Subgroups — C₆₇×#11001 #30690252

Order 17152.

1306 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3449	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#416×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1740	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₆₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	#537×C₆₇	C₄	1	4	yes	complemented
10	#536×C₆₇	—	2	2	no	—
11	#535×C₆₇	—	2	2	no	—
12	#534×C₆₇	—	2	2	no	—
13	#532×C₆₇	—	2	2	no	—
14	#525×C₆₇	—	2	4	no	—
15	C₂×#117×C₆₇	C₂²	1	2	yes	—
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	#421×C₆₇	—	2	4	no	—
18	#421×C₆₇	—	2	4	no	—
19	C₂×#104×C₆₇	—	2	4	no	—
20	#416×C₆₇	—	2	4	no	—
21	#416×C₆₇	—	2	4	no	—
22	C₂×#103×C₆₇	—	2	2	no	—
23	C₂×#102×C₆₇	—	2	2	no	—
24	#342×C₆₇	—	2	2	no	—
25	#336×C₆₇	—	2	2	no	—
26	C₂×#41×C₆₇	—	2	2	no	—
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	—	2	2	no	—
30	C₂×#41×C₆₇	—	2	2	no	—
31	C₂×#41×C₆₇	—	2	2	no	—
32	C₂×#41×C₆₇	—	2	2	no	—
33	C₂×#41×C₆₇	—	2	2	no	—
34	C₂×#41×C₆₇	—	2	2	no	—
35	C₂×#41×C₆₇	—	2	2	no	—
36	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×#41×C₆₇	D₄	1	4	yes	—
38	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	D₄	1	2	yes	—
41	C₂³×C₄×C₆₇	—	2	2	no	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₆₇	—	2	1	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	D₄	1	2	yes	complemented
47	C₂²×C₈×C₆₇	—	2	1	no	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	1	no	—
51	C₂²×C₈×C₆₇	—	2	1	no	—
52	C₂²×C₈×C₆₇	D₄	1	2	yes	complemented
53	C₄⋊Q₈×C₆₇	—	2	2	no	—
54	#127×C₆₇	—	2	2	no	—
55	#124×C₆₇	—	2	4	no	—
56	#123×C₆₇	—	2	8	no	—
57	C₂²⋊Q₈×C₆₇	—	2	8	no	—
58	#121×C₆₇	—	2	8	no	—
59	#121×C₆₇	—	2	8	no	—
60	#120×C₆₇	—	2	8	no	—
61	C₄×D₄×C₆₇	—	2	8	no	—
62	#117×C₆₇	—	2	2	no	—
63	#117×C₆₇	—	2	2	no	—
64	#117×C₆₇	—	2	2	no	—
65	#117×C₆₇	—	2	2	no	—
66	#117×C₆₇	—	2	2	no	—
67	#117×C₆₇	D₄	1	4	yes	complemented
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	D₄	1	2	yes	—
74	#104×C₆₇	—	2	16	no	—
75	#103×C₆₇	—	2	8	no	—
76	#102×C₆₇	—	2	8	no	—
77	C₂⁴×C₆₇	—	2	2	no	—
78	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
79	#41×C₆₇	—	2	4	no	—
80	#41×C₆₇	—	2	4	no	—
81	#41×C₆₇	—	2	4	no	—
82	#41×C₆₇	—	2	4	no	—
83	#41×C₆₇	—	2	4	no	—
84	#41×C₆₇	—	2	4	no	—
85	#41×C₆₇	—	2	4	no	—
86	#41×C₆₇	—	2	4	no	—
87	#41×C₆₇	—	4	2	no	—
88	#41×C₆₇	—	4	2	no	—
89	#41×C₆₇	—	4	2	no	—
90	#41×C₆₇	—	4	2	no	—
91	#41×C₆₇	—	4	2	no	—
92	#41×C₆₇	—	4	2	no	—
93	#41×C₆₇	—	4	2	no	—
94	#41×C₆₇	—	4	2	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	8	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	4	2	no	—
110	C₂×D₄×C₆₇	—	4	2	no	—
111	C₂×D₄×C₆₇	—	4	2	no	—
112	C₂×D₄×C₆₇	—	4	2	no	—
113	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	4	2	no	—
118	C₂²×C₄×C₆₇	—	4	2	no	—
119	C₂²×C₄×C₆₇	—	4	2	no	—
120	C₂²×C₄×C₆₇	—	4	2	no	—
121	C₂²×C₄×C₆₇	—	4	2	no	—
122	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	#41	1	2	yes	—
133	M₁₆×C₆₇	—	2	4	no	—
134	M₁₆×C₆₇	—	4	2	no	—
135	C₂×C₈×C₆₇	—	2	4	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	4	1	no	—
139	C₂×C₈×C₆₇	—	4	1	no	—
140	C₂×C₈×C₆₇	—	4	2	no	—
141	C₂×C₈×C₆₇	—	4	1	no	—
142	C₂×C₈×C₆₇	—	4	1	no	—
143	C₄⋊C₄×C₆₇	—	2	8	no	—
144	C₄⋊C₄×C₆₇	—	2	4	no	—
145	C₄⋊C₄×C₆₇	—	2	4	no	—
146	C₄⋊C₄×C₆₇	—	4	2	no	—
147	C₂²⋊C₄×C₆₇	—	2	16	no	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	4	2	no	—
151	C₄²×C₆₇	—	2	2	no	—
152	C₄²×C₆₇	—	2	2	no	—
153	C₄²×C₆₇	—	2	4	no	—
154	C₄²×C₆₇	—	4	2	no	—
155	C₂³×C₆₇	—	2	4	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	8	4	no	—
160	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂³×C₆₇	#120	1	2	yes	—
162	C₂³×C₆₇	#121	1	2	yes	—
163	C₂³×C₆₇	#124	1	1	yes	characteristic
164	C₂³×C₆₇	#127	1	1	yes	characteristic
165	Q₈×C₆₇	—	2	4	no	—
166	Q₈×C₆₇	—	2	4	no	—
167	Q₈×C₆₇	—	4	2	no	—
168	Q₈×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	2	4	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	—	4	4	no	—
172	D₄×C₆₇	—	4	4	no	—
173	D₄×C₆₇	—	4	4	no	—
174	D₄×C₆₇	—	4	4	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	8	8	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	4	no	—
185	C₂×C₄×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	—	4	4	no	—
187	C₂×C₄×C₆₇	—	4	4	no	—
188	C₂×C₄×C₆₇	—	8	4	no	—
189	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₆₇	#120	1	2	yes	—
194	C₂×C₄×C₆₇	#120	1	2	yes	—
195	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
196	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
197	C₂×C₄×C₆₇	#121	1	2	yes	—
198	C₂×C₄×C₆₇	#121	1	2	yes	—
199	C₂×C₄×C₆₇	#121	1	2	yes	—
200	C₂×C₄×C₆₇	#121	1	2	yes	—
201	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
202	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
203	C₂×C₄×C₆₇	#123	1	2	yes	—
204	C₂×C₄×C₆₇	#123	1	2	yes	—
205	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
206	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
207	C₈×C₆₇	—	4	2	no	—
208	C₈×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	2	4	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	1	no	—
213	C₂²×C₆₇	—	4	1	no	—
214	C₂²×C₆₇	—	4	8	no	—
215	C₂²×C₆₇	—	4	4	no	—
216	C₂²×C₆₇	—	8	4	no	—
217	C₂²×C₆₇	—	8	4	no	—
218	C₂²×C₆₇	#379	1	1	yes	characteristic
219	C₂²×C₆₇	#390	1	1	yes	characteristic
220	C₂²×C₆₇	#390	1	1	yes	characteristic
221	C₂²×C₆₇	#392	1	2	yes	—
222	C₂²×C₆₇	#394	1	2	yes	—
223	C₄×C₆₇	—	4	4	no	—
224	C₄×C₆₇	—	4	4	no	—
225	C₄×C₆₇	—	8	4	no	—
226	C₄×C₆₇	#386	1	2	yes	—
227	C₄×C₆₇	#386	1	2	yes	—
228	C₄×C₆₇	#392	1	1	yes	characteristic
229	C₄×C₆₇	#392	1	1	yes	characteristic
230	C₄×C₆₇	#394	1	1	yes	characteristic
231	C₄×C₆₇	#394	1	1	yes	characteristic
232	#11001	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂×C₆₇	—	2	2	no	—
234	C₂×C₆₇	—	2	2	no	—
235	C₂×C₆₇	—	2	1	no	—
236	C₂×C₆₇	—	2	1	no	—
237	C₂×C₆₇	—	8	4	no	—
238	C₂×C₆₇	#1438	1	1	yes	characteristic
239	C₂×C₆₇	#1794	1	2	yes	—
240	#3449	C₂×C₆₇	1	1	yes	characteristic
241	C₂×#421	C₂×C₆₇	1	2	yes	complemented
242	C₂×#416	C₂×C₆₇	1	2	yes	complemented
243	#1740	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#1729	C₂×C₆₇	1	1	yes	characteristic, complemented
245	C₆₇	#11001	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²×#41	C₂²×C₆₇	1	2	yes	—
247	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
248	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
249	#537	C₄×C₆₇	1	4	yes	complemented
250	#536	—	2	2	no	—
251	#535	—	2	2	no	—
252	#534	—	2	2	no	—
253	#532	—	2	2	no	—
254	#525	—	2	4	no	—
255	C₂×#117	C₂²×C₆₇	1	2	yes	—
256	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
257	#421	—	2	4	no	—
258	#421	—	2	4	no	—
259	C₂×#104	—	2	4	no	—
260	#416	—	2	4	no	—
261	#416	—	2	4	no	—
262	C₂×#103	—	2	2	no	—
263	C₂×#102	—	2	2	no	—
264	#342	—	2	2	no	—
265	#336	—	2	2	no	—
266	C₂×#41	—	2	2	no	—
267	C₂×#41	—	2	2	no	—
268	C₂×#41	—	2	2	no	—
269	C₂×#41	—	2	2	no	—
270	C₂×#41	—	2	2	no	—
271	C₂×#41	—	2	2	no	—
272	C₂×#41	—	2	2	no	—
273	C₂×#41	—	2	2	no	—
274	C₂×#41	—	2	2	no	—
275	C₂×#41	—	2	2	no	—
276	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
277	C₂×#41	D₄×C₆₇	1	4	yes	—
278	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
279	C₂²×D₄	—	2	2	no	—
280	C₂²×D₄	D₄×C₆₇	1	2	yes	—
281	C₂³×C₄	—	2	2	no	—
282	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
283	C₂×M₁₆	—	2	1	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	D₄×C₆₇	1	2	yes	complemented
287	C₂²×C₈	—	2	1	no	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	—	2	2	no	—
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	D₄×C₆₇	1	2	yes	complemented
293	C₄⋊Q₈	—	2	2	no	—
294	#127	—	2	2	no	—
295	#124	—	2	4	no	—
296	#123	—	2	8	no	—
297	C₂²⋊Q₈	—	2	8	no	—
298	#121	—	2	8	no	—
299	#121	—	2	8	no	—
300	#120	—	2	8	no	—
301	C₄×D₄	—	2	8	no	—
302	#117	—	2	2	no	—
303	#117	—	2	2	no	—
304	#117	—	2	2	no	—
305	#117	—	2	2	no	—
306	#117	—	2	2	no	—
307	#117	D₄×C₆₇	1	4	yes	complemented
308	C₂×C₄⋊C₄	—	2	2	no	—
309	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
312	C₂×C₄²	—	2	2	no	—
313	C₂×C₄²	D₄×C₆₇	1	2	yes	—
314	#104	—	2	16	no	—
315	#103	—	2	8	no	—
316	#102	—	2	8	no	—
317	C₂⁴	—	2	2	no	—
318	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
319	#41	—	2	4	no	—
320	#41	—	2	4	no	—
321	#41	—	2	4	no	—
322	#41	—	2	4	no	—
323	#41	—	2	4	no	—
324	#41	—	2	4	no	—
325	#41	—	2	4	no	—
326	#41	—	2	4	no	—
327	#41	—	4	2	no	—
328	#41	—	4	2	no	—
329	#41	—	4	2	no	—
330	#41	—	4	2	no	—
331	#41	—	4	2	no	—
332	#41	—	4	2	no	—
333	#41	—	4	2	no	—
334	#41	—	4	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	4	no	—
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	4	2	no	—
350	C₂×D₄	—	4	2	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	—	4	2	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	—	4	2	no	—
361	C₂²×C₄	—	4	2	no	—
362	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
363	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
365	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
366	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
370	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
371	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
372	C₂²×C₄	#41×C₆₇	1	2	yes	—
373	M₁₆	—	2	4	no	—
374	M₁₆	—	4	2	no	—
375	C₂×C₈	—	2	4	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	4	1	no	—
379	C₂×C₈	—	4	1	no	—
380	C₂×C₈	—	4	2	no	—
381	C₂×C₈	—	4	1	no	—
382	C₂×C₈	—	4	1	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₂²⋊C₄	—	2	16	no	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	2	4	no	—
390	C₂²⋊C₄	—	4	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	4	no	—
394	C₄²	—	4	2	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	8	4	no	—
400	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
401	C₂³	#120×C₆₇	1	2	yes	—
402	C₂³	#121×C₆₇	1	2	yes	—
403	C₂³	#124×C₆₇	1	1	yes	characteristic
404	C₂³	#127×C₆₇	1	1	yes	characteristic
405	Q₈	—	2	4	no	—
406	Q₈	—	2	4	no	—
407	Q₈	—	4	2	no	—
408	Q₈	—	4	2	no	—
409	D₄	—	2	4	no	—
410	D₄	—	2	4	no	—
411	D₄	—	4	4	no	—
412	D₄	—	4	4	no	—
413	D₄	—	4	4	no	—
414	D₄	—	4	4	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	8	8	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	8	4	no	—
429	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
430	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
431	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
433	C₂×C₄	#120×C₆₇	1	2	yes	—
434	C₂×C₄	#120×C₆₇	1	2	yes	—
435	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
436	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
437	C₂×C₄	#121×C₆₇	1	2	yes	—
438	C₂×C₄	#121×C₆₇	1	2	yes	—
439	C₂×C₄	#121×C₆₇	1	2	yes	—
440	C₂×C₄	#121×C₆₇	1	2	yes	—
441	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
442	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
443	C₂×C₄	#123×C₆₇	1	2	yes	—
444	C₂×C₄	#123×C₆₇	1	2	yes	—
445	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
446	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
447	C₈	—	4	2	no	—
448	C₈	—	4	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	1	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	4	8	no	—
455	C₂²	—	4	4	no	—
456	C₂²	—	8	4	no	—
457	C₂²	—	8	4	no	—
458	C₂²	#379×C₆₇	1	1	yes	characteristic
459	C₂²	#390×C₆₇	1	1	yes	characteristic
460	C₂²	#390×C₆₇	1	1	yes	characteristic
461	C₂²	#392×C₆₇	1	2	yes	—
462	C₂²	#394×C₆₇	1	2	yes	—
463	C₄	—	4	4	no	—
464	C₄	—	4	4	no	—
465	C₄	—	8	4	no	—
466	C₄	#386×C₆₇	1	2	yes	—
467	C₄	#386×C₆₇	1	2	yes	—
468	C₄	#392×C₆₇	1	1	yes	characteristic
469	C₄	#392×C₆₇	1	1	yes	characteristic
470	C₄	#394×C₆₇	1	1	yes	characteristic
471	C₄	#394×C₆₇	1	1	yes	characteristic
472	C₂	—	2	2	no	—
473	C₂	—	2	2	no	—
474	C₂	—	2	1	no	—
475	C₂	—	2	1	no	—
476	C₂	—	8	4	no	—
477	C₂	C₆₇×#1438	1	1	yes	characteristic
478	C₂	C₆₇×#1794	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	#379
prime radical	C₆₇×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#421×C₆₇	C₂	semidirect
3. C₂×#416×C₆₇	C₂	semidirect
4. C₆₇×#1740	C₂	semidirect
5. C₆₇×#1729	C₂	semidirect
9. #537×C₆₇	C₄	semidirect
46. C₂×M₁₆×C₆₇	D₄	semidirect
52. C₂²×C₈×C₆₇	D₄	semidirect
67. #117×C₆₇	D₄	semidirect
232. #11001	C₆₇	direct (N × H)
241. C₂×#421	C₂×C₆₇	semidirect
242. C₂×#416	C₂×C₆₇	semidirect
243. #1740	C₂×C₆₇	semidirect
244. #1729	C₂×C₆₇	semidirect
245. C₆₇	#11001	direct (N × H)
249. #537	C₄×C₆₇	semidirect
286. C₂×M₁₆	D₄×C₆₇	semidirect
292. C₂²×C₈	D₄×C₆₇	semidirect
307. #117	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3449	2
C₂×#421×C₆₇	2
C₂×#416×C₆₇	2
C₆₇×#1740	2
C₆₇×#1729	2
#11001	67

Subgroups — C67×#11001 #30690252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11001 #30690252