Subgroups — C₆₇×#32951 #30712202

Order 17152.

1298 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3058	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3021	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1914	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1903	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1814	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1813	C₂	1	1	yes	maximal, characteristic, complemented
9	#585×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	—
11	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	#568×C₆₇	—	2	4	no	—
13	#568×C₆₇	—	2	2	no	—
14	#568×C₆₇	C₂²	1	2	yes	—
15	#568×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#131×C₆₇	—	2	4	no	—
17	C₂×#131×C₆₇	—	2	2	no	—
18	C₂×#131×C₆₇	C₂²	1	2	yes	—
19	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
20	#535×C₆₇	C₂²	1	2	yes	—
21	#472×C₆₇	—	2	4	no	—
22	#472×C₆₇	—	2	4	no	—
23	#469×C₆₇	—	2	2	no	—
24	#466×C₆₇	—	2	2	no	—
25	#454×C₆₇	—	2	4	no	—
26	#453×C₆₇	—	2	4	no	—
27	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
28	#427×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	#421×C₆₇	C₂²	1	4	yes	—
30	#418×C₆₇	C₂²	1	2	yes	complemented
31	#413×C₆₇	C₂²	1	2	yes	—
32	C₂×#100×C₆₇	C₂²	1	4	yes	complemented
33	#344×C₆₇	C₂²	1	4	yes	complemented
34	#338×C₆₇	C₂²	1	2	yes	complemented
35	2₋¹⁺⁴×C₆₇	—	2	2	no	—
36	2₋¹⁺⁴×C₆₇	D₄	1	2	yes	—
37	H(5,𝔽₂)×C₆₇	—	2	2	no	—
38	H(5,𝔽₂)×C₆₇	—	2	2	no	—
39	H(5,𝔽₂)×C₆₇	D₄	1	2	yes	—
40	C₂×#41×C₆₇	—	2	4	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
45	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
46	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
47	C₂×SD₁₆×C₆₇	—	2	4	no	—
48	C₂×D₈×C₆₇	—	2	4	no	—
49	C₂×D₈×C₆₇	—	2	1	no	—
50	C₂×D₈×C₆₇	—	2	1	no	—
51	C₂×D₈×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	—	2	2	no	—
53	#131×C₆₇	—	2	4	no	—
54	#131×C₆₇	—	2	8	no	—
55	#131×C₆₇	—	2	4	no	—
56	#131×C₆₇	—	2	4	no	—
57	#131×C₆₇	—	2	4	no	—
58	#131×C₆₇	—	2	2	no	—
59	#131×C₆₇	—	2	2	no	—
60	#131×C₆₇	—	2	4	no	—
61	#131×C₆₇	—	2	2	no	—
62	#131×C₆₇	—	2	2	no	—
63	#131×C₆₇	D₄	1	4	yes	complemented
64	#131×C₆₇	D₄	1	4	yes	complemented
65	C₂×M₁₆×C₆₇	—	2	8	no	—
66	C₂×M₁₆×C₆₇	—	2	4	no	—
67	C₂×M₁₆×C₆₇	—	2	2	no	—
68	C₂×M₁₆×C₆₇	—	2	4	no	—
69	C₂×M₁₆×C₆₇	—	2	2	no	—
70	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
72	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	—	2	4	no	—
75	C₂²×C₈×C₆₇	—	2	2	no	—
76	C₂²×C₈×C₆₇	D₄	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	#127×C₆₇	—	2	2	no	—
79	#124×C₆₇	—	2	2	no	—
80	#121×C₆₇	—	2	4	no	—
81	#120×C₆₇	—	2	4	no	—
82	#117×C₆₇	C₂³	1	2	yes	—
83	#108×C₆₇	—	2	2	no	—
84	#108×C₆₇	—	2	2	no	—
85	#107×C₆₇	—	2	1	no	—
86	#106×C₆₇	—	2	1	no	—
87	#104×C₆₇	—	2	8	no	—
88	#102×C₆₇	—	2	4	no	—
89	#101×C₆₇	—	2	2	no	—
90	#100×C₆₇	—	2	2	no	—
91	#100×C₆₇	—	2	8	no	—
92	C₂⁴×C₆₇	—	2	2	no	—
93	#41×C₆₇	—	2	4	no	—
94	#41×C₆₇	—	2	8	no	—
95	#41×C₆₇	—	2	4	no	—
96	#41×C₆₇	—	2	4	no	—
97	#41×C₆₇	—	2	4	no	—
98	#41×C₆₇	—	2	4	no	—
99	#41×C₆₇	—	2	2	no	—
100	#41×C₆₇	—	2	2	no	—
101	#41×C₆₇	—	2	2	no	—
102	#41×C₆₇	—	2	2	no	—
103	#41×C₆₇	C₂×D₄	1	4	yes	—
104	#41×C₆₇	C₂×D₄	1	4	yes	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×Q₈×C₆₇	—	2	4	no	—
107	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×Q₈×C₆₇	#41	1	2	yes	—
109	C₂×D₄×C₆₇	—	2	8	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	4	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	2	4	no	—
114	C₂×D₄×C₆₇	—	2	4	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	4	no	—
118	C₂×D₄×C₆₇	—	4	4	no	—
119	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×D₄×C₆₇	#41	1	2	yes	—
121	C₂×D₄×C₆₇	#41	1	2	yes	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	SD₁₆×C₆₇	—	4	8	no	—
127	D₈×C₆₇	—	4	8	no	—
128	D₈×C₆₇	—	4	1	no	—
129	D₈×C₆₇	—	4	1	no	—
130	D₈×C₆₇	—	4	2	no	—
131	M₁₆×C₆₇	—	2	4	no	—
132	M₁₆×C₆₇	—	2	8	no	—
133	M₁₆×C₆₇	—	2	2	no	—
134	M₁₆×C₆₇	—	2	2	no	—
135	M₁₆×C₆₇	—	2	4	no	—
136	M₁₆×C₆₇	—	2	4	no	—
137	M₁₆×C₆₇	—	2	4	no	—
138	M₁₆×C₆₇	—	2	4	no	—
139	M₁₆×C₆₇	—	2	4	no	—
140	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
141	C₂×C₈×C₆₇	—	2	4	no	—
142	C₂×C₈×C₆₇	—	2	8	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	2	4	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	4	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	4	no	—
149	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
150	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₂²⋊C₄×C₆₇	—	2	2	no	—
156	C₂²⋊C₄×C₆₇	—	4	4	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	—	4	4	no	—
162	C₂³×C₆₇	—	16	2	no	—
163	C₂³×C₆₇	#121	1	2	yes	—
164	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
165	Q₈×C₆₇	—	2	4	no	—
166	Q₈×C₆₇	—	2	2	no	—
167	Q₈×C₆₇	—	2	2	no	—
168	Q₈×C₆₇	#121	1	4	yes	—
169	D₄×C₆₇	—	2	8	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	—	2	4	no	—
172	D₄×C₆₇	—	2	4	no	—
173	D₄×C₆₇	—	2	4	no	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	2	2	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	4	4	no	—
178	D₄×C₆₇	—	4	2	no	—
179	D₄×C₆₇	—	16	4	no	—
180	D₄×C₆₇	#121	1	4	yes	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	#121	1	2	yes	—
190	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
194	C₈×C₆₇	—	2	4	no	—
195	C₈×C₆₇	—	2	2	no	—
196	C₈×C₆₇	—	4	4	no	—
197	C₈×C₆₇	#120	1	1	yes	characteristic
198	C₈×C₆₇	#120	1	2	yes	—
199	C₈×C₆₇	#120	1	1	yes	characteristic
200	C₂²×C₆₇	—	2	2	no	—
201	C₂²×C₆₇	—	2	4	no	—
202	C₂²×C₆₇	—	4	1	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	8	4	no	—
206	C₂²×C₆₇	—	8	4	no	—
207	C₂²×C₆₇	—	8	2	no	—
208	C₂²×C₆₇	—	16	4	no	—
209	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
210	C₂²×C₆₇	#525	1	2	yes	—
211	C₄×C₆₇	—	2	4	no	—
212	C₄×C₆₇	—	2	2	no	—
213	C₄×C₆₇	—	8	2	no	—
214	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
215	C₄×C₆₇	C₂×#121	1	2	yes	—
216	C₄×C₆₇	#525	1	1	yes	characteristic
217	#32951	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	2	1	no	—
219	C₂×C₆₇	—	2	2	no	—
220	C₂×C₆₇	—	4	2	no	—
221	C₂×C₆₇	—	4	4	no	—
222	C₂×C₆₇	—	16	2	no	—
223	C₂×C₆₇	#2847	1	1	yes	characteristic
224	#3573	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₂×#472	C₂×C₆₇	1	2	yes	complemented
226	#3058	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#3021	C₂×C₆₇	1	2	yes	complemented
228	#1914	C₂×C₆₇	1	2	yes	complemented
229	#1903	C₂×C₆₇	1	4	yes	complemented
230	#1814	C₂×C₆₇	1	2	yes	complemented
231	#1813	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₆₇	#32951	1	1	yes	characteristic, direct product factor, Sylow
233	#585	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
235	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
236	#568	—	2	4	no	—
237	#568	—	2	2	no	—
238	#568	C₂²×C₆₇	1	2	yes	—
239	#568	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#131	—	2	4	no	—
241	C₂×#131	—	2	2	no	—
242	C₂×#131	C₂²×C₆₇	1	2	yes	—
243	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
244	#535	C₂²×C₆₇	1	2	yes	—
245	#472	—	2	4	no	—
246	#472	—	2	4	no	—
247	#469	—	2	2	no	—
248	#466	—	2	2	no	—
249	#454	—	2	4	no	—
250	#453	—	2	4	no	—
251	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
252	#427	C₂²×C₆₇	1	1	yes	characteristic, complemented
253	#421	C₂²×C₆₇	1	4	yes	—
254	#418	C₂²×C₆₇	1	2	yes	complemented
255	#413	C₂²×C₆₇	1	2	yes	—
256	C₂×#100	C₂²×C₆₇	1	4	yes	complemented
257	#344	C₂²×C₆₇	1	4	yes	complemented
258	#338	C₂²×C₆₇	1	2	yes	complemented
259	2₋¹⁺⁴	—	2	2	no	—
260	2₋¹⁺⁴	D₄×C₆₇	1	2	yes	—
261	H(5,𝔽₂)	—	2	2	no	—
262	H(5,𝔽₂)	—	2	2	no	—
263	H(5,𝔽₂)	D₄×C₆₇	1	2	yes	—
264	C₂×#41	—	2	4	no	—
265	C₂×#41	—	2	2	no	—
266	C₂×#41	C₂³×C₆₇	1	2	yes	—
267	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
268	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
269	AGL(1,ℤ/8)	—	2	8	no	—
270	AGL(1,ℤ/8)	—	2	8	no	—
271	C₂×SD₁₆	—	2	4	no	—
272	C₂×D₈	—	2	4	no	—
273	C₂×D₈	—	2	1	no	—
274	C₂×D₈	—	2	1	no	—
275	C₂×D₈	—	2	2	no	—
276	C₂×D₈	—	2	2	no	—
277	#131	—	2	4	no	—
278	#131	—	2	8	no	—
279	#131	—	2	4	no	—
280	#131	—	2	4	no	—
281	#131	—	2	4	no	—
282	#131	—	2	2	no	—
283	#131	—	2	2	no	—
284	#131	—	2	4	no	—
285	#131	—	2	2	no	—
286	#131	—	2	2	no	—
287	#131	D₄×C₆₇	1	4	yes	complemented
288	#131	D₄×C₆₇	1	4	yes	complemented
289	C₂×M₁₆	—	2	8	no	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	2	2	no	—
292	C₂×M₁₆	—	2	4	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
295	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
296	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
297	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
298	C₂²×C₈	—	2	4	no	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	D₄×C₆₇	1	2	yes	—
301	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
302	#127	—	2	2	no	—
303	#124	—	2	2	no	—
304	#121	—	2	4	no	—
305	#120	—	2	4	no	—
306	#117	C₂³×C₆₇	1	2	yes	—
307	#108	—	2	2	no	—
308	#108	—	2	2	no	—
309	#107	—	2	1	no	—
310	#106	—	2	1	no	—
311	#104	—	2	8	no	—
312	#102	—	2	4	no	—
313	#101	—	2	2	no	—
314	#100	—	2	2	no	—
315	#100	—	2	8	no	—
316	C₂⁴	—	2	2	no	—
317	#41	—	2	4	no	—
318	#41	—	2	8	no	—
319	#41	—	2	4	no	—
320	#41	—	2	4	no	—
321	#41	—	2	4	no	—
322	#41	—	2	4	no	—
323	#41	—	2	2	no	—
324	#41	—	2	2	no	—
325	#41	—	2	2	no	—
326	#41	—	2	2	no	—
327	#41	C₂×D₄×C₆₇	1	4	yes	—
328	#41	C₂×D₄×C₆₇	1	4	yes	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
332	C₂×Q₈	#41×C₆₇	1	2	yes	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	4	no	—
338	C₂×D₄	—	2	4	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	4	4	no	—
343	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂×D₄	#41×C₆₇	1	2	yes	—
345	C₂×D₄	#41×C₆₇	1	2	yes	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
350	SD₁₆	—	4	8	no	—
351	D₈	—	4	8	no	—
352	D₈	—	4	1	no	—
353	D₈	—	4	1	no	—
354	D₈	—	4	2	no	—
355	M₁₆	—	2	4	no	—
356	M₁₆	—	2	8	no	—
357	M₁₆	—	2	2	no	—
358	M₁₆	—	2	2	no	—
359	M₁₆	—	2	4	no	—
360	M₁₆	—	2	4	no	—
361	M₁₆	—	2	4	no	—
362	M₁₆	—	2	4	no	—
363	M₁₆	—	2	4	no	—
364	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
365	C₂×C₈	—	2	4	no	—
366	C₂×C₈	—	2	8	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	4	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	4	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	4	no	—
373	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
374	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
376	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
378	C₄⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	2	2	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₄²	—	2	2	no	—
382	C₂³	—	2	4	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	4	no	—
386	C₂³	—	16	2	no	—
387	C₂³	#121×C₆₇	1	2	yes	—
388	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
389	Q₈	—	2	4	no	—
390	Q₈	—	2	2	no	—
391	Q₈	—	2	2	no	—
392	Q₈	#121×C₆₇	1	4	yes	—
393	D₄	—	2	8	no	—
394	D₄	—	2	4	no	—
395	D₄	—	2	4	no	—
396	D₄	—	2	4	no	—
397	D₄	—	2	4	no	—
398	D₄	—	2	2	no	—
399	D₄	—	2	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	4	no	—
402	D₄	—	4	2	no	—
403	D₄	—	16	4	no	—
404	D₄	#121×C₆₇	1	4	yes	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	#121×C₆₇	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
418	C₈	—	2	4	no	—
419	C₈	—	2	2	no	—
420	C₈	—	4	4	no	—
421	C₈	#120×C₆₇	1	1	yes	characteristic
422	C₈	#120×C₆₇	1	2	yes	—
423	C₈	#120×C₆₇	1	1	yes	characteristic
424	C₂²	—	2	2	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	4	no	—
430	C₂²	—	8	4	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	16	4	no	—
433	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
434	C₂²	#525×C₆₇	1	2	yes	—
435	C₄	—	2	4	no	—
436	C₄	—	2	2	no	—
437	C₄	—	8	2	no	—
438	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
439	C₄	C₂×#121×C₆₇	1	2	yes	—
440	C₄	#525×C₆₇	1	1	yes	characteristic
441	C₂	—	2	1	no	—
442	C₂	—	2	2	no	—
443	C₂	—	4	2	no	—
444	C₂	—	4	4	no	—
445	C₂	—	16	2	no	—
446	C₂	C₆₇×#2847	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3573	C₂	semidirect
2. C₂×#472×C₆₇	C₂	semidirect
3. C₆₇×#3058	C₂	semidirect
4. C₆₇×#3021	C₂	semidirect
5. C₆₇×#1914	C₂	semidirect
6. C₆₇×#1903	C₂	semidirect
7. C₆₇×#1814	C₂	semidirect
8. C₆₇×#1813	C₂	semidirect
11. C₂²×D₈×C₆₇	C₂²	semidirect
27. C₂×#108×C₆₇	C₂²	semidirect
28. #427×C₆₇	C₂²	semidirect
30. #418×C₆₇	C₂²	semidirect
32. C₂×#100×C₆₇	C₂²	semidirect
33. #344×C₆₇	C₂²	semidirect
34. #338×C₆₇	C₂²	semidirect
63. #131×C₆₇	D₄	semidirect
64. #131×C₆₇	D₄	semidirect
217. #32951	C₆₇	direct (N × H)
224. #3573	C₂×C₆₇	semidirect
225. C₂×#472	C₂×C₆₇	semidirect
226. #3058	C₂×C₆₇	semidirect
227. #3021	C₂×C₆₇	semidirect
228. #1914	C₂×C₆₇	semidirect
229. #1903	C₂×C₆₇	semidirect
230. #1814	C₂×C₆₇	semidirect
231. #1813	C₂×C₆₇	semidirect
232. C₆₇	#32951	direct (N × H)
235. C₂²×D₈	C₂²×C₆₇	semidirect
251. C₂×#108	C₂²×C₆₇	semidirect
252. #427	C₂²×C₆₇	semidirect
254. #418	C₂²×C₆₇	semidirect
256. C₂×#100	C₂²×C₆₇	semidirect
257. #344	C₂²×C₆₇	semidirect
258. #338	C₂²×C₆₇	semidirect
287. #131	D₄×C₆₇	semidirect
288. #131	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×#3573	2
C₂×#472×C₆₇	2
C₆₇×#3058	2
C₆₇×#3021	2
C₆₇×#1914	2
C₆₇×#1903	2
C₆₇×#1814	2
C₆₇×#1813	2
#32951	67

Subgroups — C67×#32951 #30712202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32951 #30712202