Subgroups — C₆₇×#11207 #30690458

Order 17152.

958 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#389×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#340×C₆₇	C₂	1	2	yes	maximal
4	C₂×#340×C₆₇	C₂	1	2	yes	maximal
5	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#389×C₆₇	—	2	2	no	—
15	#389×C₆₇	C₄	1	4	yes	complemented
16	#384×C₆₇	—	2	4	no	—
17	C₂×#95×C₆₇	—	2	1	no	—
18	#340×C₆₇	—	2	4	no	—
19	#340×C₆₇	—	2	4	no	—
20	#336×C₆₇	—	2	2	no	—
21	C₂³×C₄×C₆₇	—	2	1	no	—
22	C₂³×C₄×C₆₇	—	2	2	no	—
23	C₂³×C₄×C₆₇	—	2	1	no	—
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	—	2	2	no	—
33	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
59	C₂×C₄²×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₇	D₄	1	2	yes	—
63	#95×C₆₇	—	2	4	no	—
64	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	4	1	no	—
79	C₂²×C₄×C₆₇	—	4	1	no	—
80	C₂²×C₄×C₆₇	—	4	1	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	—	4	2	no	—
84	C₂²×C₄×C₆₇	—	4	1	no	—
85	C₂²×C₄×C₆₇	—	4	2	no	—
86	C₂²×C₄×C₆₇	—	4	1	no	—
87	C₂²×C₄×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	1	no	—
94	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	D₈	1	2	yes	—
97	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
98	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
99	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
106	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	#41	1	2	yes	—
109	C₂²×C₄×C₆₇	#41	1	2	yes	—
110	C₂×C₈×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₂×C₈×C₆₇	—	4	2	no	—
114	C₂×C₈×C₆₇	—	4	2	no	—
115	C₂×C₈×C₆₇	—	4	2	no	—
116	C₂×C₈×C₆₇	—	4	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₄⋊C₄×C₆₇	—	4	8	no	—
124	C₄⋊C₄×C₆₇	—	4	8	no	—
125	C₄⋊C₄×C₆₇	—	4	8	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	complemented
130	C₄²×C₆₇	—	2	4	no	—
131	C₄²×C₆₇	—	2	4	no	—
132	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂³×C₆₇	#120	1	1	yes	characteristic
134	C₂³×C₆₇	#121	1	1	yes	characteristic
135	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂³×C₆₇	#123	1	1	yes	characteristic
137	C₂³×C₆₇	#124	1	1	yes	characteristic
138	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
139	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
140	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
141	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
142	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
143	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂³×C₆₇	#137	1	1	yes	characteristic
146	C₂³×C₆₇	#137	1	1	yes	characteristic
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	4	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	—	4	4	no	—
172	C₂×C₄×C₆₇	—	8	1	no	—
173	C₂×C₄×C₆₇	—	8	1	no	—
174	C₂×C₄×C₆₇	—	8	2	no	—
175	C₂×C₄×C₆₇	—	8	2	no	—
176	C₂×C₄×C₆₇	—	8	1	no	—
177	C₂×C₄×C₆₇	—	8	1	no	—
178	C₂×C₄×C₆₇	—	8	2	no	—
179	C₂×C₄×C₆₇	—	8	2	no	—
180	C₂×C₄×C₆₇	#102	1	4	yes	—
181	C₂×C₄×C₆₇	#103	1	4	yes	—
182	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
186	C₂×C₄×C₆₇	#121	1	2	yes	—
187	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
188	C₂×C₄×C₆₇	#121	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
190	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
191	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
192	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
193	C₂×C₄×C₆₇	#123	1	2	yes	—
194	C₂×C₄×C₆₇	#125	1	2	yes	—
195	C₂×C₄×C₆₇	#125	1	2	yes	—
196	C₂×C₄×C₆₇	#126	1	2	yes	—
197	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
198	C₈×C₆₇	—	4	4	no	—
199	C₂²×C₆₇	#380	1	1	yes	characteristic
200	C₂²×C₆₇	#386	1	1	yes	characteristic
201	C₂²×C₆₇	#391	1	1	yes	characteristic
202	C₂²×C₆₇	#394	1	1	yes	characteristic
203	C₂²×C₆₇	#395	1	1	yes	characteristic
204	C₂²×C₆₇	#398	1	1	yes	characteristic
205	C₂²×C₆₇	#400	1	1	yes	characteristic
206	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
207	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
208	C₂²×C₆₇	#417	1	1	yes	characteristic
209	C₂²×C₆₇	#417	1	1	yes	characteristic
210	C₂²×C₆₇	#447	1	1	yes	characteristic
211	C₂²×C₆₇	#448	1	1	yes	characteristic
212	C₂²×C₆₇	#450	1	1	yes	characteristic
213	C₂²×C₆₇	#451	1	1	yes	characteristic
214	C₂²×C₆₇	#459	1	1	yes	characteristic
215	C₂²×C₆₇	#460	1	1	yes	characteristic
216	C₂²×C₆₇	#461	1	1	yes	characteristic
217	C₂²×C₆₇	#462	1	1	yes	characteristic
218	C₂²×C₆₇	#474	1	1	yes	characteristic
219	C₂²×C₆₇	#475	1	1	yes	characteristic
220	C₂²×C₆₇	#476	1	1	yes	characteristic
221	C₂²×C₆₇	#477	1	1	yes	characteristic
222	C₂²×C₆₇	#480	1	1	yes	characteristic
223	C₂²×C₆₇	#481	1	1	yes	characteristic
224	C₂²×C₆₇	#483	1	1	yes	characteristic
225	C₂²×C₆₇	#484	1	1	yes	characteristic
226	C₂²×C₆₇	#486	1	1	yes	characteristic
227	C₂²×C₆₇	#487	1	1	yes	characteristic
228	C₂²×C₆₇	#490	1	1	yes	characteristic
229	C₂²×C₆₇	#491	1	1	yes	characteristic
230	C₂²×C₆₇	#498	1	1	yes	characteristic
231	C₂²×C₆₇	#500	1	1	yes	characteristic
232	C₂²×C₆₇	#501	1	1	yes	characteristic
233	C₂²×C₆₇	#501	1	1	yes	characteristic
234	C₄×C₆₇	—	2	4	no	—
235	C₄×C₆₇	—	2	4	no	—
236	C₄×C₆₇	—	4	4	no	—
237	C₄×C₆₇	—	8	2	no	—
238	C₄×C₆₇	—	8	4	no	—
239	C₄×C₆₇	#387	1	2	yes	—
240	C₄×C₆₇	#389	1	2	yes	—
241	C₄×C₆₇	#396	1	2	yes	—
242	C₄×C₆₇	#398	1	2	yes	—
243	#11207	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂×C₆₇	#1441	1	1	yes	characteristic
245	C₂×C₆₇	#1802	1	1	yes	characteristic
246	C₂×C₆₇	#1803	1	1	yes	characteristic
247	C₂×C₆₇	#1872	1	1	yes	characteristic
248	C₂×C₆₇	#1874	1	1	yes	characteristic
249	C₂×C₆₇	#1990	1	1	yes	characteristic
250	C₂×C₆₇	#1992	1	1	yes	characteristic
251	C₂×C₆₇	#2041	1	1	yes	characteristic
252	C₂×C₆₇	#2043	1	1	yes	characteristic
253	C₂×C₆₇	#2057	1	1	yes	characteristic
254	C₂×C₆₇	#2058	1	1	yes	characteristic
255	C₂×C₆₇	#2083	1	1	yes	characteristic
256	C₂×C₆₇	#2084	1	1	yes	characteristic
257	C₂×C₆₇	#2099	1	1	yes	characteristic
258	C₂×C₆₇	#2100	1	1	yes	characteristic
259	C₂×#389	C₂×C₆₇	1	1	yes	characteristic
260	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
261	C₂×#340	C₂×C₆₇	1	2	yes	—
262	C₂×#340	C₂×C₆₇	1	2	yes	—
263	C₂×#336	C₂×C₆₇	1	1	yes	characteristic
264	C₆₇	#11207	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
266	C₂²×C₄⋊C₄	—	2	1	no	—
267	C₂²×C₄⋊C₄	—	2	1	no	—
268	C₂²×C₄⋊C₄	—	2	1	no	—
269	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
270	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
271	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
273	#389	—	2	2	no	—
274	#389	C₄×C₆₇	1	4	yes	complemented
275	#384	—	2	4	no	—
276	C₂×#95	—	2	1	no	—
277	#340	—	2	4	no	—
278	#340	—	2	4	no	—
279	#336	—	2	2	no	—
280	C₂³×C₄	—	2	1	no	—
281	C₂³×C₄	—	2	2	no	—
282	C₂³×C₄	—	2	1	no	—
283	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
284	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	—	2	2	no	—
290	C₂²×C₈	—	2	2	no	—
291	C₂²×C₈	—	2	2	no	—
292	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	4	no	—
303	C₂×C₄⋊C₄	—	2	4	no	—
304	C₂×C₄⋊C₄	—	2	4	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	4	2	no	—
308	C₂×C₄⋊C₄	—	4	2	no	—
309	C₂×C₄⋊C₄	—	4	2	no	—
310	C₂×C₄⋊C₄	—	4	2	no	—
311	C₂×C₄⋊C₄	—	4	2	no	—
312	C₂×C₄⋊C₄	—	4	2	no	—
313	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
314	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
315	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
316	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
317	C₂×C₄⋊C₄	Q₈×C₆₇	1	2	yes	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
321	C₂×C₄²	D₄×C₆₇	1	2	yes	—
322	#95	—	2	4	no	—
323	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	4	1	no	—
338	C₂²×C₄	—	4	1	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	4	1	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	4	1	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	—	4	2	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
354	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
355	C₂²×C₄	D₈×C₆₇	1	2	yes	—
356	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
357	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
358	C₂²×C₄	Q₁₆×C₆₇	1	2	yes	—
359	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
365	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	#41×C₆₇	1	2	yes	—
368	C₂²×C₄	#41×C₆₇	1	2	yes	—
369	C₂×C₈	—	2	4	no	—
370	C₂×C₈	—	2	4	no	—
371	C₂×C₈	—	2	4	no	—
372	C₂×C₈	—	4	2	no	—
373	C₂×C₈	—	4	2	no	—
374	C₂×C₈	—	4	2	no	—
375	C₂×C₈	—	4	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	4	no	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	4	8	no	—
383	C₄⋊C₄	—	4	8	no	—
384	C₄⋊C₄	—	4	8	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	complemented
389	C₄²	—	2	4	no	—
390	C₄²	—	2	4	no	—
391	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
392	C₂³	#120×C₆₇	1	1	yes	characteristic
393	C₂³	#121×C₆₇	1	1	yes	characteristic
394	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
395	C₂³	#123×C₆₇	1	1	yes	characteristic
396	C₂³	#124×C₆₇	1	1	yes	characteristic
397	C₂³	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
398	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
399	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
400	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
401	C₂³	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
402	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
403	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
404	C₂³	#137×C₆₇	1	1	yes	characteristic
405	C₂³	#137×C₆₇	1	1	yes	characteristic
406	C₂×C₄	—	2	2	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	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	8	1	no	—
432	C₂×C₄	—	8	2	no	—
433	C₂×C₄	—	8	2	no	—
434	C₂×C₄	—	8	1	no	—
435	C₂×C₄	—	8	1	no	—
436	C₂×C₄	—	8	2	no	—
437	C₂×C₄	—	8	2	no	—
438	C₂×C₄	—	8	1	no	—
439	C₂×C₄	#102×C₆₇	1	4	yes	—
440	C₂×C₄	#103×C₆₇	1	4	yes	—
441	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
444	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
445	C₂×C₄	#121×C₆₇	1	2	yes	—
446	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
447	C₂×C₄	#121×C₆₇	1	2	yes	—
448	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
449	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
450	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
451	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
452	C₂×C₄	#123×C₆₇	1	2	yes	—
453	C₂×C₄	#125×C₆₇	1	2	yes	—
454	C₂×C₄	#125×C₆₇	1	2	yes	—
455	C₂×C₄	#126×C₆₇	1	2	yes	—
456	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
457	C₈	—	4	4	no	—
458	C₂²	#380×C₆₇	1	1	yes	characteristic
459	C₂²	#386×C₆₇	1	1	yes	characteristic
460	C₂²	#391×C₆₇	1	1	yes	characteristic
461	C₂²	#394×C₆₇	1	1	yes	characteristic
462	C₂²	#395×C₆₇	1	1	yes	characteristic
463	C₂²	#398×C₆₇	1	1	yes	characteristic
464	C₂²	#400×C₆₇	1	1	yes	characteristic
465	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
466	C₂²	C₂×#103×C₆₇	1	1	yes	characteristic
467	C₂²	#417×C₆₇	1	1	yes	characteristic
468	C₂²	#417×C₆₇	1	1	yes	characteristic
469	C₂²	#447×C₆₇	1	1	yes	characteristic
470	C₂²	#448×C₆₇	1	1	yes	characteristic
471	C₂²	#450×C₆₇	1	1	yes	characteristic
472	C₂²	#451×C₆₇	1	1	yes	characteristic
473	C₂²	#459×C₆₇	1	1	yes	characteristic
474	C₂²	#460×C₆₇	1	1	yes	characteristic
475	C₂²	#461×C₆₇	1	1	yes	characteristic
476	C₂²	#462×C₆₇	1	1	yes	characteristic
477	C₂²	#474×C₆₇	1	1	yes	characteristic
478	C₂²	#475×C₆₇	1	1	yes	characteristic
479	C₂²	#476×C₆₇	1	1	yes	characteristic
480	C₂²	#477×C₆₇	1	1	yes	characteristic
481	C₂²	#480×C₆₇	1	1	yes	characteristic
482	C₂²	#481×C₆₇	1	1	yes	characteristic
483	C₂²	#483×C₆₇	1	1	yes	characteristic
484	C₂²	#484×C₆₇	1	1	yes	characteristic
485	C₂²	#486×C₆₇	1	1	yes	characteristic
486	C₂²	#487×C₆₇	1	1	yes	characteristic
487	C₂²	#490×C₆₇	1	1	yes	characteristic
488	C₂²	#491×C₆₇	1	1	yes	characteristic
489	C₂²	#498×C₆₇	1	1	yes	characteristic
490	C₂²	#500×C₆₇	1	1	yes	characteristic
491	C₂²	#501×C₆₇	1	1	yes	characteristic
492	C₂²	#501×C₆₇	1	1	yes	characteristic
493	C₄	—	2	4	no	—
494	C₄	—	2	4	no	—
495	C₄	—	4	4	no	—
496	C₄	—	8	4	no	—
497	C₄	—	8	2	no	—
498	C₄	#387×C₆₇	1	2	yes	—
499	C₄	#389×C₆₇	1	2	yes	—
500	C₄	#396×C₆₇	1	2	yes	—
501	C₄	#398×C₆₇	1	2	yes	—
502	C₂	C₆₇×#1441	1	1	yes	characteristic
503	C₂	C₆₇×#1802	1	1	yes	characteristic
504	C₂	C₆₇×#1803	1	1	yes	characteristic
505	C₂	C₆₇×#1872	1	1	yes	characteristic
506	C₂	C₆₇×#1874	1	1	yes	characteristic
507	C₂	C₆₇×#1990	1	1	yes	characteristic
508	C₂	C₆₇×#1992	1	1	yes	characteristic
509	C₂	C₆₇×#2041	1	1	yes	characteristic
510	C₂	C₆₇×#2043	1	1	yes	characteristic
511	C₂	C₆₇×#2057	1	1	yes	characteristic
512	C₂	C₆₇×#2058	1	1	yes	characteristic
513	C₂	C₆₇×#2083	1	1	yes	characteristic
514	C₂	C₆₇×#2084	1	1	yes	characteristic
515	C₂	C₆₇×#2099	1	1	yes	characteristic
516	C₂	C₆₇×#2100	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. #389×C₆₇	C₄	semidirect
129. C₄⋊C₄×C₆₇	C₄⋊C₄	semidirect
243. #11207	C₆₇	direct (N × H)
264. C₆₇	#11207	direct (N × H)
274. #389	C₄×C₆₇	semidirect
388. C₄⋊C₄	C₄⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#389×C₆₇	2
C₂×#384×C₆₇	2
C₂×#340×C₆₇	2
C₂×#336×C₆₇	2
#11207	67

Subgroups — C67×#11207 #30690458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11207 #30690458