Subgroups — C₆₇×#33579 #30712830

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2211	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2210	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2207	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2206	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₆₇	—	2	1	no	—
13	C₂×#135×C₆₇	C₂²	1	2	yes	—
14	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
15	#532×C₆₇	—	2	1	no	—
16	C₄×#41×C₆₇	—	2	1	no	—
17	#508×C₆₇	—	2	8	no	—
18	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
19	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
20	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
21	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
23	#495×C₆₇	—	2	1	no	—
24	#495×C₆₇	C₂²	1	2	yes	—
25	#494×C₆₇	C₂²	1	1	yes	characteristic
26	#493×C₆₇	C₂²	1	1	yes	characteristic
27	#492×C₆₇	—	2	1	no	—
28	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×SD₁₆×C₆₇	—	2	1	no	—
31	C₄×SD₁₆×C₆₇	—	2	1	no	—
32	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
34	#427×C₆₇	C₂²	1	1	yes	characteristic
35	#416×C₆₇	—	2	1	no	—
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
40	#358×C₆₇	C₂²	1	2	yes	—
41	#358×C₆₇	C₂²	1	2	yes	—
42	#357×C₆₇	C₂²	1	2	yes	—
43	#357×C₆₇	C₂²	1	2	yes	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂×#41×C₆₇	—	2	2	no	—
46	#135×C₆₇	—	2	4	no	—
47	#135×C₆₇	—	2	2	no	—
48	#135×C₆₇	—	2	4	no	—
49	#135×C₆₇	—	4	4	no	—
50	#135×C₆₇	D₄	1	4	yes	complemented
51	C₂×Q₁₆×C₆₇	—	2	1	no	—
52	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
53	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×SD₁₆×C₆₇	—	2	1	no	—
55	C₂×SD₁₆×C₆₇	—	2	2	no	—
56	C₂×SD₁₆×C₆₇	—	2	2	no	—
57	C₂×D₈×C₆₇	—	2	1	no	—
58	C₂×D₈×C₆₇	C₂³	1	2	yes	—
59	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₆₇	—	2	1	no	—
63	#127×C₆₇	—	2	1	no	—
64	#124×C₆₇	—	2	2	no	—
65	#121×C₆₇	—	4	1	no	—
66	#121×C₆₇	—	4	1	no	—
67	C₄×Q₈×C₆₇	—	2	1	no	—
68	C₄×Q₈×C₆₇	—	2	1	no	—
69	C₄×D₄×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	—	2	1	no	—
71	C₄×D₄×C₆₇	—	4	1	no	—
72	C₄×D₄×C₆₇	—	4	1	no	—
73	C₄×D₄×C₆₇	—	4	1	no	—
74	C₄×D₄×C₆₇	—	4	1	no	—
75	#117×C₆₇	—	2	1	no	—
76	#117×C₆₇	—	4	1	no	—
77	C₂×C₄²×C₆₇	—	4	1	no	—
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	Q₃₂×C₆₇	—	2	4	no	—
80	SD₃₂×C₆₇	—	2	4	no	—
81	SD₃₂×C₆₇	—	2	4	no	—
82	D₁₆×C₆₇	—	2	4	no	—
83	C₂×C₁₆×C₆₇	—	2	4	no	—
84	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
85	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
86	#107×C₆₇	C₂³	1	1	yes	characteristic
87	#107×C₆₇	C₂³	1	1	yes	characteristic
88	#106×C₆₇	—	2	1	no	—
89	#103×C₆₇	—	2	1	no	—
90	#103×C₆₇	—	2	1	no	—
91	#102×C₆₇	—	2	1	no	—
92	#102×C₆₇	—	2	1	no	—
93	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
94	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
95	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	#41×C₆₇	—	2	4	no	—
98	#41×C₆₇	—	4	2	no	—
99	#41×C₆₇	—	4	4	no	—
100	#41×C₆₇	—	4	4	no	—
101	C₂×Q₈×C₆₇	—	2	1	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	—	4	2	no	—
107	C₂×D₄×C₆₇	—	4	1	no	—
108	C₂×D₄×C₆₇	—	4	1	no	—
109	C₂²×C₄×C₆₇	—	4	1	no	—
110	C₂²×C₄×C₆₇	—	4	1	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	—	4	2	no	—
113	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	Q₁₆×C₆₇	—	2	2	no	—
117	Q₁₆×C₆₇	—	2	1	no	—
118	Q₁₆×C₆₇	—	2	2	no	—
119	Q₁₆×C₆₇	—	4	2	no	—
120	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
121	SD₁₆×C₆₇	—	2	4	no	—
122	SD₁₆×C₆₇	—	4	4	no	—
123	SD₁₆×C₆₇	—	4	4	no	—
124	D₈×C₆₇	—	2	2	no	—
125	D₈×C₆₇	—	2	1	no	—
126	D₈×C₆₇	—	2	2	no	—
127	D₈×C₆₇	—	4	2	no	—
128	D₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₆₇	#41	1	2	yes	—
135	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	4	1	no	—
139	C₄⋊C₄×C₆₇	—	4	1	no	—
140	C₄⋊C₄×C₆₇	—	4	1	no	—
141	C₄⋊C₄×C₆₇	—	4	1	no	—
142	C₂²⋊C₄×C₆₇	—	4	1	no	—
143	C₂²⋊C₄×C₆₇	—	4	1	no	—
144	C₂²⋊C₄×C₆₇	—	4	1	no	—
145	C₂²⋊C₄×C₆₇	—	4	2	no	—
146	C₂²⋊C₄×C₆₇	—	4	2	no	—
147	C₄²×C₆₇	—	4	1	no	—
148	C₄²×C₆₇	—	4	1	no	—
149	C₄²×C₆₇	—	4	1	no	—
150	C₄²×C₆₇	D₈	1	1	yes	characteristic
151	C₄²×C₆₇	D₈	1	1	yes	characteristic
152	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₁₆×C₆₇	—	2	2	no	—
155	C₁₆×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
159	Q₈×C₆₇	—	2	2	no	—
160	Q₈×C₆₇	—	4	1	no	—
161	Q₈×C₆₇	—	4	2	no	—
162	Q₈×C₆₇	—	4	2	no	—
163	D₄×C₆₇	—	2	2	no	—
164	D₄×C₆₇	—	4	4	no	—
165	D₄×C₆₇	—	4	2	no	—
166	D₄×C₆₇	—	4	1	no	—
167	D₄×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	8	2	no	—
169	D₄×C₆₇	—	8	4	no	—
170	D₄×C₆₇	—	8	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	8	2	no	—
179	C₂×C₄×C₆₇	—	8	4	no	—
180	C₂×C₄×C₆₇	—	8	2	no	—
181	C₂×C₄×C₆₇	#121	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
187	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	C₈×C₆₇	—	2	1	no	—
190	C₈×C₆₇	—	2	1	no	—
191	C₈×C₆₇	#121	1	2	yes	—
192	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₂²×C₆₇	—	2	1	no	—
195	C₂²×C₆₇	—	2	1	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	8	1	no	—
198	C₂²×C₆₇	—	8	1	no	—
199	C₂²×C₆₇	—	8	2	no	—
200	C₂²×C₆₇	—	8	2	no	—
201	C₂²×C₆₇	—	8	2	no	—
202	C₂²×C₆₇	#459	1	2	yes	—
203	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
204	C₄×C₆₇	—	2	1	no	—
205	C₄×C₆₇	—	2	1	no	—
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	8	1	no	—
209	C₄×C₆₇	—	8	1	no	—
210	C₄×C₆₇	—	8	2	no	—
211	C₄×C₆₇	#459	1	2	yes	—
212	C₄×C₆₇	#508	1	2	yes	complemented
213	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
214	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	#33579	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂×C₆₇	—	2	2	no	—
217	C₂×C₆₇	—	8	2	no	—
218	C₂×C₆₇	—	16	2	no	—
219	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
221	C₂×C₆₇	#3414	1	1	yes	characteristic
222	C₂×#508	C₂×C₆₇	1	2	yes	complemented
223	#3147	C₂×C₆₇	1	1	yes	characteristic, complemented
224	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#2211	C₂×C₆₇	1	2	yes	complemented
226	#2210	C₂×C₆₇	1	2	yes	complemented
227	#2209	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#2208	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#2207	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#2206	C₂×C₆₇	1	1	yes	characteristic, complemented
231	C₂×C₄⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#2138	C₂×C₆₇	1	2	yes	complemented
233	C₆₇	#33579	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×#135	—	2	1	no	—
235	C₂×#135	C₂²×C₆₇	1	2	yes	—
236	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
237	#532	—	2	1	no	—
238	C₄×#41	—	2	1	no	—
239	#508	—	2	8	no	—
240	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
241	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
242	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
243	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
244	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
245	#495	—	2	1	no	—
246	#495	C₂²×C₆₇	1	2	yes	—
247	#494	C₂²×C₆₇	1	1	yes	characteristic
248	#493	C₂²×C₆₇	1	1	yes	characteristic
249	#492	—	2	1	no	—
250	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
251	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
252	C₄×SD₁₆	—	2	1	no	—
253	C₄×SD₁₆	—	2	1	no	—
254	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
256	#427	C₂²×C₆₇	1	1	yes	characteristic
257	#416	—	2	1	no	—
258	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
259	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
260	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
261	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
262	#358	C₂²×C₆₇	1	2	yes	—
263	#358	C₂²×C₆₇	1	2	yes	—
264	#357	C₂²×C₆₇	1	2	yes	—
265	#357	C₂²×C₆₇	1	2	yes	—
266	C₂×#41	—	2	1	no	—
267	C₂×#41	—	2	2	no	—
268	#135	—	2	4	no	—
269	#135	—	2	2	no	—
270	#135	—	2	4	no	—
271	#135	—	4	4	no	—
272	#135	D₄×C₆₇	1	4	yes	complemented
273	C₂×Q₁₆	—	2	1	no	—
274	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
275	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
276	C₂×SD₁₆	—	2	1	no	—
277	C₂×SD₁₆	—	2	2	no	—
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×D₈	—	2	1	no	—
280	C₂×D₈	C₂³×C₆₇	1	2	yes	—
281	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
282	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
284	C₄⋊Q₈	—	2	1	no	—
285	#127	—	2	1	no	—
286	#124	—	2	2	no	—
287	#121	—	4	1	no	—
288	#121	—	4	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	—	2	1	no	—
293	C₄×D₄	—	4	1	no	—
294	C₄×D₄	—	4	1	no	—
295	C₄×D₄	—	4	1	no	—
296	C₄×D₄	—	4	1	no	—
297	#117	—	2	1	no	—
298	#117	—	4	1	no	—
299	C₂×C₄²	—	4	1	no	—
300	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
301	Q₃₂	—	2	4	no	—
302	SD₃₂	—	2	4	no	—
303	SD₃₂	—	2	4	no	—
304	D₁₆	—	2	4	no	—
305	C₂×C₁₆	—	2	4	no	—
306	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
307	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
308	#107	C₂³×C₆₇	1	1	yes	characteristic
309	#107	C₂³×C₆₇	1	1	yes	characteristic
310	#106	—	2	1	no	—
311	#103	—	2	1	no	—
312	#103	—	2	1	no	—
313	#102	—	2	1	no	—
314	#102	—	2	1	no	—
315	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
316	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
317	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
318	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
319	#41	—	2	4	no	—
320	#41	—	4	2	no	—
321	#41	—	4	4	no	—
322	#41	—	4	4	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂×D₄	—	4	1	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
336	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	Q₁₆	—	2	2	no	—
339	Q₁₆	—	2	1	no	—
340	Q₁₆	—	2	2	no	—
341	Q₁₆	—	4	2	no	—
342	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
343	SD₁₆	—	2	4	no	—
344	SD₁₆	—	4	4	no	—
345	SD₁₆	—	4	4	no	—
346	D₈	—	2	2	no	—
347	D₈	—	2	1	no	—
348	D₈	—	2	2	no	—
349	D₈	—	4	2	no	—
350	D₈	C₂×D₄×C₆₇	1	2	yes	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
354	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
355	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
356	C₂×C₈	#41×C₆₇	1	2	yes	—
357	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₄⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₂²⋊C₄	—	4	1	no	—
366	C₂²⋊C₄	—	4	1	no	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	4	2	no	—
369	C₄²	—	4	1	no	—
370	C₄²	—	4	1	no	—
371	C₄²	—	4	1	no	—
372	C₄²	D₈×C₆₇	1	1	yes	characteristic
373	C₄²	D₈×C₆₇	1	1	yes	characteristic
374	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
375	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
376	C₁₆	—	2	2	no	—
377	C₁₆	—	2	2	no	—
378	C₂³	—	4	1	no	—
379	C₂³	—	4	2	no	—
380	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
381	Q₈	—	2	2	no	—
382	Q₈	—	4	1	no	—
383	Q₈	—	4	2	no	—
384	Q₈	—	4	2	no	—
385	D₄	—	2	2	no	—
386	D₄	—	4	4	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	1	no	—
389	D₄	—	4	2	no	—
390	D₄	—	8	2	no	—
391	D₄	—	8	4	no	—
392	D₄	—	8	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	4	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	8	2	no	—
401	C₂×C₄	—	8	4	no	—
402	C₂×C₄	—	8	2	no	—
403	C₂×C₄	#121×C₆₇	1	2	yes	—
404	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
405	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
406	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
409	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
410	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
411	C₈	—	2	1	no	—
412	C₈	—	2	1	no	—
413	C₈	#121×C₆₇	1	2	yes	—
414	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
415	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
416	C₂²	—	2	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	8	1	no	—
420	C₂²	—	8	1	no	—
421	C₂²	—	8	2	no	—
422	C₂²	—	8	2	no	—
423	C₂²	—	8	2	no	—
424	C₂²	#459×C₆₇	1	2	yes	—
425	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	4	2	no	—
430	C₄	—	8	2	no	—
431	C₄	—	8	1	no	—
432	C₄	—	8	1	no	—
433	C₄	#459×C₆₇	1	2	yes	—
434	C₄	#508×C₆₇	1	2	yes	complemented
435	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
436	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
437	C₂	—	2	2	no	—
438	C₂	—	8	2	no	—
439	C₂	—	16	2	no	—
440	C₂	C₂×#459×C₆₇	1	1	yes	characteristic
441	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
442	C₂	C₆₇×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₆₇	C₂	semidirect
2. C₆₇×#3147	C₂	semidirect
3. C₄×#135×C₆₇	C₂	semidirect
4. C₆₇×#2211	C₂	semidirect
5. C₆₇×#2210	C₂	semidirect
6. C₆₇×#2209	C₂	semidirect
7. C₆₇×#2208	C₂	semidirect
8. C₆₇×#2207	C₂	semidirect
9. C₆₇×#2206	C₂	semidirect
10. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
11. C₆₇×#2138	C₂	semidirect
50. #135×C₆₇	D₄	semidirect
212. C₄×C₆₇	#508	semidirect
215. #33579	C₆₇	direct (N × H)
222. C₂×#508	C₂×C₆₇	semidirect
223. #3147	C₂×C₆₇	semidirect
224. C₄×#135	C₂×C₆₇	semidirect
225. #2211	C₂×C₆₇	semidirect
226. #2210	C₂×C₆₇	semidirect
227. #2209	C₂×C₆₇	semidirect
228. #2208	C₂×C₆₇	semidirect
229. #2207	C₂×C₆₇	semidirect
230. #2206	C₂×C₆₇	semidirect
231. C₂×C₄⋊C₁₆	C₂×C₆₇	semidirect
232. #2138	C₂×C₆₇	semidirect
233. C₆₇	#33579	direct (N × H)
272. #135	D₄×C₆₇	semidirect
434. C₄	#508×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₆₇×#3147	2
C₄×#135×C₆₇	2
C₆₇×#2211	2
C₆₇×#2210	2
C₆₇×#2209	2
C₆₇×#2208	2
C₆₇×#2207	2
C₆₇×#2206	2
C₂×C₄⋊C₁₆×C₆₇	2
C₆₇×#2138	2
#33579	67

Subgroups — C67×#33579 #30712830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33579 #30712830