₆₇	—	4	2	no	—
191	C₂³×C₆₇	—	4	1	no	—
192	C₂³×C₆₇	—	4	2	no	—
193	C₂³×C₆₇	—	4	1	no	—
194	C₂³×C₆₇	—	8	4	no	—
195	C₂³×C₆₇	—	8	4	no	—
196	C₂³×C₆₇	—	8	8	no	—
197	C₂³×C₆₇	—	8	8	no	—
198	C₂³×C₆₇	#120	1	2	yes	—
199	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	D₄×C₆₇	—	2	16	no	—
201	D₄×C₆₇	—	2	4	no	—
202	D₄×C₆₇	—	2	4	no	—
203	D₄×C₆₇	—	2	4	no	—
204	D₄×C₆₇	—	4	2	no	—
205	D₄×C₆₇	—	4	1	no	—
206	D₄×C₆₇	—	4	1	no	—
207	D₄×C₆₇	—	8	4	no	—
208	D₄×C₆₇	#123	1	8	yes	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	4	1	no	—
224	C₂×C₄×C₆₇	—	4	1	no	—
225	C₂×C₄×C₆₇	—	4	2	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	2	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	8	8	no	—
230	C₂×C₄×C₆₇	—	8	8	no	—
231	C₂×C₄×C₆₇	#120	1	2	yes	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
235	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
236	C₈×C₆₇	—	4	2	no	—
237	C₈×C₆₇	—	4	1	no	—
238	C₈×C₆₇	—	4	1	no	—
239	C₂²×C₆₇	—	2	2	no	—
240	C₂²×C₆₇	—	2	8	no	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	2	2	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	—	4	4	no	—
246	C₂²×C₆₇	—	4	2	no	—
247	C₂²×C₆₇	—	4	4	no	—
248	C₂²×C₆₇	—	4	4	no	—
249	C₂²×C₆₇	—	4	4	no	—
250	C₂²×C₆₇	—	4	4	no	—
251	C₂²×C₆₇	—	8	16	no	—
252	C₂²×C₆₇	—	8	1	no	—
253	C₂²×C₆₇	—	8	2	no	—
254	C₂²×C₆₇	—	8	2	no	—
255	C₂²×C₆₇	—	8	1	no	—
256	C₂²×C₆₇	—	8	1	no	—
257	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
258	C₂²×C₆₇	#525	1	2	yes	—
259	C₂²×C₆₇	#576	1	2	yes	—
260	C₂²×C₆₇	#576	1	1	yes	characteristic
261	C₂²×C₆₇	#577	1	1	yes	characteristic
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	4	2	no	—
264	C₄×C₆₇	—	4	2	no	—
265	C₄×C₆₇	—	4	2	no	—
266	C₄×C₆₇	—	4	2	no	—
267	C₄×C₆₇	—	8	2	no	—
268	C₄×C₆₇	—	8	1	no	—
269	C₄×C₆₇	C₂×#123	1	2	yes	—
270	C₄×C₆₇	#525	1	1	yes	characteristic
271	C₄×C₆₇	#525	1	1	yes	characteristic
272	#33014	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂×C₆₇	—	4	2	no	—
274	C₂×C₆₇	—	4	8	no	—
275	C₂×C₆₇	—	16	1	no	—
276	C₂×C₆₇	#2849	1	1	yes	characteristic
277	C₂×C₆₇	#3008	1	1	yes	characteristic
278	C₂×C₆₇	#3012	1	1	yes	characteristic
279	C₂×C₆₇	#3118	1	2	yes	—
280	C₂×C₆₇	#3122	1	2	yes	—
281	C₂×#518	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₂×#482	C₂×C₆₇	1	1	yes	characteristic, complemented
283	C₂×#447	C₂×C₆₇	1	1	yes	characteristic
284	#2624	C₂×C₆₇	1	1	yes	characteristic, complemented
285	#1879	C₂×C₆₇	1	2	yes	complemented
286	#1879	C₂×C₆₇	1	2	yes	complemented
287	#1864	C₂×C₆₇	1	2	yes	complemented
288	#1862	C₂×C₆₇	1	2	yes	complemented
289	#1825	C₂×C₆₇	1	1	yes	characteristic, complemented
290	#1818	C₂×C₆₇	1	1	yes	characteristic, complemented
291	#1786	C₂×C₆₇	1	1	yes	characteristic, complemented
292	C₆₇	#33014	1	1	yes	characteristic, direct product factor, Sylow
293	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
294	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
295	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
296	#518	—	2	4	no	—
297	#518	—	2	4	no	—
298	C₂×C₄×D₄	—	2	1	no	—
299	C₂×C₄×D₄	—	2	1	no	—
300	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
301	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
302	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
304	C₂²×C₂²⋊C₄	—	2	2	no	—
305	#482	—	2	4	no	—
306	#447	—	2	4	no	—
307	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
308	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
309	C₂×#102	C₂²×C₆₇	1	2	yes	—
310	C₂×#102	C₂²×C₆₇	1	2	yes	—
311	C₂×#102	C₂²×C₆₇	1	2	yes	—
312	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
313	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
314	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
316	#399	—	2	1	no	—
317	#397	—	2	1	no	—
318	#394	—	2	2	no	—
319	#390	C₂²×C₆₇	1	2	yes	complemented
320	#389	C₂²×C₆₇	1	2	yes	complemented
321	#380	C₂²×C₆₇	1	1	yes	characteristic
322	#340	C₂²×C₆₇	1	2	yes	complemented
323	#340	C₂²×C₆₇	1	2	yes	complemented
324	#336	C₂²×C₆₇	1	2	yes	complemented
325	#336	C₂²×C₆₇	1	2	yes	complemented
326	C₂⁵	—	2	1	no	—
327	C₂²×D₄	—	2	4	no	—
328	C₂²×D₄	—	2	4	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	—	2	2	no	—
332	C₂²×D₄	D₄×C₆₇	1	4	yes	—
333	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
334	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	—	2	2	no	—
337	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
338	C₂×D₈	—	2	4	no	—
339	C₂×D₈	—	2	4	no	—
340	C₂×D₈	—	2	4	no	—
341	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
342	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
343	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
344	#121	—	2	2	no	—
345	#121	—	2	2	no	—
346	C₄×D₄	—	2	8	no	—
347	C₄×D₄	—	2	8	no	—
348	C₄×D₄	—	2	4	no	—
349	C₄×D₄	—	2	4	no	—
350	C₄×D₄	—	2	4	no	—
351	C₄×D₄	—	2	4	no	—
352	#117	—	2	2	no	—
353	#117	—	2	2	no	—
354	#117	—	2	2	no	—
355	#117	D₄×C₆₇	1	4	yes	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
361	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
363	C₂×C₂²⋊C₄	—	2	16	no	—
364	C₂×C₂²⋊C₄	—	2	1	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	—	2	2	no	—
367	C₂×C₂²⋊C₄	—	2	1	no	—
368	C₂×C₂²⋊C₄	—	2	1	no	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
374	#107	—	2	2	no	—
375	#106	—	2	2	no	—
376	#102	—	2	4	no	—
377	#102	—	2	4	no	—
378	#102	—	2	4	no	—
379	#102	—	2	2	no	—
380	#102	—	2	2	no	—
381	C₂²⋊C₈	—	2	2	no	—
382	C₂²⋊C₈	—	2	2	no	—
383	#95	—	2	1	no	—
384	#95	—	2	2	no	—
385	C₂⁴	—	2	8	no	—
386	C₂⁴	—	2	2	no	—
387	C₂⁴	—	2	4	no	—
388	C₂⁴	—	2	1	no	—
389	C₂⁴	—	8	1	no	—
390	C₂⁴	—	8	2	no	—
391	C₂⁴	—	8	1	no	—
392	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
393	C₂×D₄	—	2	8	no	—
394	C₂×D₄	—	2	8	no	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	2	16	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	4	2	no	—
406	C₂×D₄	—	8	2	no	—
407	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
408	C₂×D₄	#41×C₆₇	1	4	yes	—
409	C₂×D₄	#41×C₆₇	1	4	yes	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	8	2	no	—
419	C₂²×C₄	—	8	2	no	—
420	C₂²×C₄	—	8	2	no	—
421	C₂²×C₄	—	8	2	no	—
422	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
423	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
427	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
428	D₈	—	4	8	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	8	2	no	—
445	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
446	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	4	2	no	—
452	C₂²⋊C₄	—	8	2	no	—
453	C₂²⋊C₄	—	8	2	no	—
454	C₂²⋊C₄	—	8	1	no	—
455	C₄²	—	2	4	no	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	2	no	—
458	C₂³	—	2	16	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	4	no	—
463	C₂³	—	2	1	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	1	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	2	no	—
468	C₂³	—	4	2	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	2	no	—
472	C₂³	—	4	1	no	—
473	C₂³	—	4	2	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	8	4	no	—
476	C₂³	—	8	4	no	—
477	C₂³	—	8	8	no	—
478	C₂³	—	8	8	no	—
479	C₂³	#120×C₆₇	1	2	yes	—
480	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
481	D₄	—	2	16	no	—
482	D₄	—	2	4	no	—
483	D₄	—	2	4	no	—
484	D₄	—	2	4	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	1	no	—
487	D₄	—	4	1	no	—
488	D₄	—	8	4	no	—
489	D₄	#123×C₆₇	1	8	yes	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	8	8	no	—
511	C₂×C₄	—	8	8	no	—
512	C₂×C₄	#120×C₆₇	1	2	yes	—
513	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
516	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
517	C₈	—	4	2	no	—
518	C₈	—	4	1	no	—
519	C₈	—	4	1	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	2	8	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	8	16	no	—
533	C₂²	—	8	1	no	—
534	C₂²	—	8	1	no	—
535	C₂²	—	8	2	no	—
536	C₂²	—	8	2	no	—
537	C₂²	—	8	1	no	—
538	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
539	C₂²	#525×C₆₇	1	2	yes	—
540	C₂²	#576×C₆₇	1	1	yes	characteristic
541	C₂²	#576×C₆₇	1	2	yes	—
542	C₂²	#577×C₆₇	1	1	yes	characteristic
543	C₄	—	2	2	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	8	2	no	—
549	C₄	—	8	1	no	—
550	C₄	C₂×#123×C₆₇	1	2	yes	—
551	C₄	#525×C₆₇	1	1	yes	characteristic
552	C₄	#525×C₆₇	1	1	yes	characteristic
553	C₂	—	4	2	no	—
554	C₂	—	4	8	no	—
555	C₂	—	16	1	no	—
556	C₂	C₆₇×#2849	1	1	yes	characteristic
557	C₂	C₆₇×#3008	1	1	yes	characteristic
558	C₂	C₆₇×#3012	1	1	yes	characteristic
559	C₂	C₆₇×#3118	1	2	yes	—
560	C₂	C₆₇×#3122	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#447×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#518×C₆₇	C₂	semidirect
2. C₂×#482×C₆₇	C₂	semidirect
4. C₆₇×#2624	C₂	semidirect
5. C₆₇×#1879	C₂	semidirect
6. C₆₇×#1879	C₂	semidirect
7. C₆₇×#1864	C₂	semidirect
8. C₆₇×#1862	C₂	semidirect
9. C₆₇×#1825	C₂	semidirect
10. C₆₇×#1818	C₂	semidirect
11. C₆₇×#1786	C₂	semidirect
26. C₂×#107×C₆₇	C₂²	semidirect
27. C₂×#106×C₆₇	C₂²	semidirect
31. C₂×#102×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
38. #390×C₆₇	C₂²	semidirect
39. #389×C₆₇	C₂²	semidirect
41. #340×C₆₇	C₂²	semidirect
42. #340×C₆₇	C₂²	semidirect
43. #336×C₆₇	C₂²	semidirect
44. #336×C₆₇	C₂²	semidirect
272. #33014	C₆₇	direct (N × H)
281. C₂×#518	C₂×C₆₇	semidirect
282. C₂×#482	C₂×C₆₇	semidirect
284. #2624	C₂×C₆₇	semidirect
285. #1879	C₂×C₆₇	semidirect
286. #1879	C₂×C₆₇	semidirect
287. #1864	C₂×C₆₇	semidirect
288. #1862	C₂×C₆₇	semidirect
289. #1825	C₂×C₆₇	semidirect
290. #1818	C₂×C₆₇	semidirect
291. #1786	C₂×C₆₇	semidirect
292. C₆₇	#33014	direct (N × H)
307. C₂×#107	C₂²×C₆₇	semidirect
308. C₂×#106	C₂²×C₆₇	semidirect
312. C₂×#102	C₂²×C₆₇	semidirect
313. C₂×#102	C₂²×C₆₇	semidirect
319. #390	C₂²×C₆₇	semidirect
320. #389	C₂²×C₆₇	semidirect
322. #340	C₂²×C₆₇	semidirect
323. #340	C₂²×C₆₇	semidirect
324. #336	C₂²×C₆₇	semidirect
325. #336	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#518×C₆₇	2
C₂×#482×C₆₇	2
C₂×#447×C₆₇	2
C₆₇×#2624	2
C₆₇×#1879	2
C₆₇×#1864	2
C₆₇×#1862	2
C₆₇×#1825	2
C₆₇×#1818	2
C₆₇×#1786	2
#33014	67

Subgroups — C67×#33014 #30712265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33014 #30712265