Subgroups — C₆₇×#25874 #30705125

Order 17152.

1306 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3496	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3142	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2566	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×#1792	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1792	C₂	1	2	yes	maximal, complemented
9	C₄×#100×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
11	#547×C₆₇	—	2	2	no	—
12	#546×C₆₇	—	2	4	no	—
13	D₄²×C₆₇	—	2	2	no	—
14	#537×C₆₇	C₂²	1	2	yes	—
15	#534×C₆₇	—	2	2	no	—
16	#532×C₆₇	C₂²	1	2	yes	—
17	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×#121×C₆₇	—	2	2	no	—
19	C₂×C₄×D₄×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	complemented
24	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
25	#417×C₆₇	C₂²	1	4	yes	complemented
26	#417×C₆₇	C₂²	1	4	yes	complemented
27	C₂×#100×C₆₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
29	#396×C₆₇	—	2	2	no	—
30	#390×C₆₇	C₂²	1	1	yes	characteristic
31	#390×C₆₇	C₂²	1	1	yes	characteristic
32	#385×C₆₇	—	2	2	no	—
33	#384×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#382×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#343×C₆₇	C₂²	1	2	yes	complemented
37	H(5,𝔽₂)×C₆₇	—	2	4	no	—
38	H(5,𝔽₂)×C₆₇	—	2	4	no	—
39	H(5,𝔽₂)×C₆₇	—	4	4	no	—
40	C₂×#41×C₆₇	—	2	4	no	—
41	C₂×#41×C₆₇	C₂³	1	4	yes	—
42	C₂²×D₄×C₆₇	—	2	4	no	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	—	2	4	no	—
45	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	D₄	1	2	yes	—
47	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
48	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
49	#127×C₆₇	D₄	1	2	yes	—
50	#125×C₆₇	—	2	2	no	—
51	#125×C₆₇	—	2	2	no	—
52	#124×C₆₇	D₄	1	4	yes	—
53	#123×C₆₇	—	2	8	no	—
54	C₂²⋊Q₈×C₆₇	—	2	4	no	—
55	#121×C₆₇	—	2	4	no	—
56	#121×C₆₇	—	2	4	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	#120×C₆₇	—	2	8	no	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	—	2	2	no	—
65	C₄×D₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×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	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	#104×C₆₇	—	4	4	no	—
82	#104×C₆₇	—	4	4	no	—
83	#103×C₆₇	—	2	4	no	—
84	#103×C₆₇	—	2	4	no	—
85	#102×C₆₇	—	2	4	no	—
86	#102×C₆₇	—	2	4	no	—
87	#100×C₆₇	—	2	4	no	—
88	#97×C₆₇	—	2	2	no	—
89	C₄×C₈×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	1	no	—
91	#95×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	—	2	4	no	—
93	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
94	#41×C₆₇	—	2	4	no	—
95	#41×C₆₇	—	2	4	no	—
96	#41×C₆₇	—	2	4	no	—
97	#41×C₆₇	—	2	4	no	—
98	#41×C₆₇	—	4	8	no	—
99	#41×C₆₇	—	4	8	no	—
100	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	4	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	—	4	8	no	—
109	C₂×D₄×C₆₇	—	4	8	no	—
110	C₂×D₄×C₆₇	—	4	8	no	—
111	C₂×D₄×C₆₇	—	4	8	no	—
112	C₂×D₄×C₆₇	—	8	4	no	—
113	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
114	C₂×D₄×C₆₇	#41	1	4	yes	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	8	2	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	M₁₆×C₆₇	—	2	4	no	—
129	M₁₆×C₆₇	—	2	4	no	—
130	C₂×C₈×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	4	2	no	—
138	C₄⋊C₄×C₆₇	—	4	2	no	—
139	C₄⋊C₄×C₆₇	—	4	1	no	—
140	C₄⋊C₄×C₆₇	—	4	1	no	—
141	C₄⋊C₄×C₆₇	—	8	2	no	—
142	C₂²⋊C₄×C₆₇	—	2	8	no	—
143	C₂²⋊C₄×C₆₇	—	2	4	no	—
144	C₂²⋊C₄×C₆₇	—	2	4	no	—
145	C₂²⋊C₄×C₆₇	—	2	4	no	—
146	C₂²⋊C₄×C₆₇	—	4	4	no	—
147	C₂²⋊C₄×C₆₇	—	8	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	C₂×D₄	1	2	yes	—
151	C₄²×C₆₇	C₂×D₄	1	2	yes	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	4	2	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	8	no	—
158	C₂³×C₆₇	—	8	4	no	—
159	C₂³×C₆₇	—	8	4	no	—
160	C₂³×C₆₇	#124	1	4	yes	—
161	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₆₇	—	2	4	no	—
163	Q₈×C₆₇	—	2	4	no	—
164	D₄×C₆₇	—	2	8	no	—
165	D₄×C₆₇	—	2	4	no	—
166	D₄×C₆₇	—	2	4	no	—
167	D₄×C₆₇	—	4	4	no	—
168	D₄×C₆₇	—	4	16	no	—
169	D₄×C₆₇	—	4	4	no	—
170	D₄×C₆₇	—	4	4	no	—
171	D₄×C₆₇	—	8	2	no	—
172	D₄×C₆₇	—	8	4	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	4	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	4	8	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	8	2	no	—
193	C₂×C₄×C₆₇	#120	1	2	yes	—
194	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
195	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
198	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
199	C₈×C₆₇	—	4	4	no	—
200	C₂²×C₆₇	—	2	1	no	—
201	C₂²×C₆₇	—	2	1	no	—
202	C₂²×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	4	no	—
207	C₂²×C₆₇	—	4	4	no	—
208	C₂²×C₆₇	—	4	4	no	—
209	C₂²×C₆₇	—	8	2	no	—
210	C₂²×C₆₇	—	16	8	no	—
211	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
212	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
213	C₂²×C₆₇	#535	1	1	yes	characteristic
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	2	1	no	—
216	C₄×C₆₇	—	2	1	no	—
217	C₄×C₆₇	—	4	2	no	—
218	C₄×C₆₇	—	4	4	no	—
219	C₄×C₆₇	—	4	2	no	—
220	C₄×C₆₇	—	8	1	no	—
221	C₄×C₆₇	—	8	1	no	—
222	C₄×C₆₇	—	8	1	no	—
223	C₄×C₆₇	#546	1	4	yes	—
224	#25874	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₆₇	—	2	1	no	—
226	C₂×C₆₇	—	2	1	no	—
227	C₂×C₆₇	—	4	4	no	—
228	C₂×C₆₇	—	8	4	no	—
229	C₂×C₆₇	#2435	1	1	yes	characteristic
230	C₂×C₆₇	#3018	1	1	yes	characteristic
231	C₂×C₆₇	#3019	1	1	yes	characteristic
232	#3496	C₂×C₆₇	1	1	yes	characteristic
233	#3142	C₂×C₆₇	1	2	yes	complemented
234	#2566	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#2040	C₂×C₆₇	1	2	yes	complemented
236	#2039	C₂×C₆₇	1	2	yes	complemented
237	#1980	C₂×C₆₇	1	2	yes	complemented
238	#1792	C₂×C₆₇	1	2	yes	complemented
239	#1792	C₂×C₆₇	1	2	yes	complemented
240	C₄×#100	C₂×C₆₇	1	1	yes	characteristic, complemented
241	C₆₇	#25874	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
243	#547	—	2	2	no	—
244	#546	—	2	4	no	—
245	D₄²	—	2	2	no	—
246	#537	C₂²×C₆₇	1	2	yes	—
247	#534	—	2	2	no	—
248	#532	C₂²×C₆₇	1	2	yes	—
249	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
250	C₂×#121	—	2	2	no	—
251	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
252	#491	—	2	2	no	—
253	#490	—	2	2	no	—
254	#488	—	2	4	no	—
255	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
256	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
257	#417	C₂²×C₆₇	1	4	yes	complemented
258	#417	C₂²×C₆₇	1	4	yes	complemented
259	C₂×#100	C₂²×C₆₇	1	2	yes	—
260	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
261	#396	—	2	2	no	—
262	#390	C₂²×C₆₇	1	1	yes	characteristic
263	#390	C₂²×C₆₇	1	1	yes	characteristic
264	#385	—	2	2	no	—
265	#384	C₂²×C₆₇	1	1	yes	characteristic, complemented
266	#382	C₂²×C₆₇	1	1	yes	characteristic, complemented
267	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
268	#343	C₂²×C₆₇	1	2	yes	complemented
269	H(5,𝔽₂)	—	2	4	no	—
270	H(5,𝔽₂)	—	2	4	no	—
271	H(5,𝔽₂)	—	4	4	no	—
272	C₂×#41	—	2	4	no	—
273	C₂×#41	C₂³×C₆₇	1	4	yes	—
274	C₂²×D₄	—	2	4	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	—	2	4	no	—
277	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
278	C₂³×C₄	D₄×C₆₇	1	2	yes	—
279	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
280	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
281	#127	D₄×C₆₇	1	2	yes	—
282	#125	—	2	2	no	—
283	#125	—	2	2	no	—
284	#124	D₄×C₆₇	1	4	yes	—
285	#123	—	2	8	no	—
286	C₂²⋊Q₈	—	2	4	no	—
287	#121	—	2	4	no	—
288	#121	—	2	4	no	—
289	#121	—	2	4	no	—
290	#121	—	2	2	no	—
291	#121	—	2	4	no	—
292	#121	—	2	2	no	—
293	#120	—	2	8	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	—	2	2	no	—
297	C₄×D₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	—	2	4	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
310	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
313	#104	—	4	4	no	—
314	#104	—	4	4	no	—
315	#103	—	2	4	no	—
316	#103	—	2	4	no	—
317	#102	—	2	4	no	—
318	#102	—	2	4	no	—
319	#100	—	2	4	no	—
320	#97	—	2	2	no	—
321	C₄×C₈	—	2	2	no	—
322	#95	—	2	1	no	—
323	#95	—	2	2	no	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
326	#41	—	2	4	no	—
327	#41	—	2	4	no	—
328	#41	—	2	4	no	—
329	#41	—	2	4	no	—
330	#41	—	4	8	no	—
331	#41	—	4	8	no	—
332	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	4	8	no	—
341	C₂×D₄	—	4	8	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	—	4	8	no	—
344	C₂×D₄	—	8	4	no	—
345	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₂×D₄	#41×C₆₇	1	4	yes	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	8	2	no	—
356	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
360	M₁₆	—	2	4	no	—
361	M₁₆	—	2	4	no	—
362	C₂×C₈	—	2	4	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₄⋊C₄	—	4	1	no	—
373	C₄⋊C₄	—	8	2	no	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	4	no	—
378	C₂²⋊C₄	—	4	4	no	—
379	C₂²⋊C₄	—	8	2	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	C₂×D₄×C₆₇	1	2	yes	—
383	C₄²	C₂×D₄×C₆₇	1	2	yes	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	8	no	—
390	C₂³	—	8	4	no	—
391	C₂³	—	8	4	no	—
392	C₂³	#124×C₆₇	1	4	yes	—
393	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
394	Q₈	—	2	4	no	—
395	Q₈	—	2	4	no	—
396	D₄	—	2	8	no	—
397	D₄	—	2	4	no	—
398	D₄	—	2	4	no	—
399	D₄	—	4	4	no	—
400	D₄	—	4	16	no	—
401	D₄	—	4	4	no	—
402	D₄	—	4	4	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	4	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	8	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	4	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	8	2	no	—
425	C₂×C₄	#120×C₆₇	1	2	yes	—
426	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
427	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
429	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
430	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
431	C₈	—	4	4	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	2	4	no	—
435	C₂²	—	4	4	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	16	8	no	—
443	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
444	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
445	C₂²	#535×C₆₇	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	4	no	—
452	C₄	—	8	1	no	—
453	C₄	—	8	1	no	—
454	C₄	—	8	1	no	—
455	C₄	#546×C₆₇	1	4	yes	—
456	C₂	—	2	1	no	—
457	C₂	—	2	1	no	—
458	C₂	—	4	4	no	—
459	C₂	—	8	4	no	—
460	C₂	C₆₇×#2435	1	1	yes	characteristic
461	C₂	C₆₇×#3018	1	1	yes	characteristic
462	C₂	C₆₇×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3496	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×#2566	C₂	semidirect
4. C₆₇×#2040	C₂	semidirect
5. C₆₇×#2039	C₂	semidirect
6. C₆₇×#1980	C₂	semidirect
7. C₆₇×#1792	C₂	semidirect
8. C₆₇×#1792	C₂	semidirect
9. C₄×#100×C₆₇	C₂	semidirect
17. C₂×#125×C₆₇	C₂²	semidirect
23. C₂×#104×C₆₇	C₂²	semidirect
24. C₂×#104×C₆₇	C₂²	semidirect
25. #417×C₆₇	C₂²	semidirect
26. #417×C₆₇	C₂²	semidirect
28. C₄×M₁₆×C₆₇	C₂²	semidirect
33. #384×C₆₇	C₂²	semidirect
34. #382×C₆₇	C₂²	semidirect
35. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
36. #343×C₆₇	C₂²	semidirect
224. #25874	C₆₇	direct (N × H)
233. #3142	C₂×C₆₇	semidirect
234. #2566	C₂×C₆₇	semidirect
235. #2040	C₂×C₆₇	semidirect
236. #2039	C₂×C₆₇	semidirect
237. #1980	C₂×C₆₇	semidirect
238. #1792	C₂×C₆₇	semidirect
239. #1792	C₂×C₆₇	semidirect
240. C₄×#100	C₂×C₆₇	semidirect
241. C₆₇	#25874	direct (N × H)
249. C₂×#125	C₂²×C₆₇	semidirect
255. C₂×#104	C₂²×C₆₇	semidirect
256. C₂×#104	C₂²×C₆₇	semidirect
257. #417	C₂²×C₆₇	semidirect
258. #417	C₂²×C₆₇	semidirect
260. C₄×M₁₆	C₂²×C₆₇	semidirect
265. #384	C₂²×C₆₇	semidirect
266. #382	C₂²×C₆₇	semidirect
267. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
268. #343	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3496	2
C₆₇×#3142	2
C₆₇×#2566	2
C₆₇×#2040	2
C₆₇×#2039	2
C₆₇×#1980	2
C₆₇×#1792	2
C₄×#100×C₆₇	2
#25874	67

Subgroups — C67×#25874 #30705125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25874 #30705125