Subgroups — C₆₇×#25873 #30705124

Order 17152.

1206 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3497	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3142	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2565	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2040	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2039	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1980	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1793	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1793	C₂	1	2	yes	maximal, complemented
9	C₄×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
11	#550×C₆₇	—	2	2	no	—
12	D₄×Q₈×C₆₇	—	2	2	no	—
13	#548×C₆₇	—	2	4	no	—
14	#537×C₆₇	—	2	2	no	—
15	#537×C₆₇	C₂²	1	2	yes	—
16	#532×C₆₇	—	2	2	no	—
17	#532×C₆₇	C₂²	1	2	yes	—
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#491×C₆₇	—	2	2	no	—
21	#490×C₆₇	—	2	2	no	—
22	#488×C₆₇	—	2	4	no	—
23	C₂×#104×C₆₇	C₂²	1	4	yes	—
24	C₂×#104×C₆₇	C₂²	1	4	yes	—
25	#417×C₆₇	C₂²	1	4	yes	—
26	#417×C₆₇	C₂²	1	4	yes	—
27	C₂×#101×C₆₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
29	#398×C₆₇	—	2	2	no	—
30	#391×C₆₇	C₂²	1	1	yes	characteristic
31	#391×C₆₇	C₂²	1	1	yes	characteristic
32	#387×C₆₇	—	2	2	no	—
33	#384×C₆₇	C₂²	1	1	yes	characteristic
34	#382×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#343×C₆₇	C₂²	1	2	yes	—
37	2₋¹⁺⁴×C₆₇	—	2	4	no	—
38	2₋¹⁺⁴×C₆₇	—	2	4	no	—
39	2₋¹⁺⁴×C₆₇	—	4	4	no	—
40	C₂×#41×C₆₇	—	2	4	no	—
41	C₂×#41×C₆₇	—	2	4	no	—
42	C₂×#41×C₆₇	C₂³	1	4	yes	—
43	C₂²×Q₈×C₆₇	—	2	4	no	—
44	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
48	#127×C₆₇	—	2	2	no	—
49	#127×C₆₇	D₄	1	2	yes	—
50	#125×C₆₇	—	2	2	no	—
51	#125×C₆₇	—	2	2	no	—
52	#124×C₆₇	—	2	4	no	—
53	#124×C₆₇	D₄	1	4	yes	—
54	C₂²⋊Q₈×C₆₇	—	2	8	no	—
55	C₂²⋊Q₈×C₆₇	—	2	4	no	—
56	#121×C₆₇	—	2	8	no	—
57	#121×C₆₇	—	2	4	no	—
58	#121×C₆₇	—	2	2	no	—
59	#121×C₆₇	—	2	4	no	—
60	#121×C₆₇	—	2	2	no	—
61	C₄×Q₈×C₆₇	—	2	2	no	—
62	C₄×Q₈×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	2	4	no	—
65	C₄×D₄×C₆₇	—	2	2	no	—
66	C₄×D₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#104×C₆₇	—	4	4	no	—
80	#104×C₆₇	—	4	4	no	—
81	#103×C₆₇	—	2	4	no	—
82	#103×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	#102×C₆₇	—	2	4	no	—
85	#101×C₆₇	—	2	4	no	—
86	#97×C₆₇	—	2	2	no	—
87	C₄×C₈×C₆₇	—	2	2	no	—
88	#95×C₆₇	—	2	1	no	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	#41×C₆₇	—	2	4	no	—
93	#41×C₆₇	—	2	4	no	—
94	#41×C₆₇	—	2	4	no	—
95	#41×C₆₇	—	2	4	no	—
96	#41×C₆₇	—	4	8	no	—
97	#41×C₆₇	—	4	8	no	—
98	#41×C₆₇	—	4	8	no	—
99	#41×C₆₇	—	4	8	no	—
100	C₂×Q₈×C₆₇	—	2	4	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	4	8	no	—
104	C₂×Q₈×C₆₇	—	4	8	no	—
105	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
106	C₂×Q₈×C₆₇	#41	1	4	yes	—
107	C₂×D₄×C₆₇	—	2	4	no	—
108	C₂×D₄×C₆₇	—	2	4	no	—
109	C₂×D₄×C₆₇	—	2	4	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
124	M₁₆×C₆₇	—	2	4	no	—
125	M₁₆×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₄⋊C₄×C₆₇	—	4	1	no	—
137	C₄⋊C₄×C₆₇	—	4	1	no	—
138	C₄⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	2	8	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	2	4	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	C₂×D₄	1	2	yes	—
147	C₄²×C₆₇	C₂×D₄	1	2	yes	—
148	C₂³×C₆₇	—	2	4	no	—
149	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₆₇	—	2	8	no	—
151	Q₈×C₆₇	—	2	4	no	—
152	Q₈×C₆₇	—	2	4	no	—
153	Q₈×C₆₇	—	4	8	no	—
154	D₄×C₆₇	—	2	4	no	—
155	D₄×C₆₇	—	2	4	no	—
156	D₄×C₆₇	—	4	4	no	—
157	D₄×C₆₇	—	4	8	no	—
158	D₄×C₆₇	—	4	4	no	—
159	D₄×C₆₇	—	4	4	no	—
160	D₄×C₆₇	—	8	2	no	—
161	D₄×C₆₇	—	8	4	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	16	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	4	no	—
173	C₂×C₄×C₆₇	—	4	4	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
183	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
184	C₂×C₄×C₆₇	#120	1	2	yes	—
185	C₂×C₄×C₆₇	#124	1	4	yes	—
186	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
188	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
189	C₈×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
196	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
197	C₂²×C₆₇	#535	1	1	yes	characteristic
198	C₄×C₆₇	—	2	4	no	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	2	1	no	—
201	C₄×C₆₇	—	2	1	no	—
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	—	4	4	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	8	1	no	—
206	C₄×C₆₇	—	8	1	no	—
207	C₄×C₆₇	—	8	1	no	—
208	C₄×C₆₇	#546	1	4	yes	—
209	#25873	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₆₇	—	2	1	no	—
211	C₂×C₆₇	—	2	1	no	—
212	C₂×C₆₇	—	8	4	no	—
213	C₂×C₆₇	#2435	1	1	yes	characteristic
214	C₂×C₆₇	#3018	1	1	yes	characteristic
215	C₂×C₆₇	#3020	1	1	yes	characteristic
216	#3497	C₂×C₆₇	1	1	yes	characteristic
217	#3142	C₂×C₆₇	1	2	yes	complemented
218	#2565	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#2040	C₂×C₆₇	1	2	yes	complemented
220	#2039	C₂×C₆₇	1	2	yes	complemented
221	#1980	C₂×C₆₇	1	2	yes	complemented
222	#1793	C₂×C₆₇	1	2	yes	complemented
223	#1793	C₂×C₆₇	1	2	yes	complemented
224	C₄×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₆₇	#25873	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
227	#550	—	2	2	no	—
228	D₄×Q₈	—	2	2	no	—
229	#548	—	2	4	no	—
230	#537	—	2	2	no	—
231	#537	C₂²×C₆₇	1	2	yes	—
232	#532	—	2	2	no	—
233	#532	C₂²×C₆₇	1	2	yes	—
234	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
236	#491	—	2	2	no	—
237	#490	—	2	2	no	—
238	#488	—	2	4	no	—
239	C₂×#104	C₂²×C₆₇	1	4	yes	—
240	C₂×#104	C₂²×C₆₇	1	4	yes	—
241	#417	C₂²×C₆₇	1	4	yes	—
242	#417	C₂²×C₆₇	1	4	yes	—
243	C₂×#101	C₂²×C₆₇	1	2	yes	—
244	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
245	#398	—	2	2	no	—
246	#391	C₂²×C₆₇	1	1	yes	characteristic
247	#391	C₂²×C₆₇	1	1	yes	characteristic
248	#387	—	2	2	no	—
249	#384	C₂²×C₆₇	1	1	yes	characteristic
250	#382	C₂²×C₆₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
252	#343	C₂²×C₆₇	1	2	yes	—
253	2₋¹⁺⁴	—	2	4	no	—
254	2₋¹⁺⁴	—	2	4	no	—
255	2₋¹⁺⁴	—	4	4	no	—
256	C₂×#41	—	2	4	no	—
257	C₂×#41	—	2	4	no	—
258	C₂×#41	C₂³×C₆₇	1	4	yes	—
259	C₂²×Q₈	—	2	4	no	—
260	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
261	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
262	C₄⋊Q₈	—	2	2	no	—
263	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
264	#127	—	2	2	no	—
265	#127	D₄×C₆₇	1	2	yes	—
266	#125	—	2	2	no	—
267	#125	—	2	2	no	—
268	#124	—	2	4	no	—
269	#124	D₄×C₆₇	1	4	yes	—
270	C₂²⋊Q₈	—	2	8	no	—
271	C₂²⋊Q₈	—	2	4	no	—
272	#121	—	2	8	no	—
273	#121	—	2	4	no	—
274	#121	—	2	2	no	—
275	#121	—	2	4	no	—
276	#121	—	2	2	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×Q₈	—	2	2	no	—
279	C₄×D₄	—	2	4	no	—
280	C₄×D₄	—	2	4	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	2	no	—
287	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×C₄²	D₄×C₆₇	1	2	yes	—
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	#104	—	4	4	no	—
296	#104	—	4	4	no	—
297	#103	—	2	4	no	—
298	#103	—	2	4	no	—
299	#102	—	2	4	no	—
300	#102	—	2	4	no	—
301	#101	—	2	4	no	—
302	#97	—	2	2	no	—
303	C₄×C₈	—	2	2	no	—
304	#95	—	2	1	no	—
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	#41	—	2	4	no	—
309	#41	—	2	4	no	—
310	#41	—	2	4	no	—
311	#41	—	2	4	no	—
312	#41	—	4	8	no	—
313	#41	—	4	8	no	—
314	#41	—	4	8	no	—
315	#41	—	4	8	no	—
316	C₂×Q₈	—	2	4	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	4	8	no	—
320	C₂×Q₈	—	4	8	no	—
321	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
322	C₂×Q₈	#41×C₆₇	1	4	yes	—
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	2	4	no	—
325	C₂×D₄	—	2	4	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
340	M₁₆	—	2	4	no	—
341	M₁₆	—	2	4	no	—
342	C₂×C₈	—	2	4	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	—	4	4	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	2	8	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	C₂×D₄×C₆₇	1	2	yes	—
363	C₄²	C₂×D₄×C₆₇	1	2	yes	—
364	C₂³	—	2	4	no	—
365	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
366	Q₈	—	2	8	no	—
367	Q₈	—	2	4	no	—
368	Q₈	—	2	4	no	—
369	Q₈	—	4	8	no	—
370	D₄	—	2	4	no	—
371	D₄	—	2	4	no	—
372	D₄	—	4	8	no	—
373	D₄	—	4	4	no	—
374	D₄	—	4	4	no	—
375	D₄	—	4	4	no	—
376	D₄	—	8	2	no	—
377	D₄	—	8	4	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	16	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
399	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
400	C₂×C₄	#120×C₆₇	1	2	yes	—
401	C₂×C₄	#124×C₆₇	1	4	yes	—
402	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
403	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
404	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
405	C₈	—	4	4	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	4	4	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	4	no	—
411	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
412	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
413	C₂²	#535×C₆₇	1	1	yes	characteristic
414	C₄	—	2	2	no	—
415	C₄	—	2	1	no	—
416	C₄	—	2	1	no	—
417	C₄	—	2	4	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	4	no	—
421	C₄	—	8	1	no	—
422	C₄	—	8	1	no	—
423	C₄	—	8	1	no	—
424	C₄	#546×C₆₇	1	4	yes	—
425	C₂	—	2	1	no	—
426	C₂	—	2	1	no	—
427	C₂	—	8	4	no	—
428	C₂	C₆₇×#2435	1	1	yes	characteristic
429	C₂	C₆₇×#3018	1	1	yes	characteristic
430	C₂	C₆₇×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3142	C₂	semidirect
3. C₆₇×#2565	C₂	semidirect
4. C₆₇×#2040	C₂	semidirect
5. C₆₇×#2039	C₂	semidirect
6. C₆₇×#1980	C₂	semidirect
7. C₆₇×#1793	C₂	semidirect
8. C₆₇×#1793	C₂	semidirect
9. C₄×#101×C₆₇	C₂	semidirect
209. #25873	C₆₇	direct (N × H)
217. #3142	C₂×C₆₇	semidirect
218. #2565	C₂×C₆₇	semidirect
219. #2040	C₂×C₆₇	semidirect
220. #2039	C₂×C₆₇	semidirect
221. #1980	C₂×C₆₇	semidirect
222. #1793	C₂×C₆₇	semidirect
223. #1793	C₂×C₆₇	semidirect
224. C₄×#101	C₂×C₆₇	semidirect
225. C₆₇	#25873	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3497	2
C₆₇×#3142	2
C₆₇×#2565	2
C₆₇×#2040	2
C₆₇×#2039	2
C₆₇×#1980	2
C₆₇×#1793	2
C₄×#101×C₆₇	2
#25873	67

Subgroups — C67×#25873 #30705124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25873 #30705124