Subgroups — C₆₇×#32945 #30712196

Order 17152.

1346 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#469×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3006	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1907	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1900	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1815	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1811	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×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	8	no	—
13	C₂×#131×C₆₇	—	2	2	no	—
14	C₂×#131×C₆₇	C₂²	1	2	yes	—
15	C₂×#131×C₆₇	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₆₇	—	2	2	no	—
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	#472×C₆₇	—	2	8	no	—
21	#469×C₆₇	—	2	2	no	—
22	#469×C₆₇	—	2	2	no	—
23	#449×C₆₇	—	2	4	no	—
24	#448×C₆₇	—	2	2	no	—
25	#447×C₆₇	—	2	2	no	—
26	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	#417×C₆₇	C₂²	1	4	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#100×C₆₇	C₂²	1	4	yes	complemented
31	#407×C₆₇	C₂²	1	2	yes	—
32	#343×C₆₇	C₂²	1	4	yes	complemented
33	#341×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#41×C₆₇	—	2	4	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
36	C₂×#41×C₆₇	—	2	1	no	—
37	C₂×#41×C₆₇	—	2	1	no	—
38	C₂×#41×C₆₇	D₄	1	2	yes	—
39	C₂×#41×C₆₇	C₂³	1	2	yes	—
40	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×D₄×C₆₇	—	2	1	no	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	—	2	1	no	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
48	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×D₈×C₆₇	—	2	4	no	—
51	C₂×D₈×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	—	2	4	no	—
53	#131×C₆₇	—	2	16	no	—
54	#131×C₆₇	—	2	4	no	—
55	#131×C₆₇	—	2	4	no	—
56	#131×C₆₇	—	2	4	no	—
57	#131×C₆₇	D₄	1	8	yes	complemented
58	C₂×M₁₆×C₆₇	—	2	16	no	—
59	C₂×M₁₆×C₆₇	—	2	1	no	—
60	C₂×M₁₆×C₆₇	—	2	4	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	—	2	4	no	—
64	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
65	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	—	2	1	no	—
69	C₂²×C₈×C₆₇	—	2	2	no	—
70	C₂²×C₈×C₆₇	—	2	2	no	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	#121×C₆₇	—	2	2	no	—
74	#121×C₆₇	—	2	2	no	—
75	#121×C₆₇	—	2	2	no	—
76	#121×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
79	#108×C₆₇	—	2	4	no	—
80	#106×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	4	no	—
82	#102×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	#100×C₆₇	—	2	8	no	—
85	C₂²⋊C₈×C₆₇	—	2	2	no	—
86	C₂²⋊C₈×C₆₇	—	2	2	no	—
87	C₂⁴×C₆₇	—	2	1	no	—
88	C₂⁴×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	#41	1	1	yes	characteristic
90	#41×C₆₇	—	2	4	no	—
91	#41×C₆₇	—	2	8	no	—
92	#41×C₆₇	—	2	4	no	—
93	#41×C₆₇	—	2	4	no	—
94	#41×C₆₇	—	2	4	no	—
95	#41×C₆₇	—	2	2	no	—
96	#41×C₆₇	—	2	2	no	—
97	#41×C₆₇	C₂×D₄	1	4	yes	—
98	#41×C₆₇	#41	1	4	yes	—
99	C₂×Q₈×C₆₇	—	2	4	no	—
100	C₂×Q₈×C₆₇	—	2	1	no	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₂×D₄×C₆₇	—	2	8	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	4	no	—
106	C₂×D₄×C₆₇	—	2	1	no	—
107	C₂×D₄×C₆₇	—	2	4	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	4	no	—
111	C₂×D₄×C₆₇	—	4	2	no	—
112	C₂×D₄×C₆₇	—	4	2	no	—
113	C₂×D₄×C₆₇	—	8	4	no	—
114	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
129	SD₁₆×C₆₇	—	4	8	no	—
130	D₈×C₆₇	—	4	8	no	—
131	D₈×C₆₇	—	4	4	no	—
132	M₁₆×C₆₇	—	2	4	no	—
133	M₁₆×C₆₇	—	2	8	no	—
134	M₁₆×C₆₇	—	2	2	no	—
135	M₁₆×C₆₇	—	2	4	no	—
136	M₁₆×C₆₇	—	2	8	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	2	no	—
145	C₂×C₈×C₆₇	—	2	8	no	—
146	C₂×C₈×C₆₇	—	2	4	no	—
147	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
148	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	2	no	—
152	C₂²⋊C₄×C₆₇	—	4	2	no	—
153	C₂²⋊C₄×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	4	no	—
160	C₂³×C₆₇	—	8	2	no	—
161	C₂³×C₆₇	—	8	2	no	—
162	C₂³×C₆₇	—	8	2	no	—
163	C₂³×C₆₇	#121	1	1	yes	characteristic
164	C₂³×C₆₇	#121	1	1	yes	characteristic
165	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
166	Q₈×C₆₇	—	2	4	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	2	no	—
174	D₄×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	4	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	8	4	no	—
178	D₄×C₆₇	—	8	4	no	—
179	D₄×C₆₇	#121	1	4	yes	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	8	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	8	2	no	—
194	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
195	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
200	C₈×C₆₇	—	2	4	no	—
201	C₈×C₆₇	—	2	2	no	—
202	C₈×C₆₇	—	4	4	no	—
203	C₈×C₆₇	#120	1	4	yes	—
204	C₂²×C₆₇	—	2	2	no	—
205	C₂²×C₆₇	—	2	1	no	—
206	C₂²×C₆₇	—	2	1	no	—
207	C₂²×C₆₇	—	2	4	no	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	4	4	no	—
211	C₂²×C₆₇	—	8	2	no	—
212	C₂²×C₆₇	—	8	4	no	—
213	C₂²×C₆₇	—	8	2	no	—
214	C₂²×C₆₇	—	8	2	no	—
215	C₂²×C₆₇	—	8	2	no	—
216	C₂²×C₆₇	—	16	4	no	—
217	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
218	C₂²×C₆₇	#525	1	2	yes	—
219	C₄×C₆₇	—	2	1	no	—
220	C₄×C₆₇	—	2	4	no	—
221	C₄×C₆₇	—	2	1	no	—
222	C₄×C₆₇	—	8	2	no	—
223	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
224	C₄×C₆₇	C₂×#121	1	2	yes	—
225	C₄×C₆₇	#525	1	1	yes	characteristic
226	#32945	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂×C₆₇	—	2	2	no	—
228	C₂×C₆₇	—	4	1	no	—
229	C₂×C₆₇	—	4	1	no	—
230	C₂×C₆₇	—	4	4	no	—
231	C₂×C₆₇	—	16	2	no	—
232	C₂×C₆₇	#2847	1	1	yes	characteristic
233	C₂×C₆₇	#3068	1	2	yes	—
234	C₂×#568	C₂×C₆₇	1	1	yes	characteristic, complemented
235	C₂×#472	C₂×C₆₇	1	2	yes	complemented
236	C₂×#469	C₂×C₆₇	1	1	yes	characteristic, complemented
237	#3006	C₂×C₆₇	1	2	yes	complemented
238	#1907	C₂×C₆₇	1	4	yes	complemented
239	#1900	C₂×C₆₇	1	2	yes	complemented
240	#1815	C₂×C₆₇	1	2	yes	complemented
241	#1811	C₂×C₆₇	1	1	yes	characteristic, complemented
242	C₆₇	#32945	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
245	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
246	#568	—	2	8	no	—
247	C₂×#131	—	2	2	no	—
248	C₂×#131	C₂²×C₆₇	1	2	yes	—
249	C₂×#131	C₂²×C₆₇	1	2	yes	—
250	C₂²×M₁₆	—	2	2	no	—
251	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
252	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×#121	C₂²×C₆₇	1	2	yes	—
254	#472	—	2	8	no	—
255	#469	—	2	2	no	—
256	#469	—	2	2	no	—
257	#449	—	2	4	no	—
258	#448	—	2	2	no	—
259	#447	—	2	2	no	—
260	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
261	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
262	#417	C₂²×C₆₇	1	4	yes	—
263	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
264	C₂×#100	C₂²×C₆₇	1	4	yes	complemented
265	#407	C₂²×C₆₇	1	2	yes	—
266	#343	C₂²×C₆₇	1	4	yes	complemented
267	#341	C₂²×C₆₇	1	2	yes	complemented
268	C₂×#41	—	2	4	no	—
269	C₂×#41	—	2	4	no	—
270	C₂×#41	—	2	1	no	—
271	C₂×#41	—	2	1	no	—
272	C₂×#41	D₄×C₆₇	1	2	yes	—
273	C₂×#41	C₂³×C₆₇	1	2	yes	—
274	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	—	2	2	no	—
277	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
278	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
279	C₂³×C₄	—	2	1	no	—
280	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
281	AGL(1,ℤ/8)	—	2	8	no	—
282	AGL(1,ℤ/8)	—	2	8	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×D₈	—	2	4	no	—
285	C₂×D₈	—	2	2	no	—
286	C₂×D₈	—	2	4	no	—
287	#131	—	2	16	no	—
288	#131	—	2	4	no	—
289	#131	—	2	4	no	—
290	#131	—	2	4	no	—
291	#131	D₄×C₆₇	1	8	yes	complemented
292	C₂×M₁₆	—	2	16	no	—
293	C₂×M₁₆	—	2	1	no	—
294	C₂×M₁₆	—	2	4	no	—
295	C₂×M₁₆	—	2	2	no	—
296	C₂×M₁₆	—	2	2	no	—
297	C₂×M₁₆	—	2	4	no	—
298	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
299	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
300	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
301	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
302	C₂²×C₈	—	2	1	no	—
303	C₂²×C₈	—	2	2	no	—
304	C₂²×C₈	—	2	2	no	—
305	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
307	#121	—	2	2	no	—
308	#121	—	2	2	no	—
309	#121	—	2	2	no	—
310	#121	—	2	2	no	—
311	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	#108	—	2	4	no	—
314	#106	—	2	2	no	—
315	#103	—	2	4	no	—
316	#102	—	2	4	no	—
317	#102	—	2	4	no	—
318	#100	—	2	8	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₂⁴	—	2	1	no	—
322	C₂⁴	—	2	2	no	—
323	C₂⁴	#41×C₆₇	1	1	yes	characteristic
324	#41	—	2	4	no	—
325	#41	—	2	8	no	—
326	#41	—	2	4	no	—
327	#41	—	2	4	no	—
328	#41	—	2	4	no	—
329	#41	—	2	2	no	—
330	#41	—	2	2	no	—
331	#41	C₂×D₄×C₆₇	1	4	yes	—
332	#41	#41×C₆₇	1	4	yes	—
333	C₂×Q₈	—	2	4	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
336	C₂×D₄	—	2	8	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂×D₄	—	4	2	no	—
347	C₂×D₄	—	8	4	no	—
348	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
363	SD₁₆	—	4	8	no	—
364	D₈	—	4	8	no	—
365	D₈	—	4	4	no	—
366	M₁₆	—	2	4	no	—
367	M₁₆	—	2	8	no	—
368	M₁₆	—	2	2	no	—
369	M₁₆	—	2	4	no	—
370	M₁₆	—	2	8	no	—
371	M₁₆	—	2	4	no	—
372	M₁₆	—	2	4	no	—
373	M₁₆	—	2	4	no	—
374	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
375	C₂×C₈	—	2	4	no	—
376	C₂×C₈	—	2	8	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	8	no	—
380	C₂×C₈	—	2	4	no	—
381	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
382	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
383	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	4	2	no	—
387	C₂²⋊C₄	—	4	2	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	4	no	—
394	C₂³	—	8	2	no	—
395	C₂³	—	8	2	no	—
396	C₂³	—	8	2	no	—
397	C₂³	#121×C₆₇	1	1	yes	characteristic
398	C₂³	#121×C₆₇	1	1	yes	characteristic
399	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
400	Q₈	—	2	4	no	—
401	Q₈	—	2	2	no	—
402	Q₈	#121×C₆₇	1	4	yes	—
403	D₄	—	2	8	no	—
404	D₄	—	2	4	no	—
405	D₄	—	2	4	no	—
406	D₄	—	2	4	no	—
407	D₄	—	2	2	no	—
408	D₄	—	4	2	no	—
409	D₄	—	4	4	no	—
410	D₄	—	4	2	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	4	no	—
413	D₄	#121×C₆₇	1	4	yes	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	8	2	no	—
428	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
429	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
434	C₈	—	2	4	no	—
435	C₈	—	2	2	no	—
436	C₈	—	4	4	no	—
437	C₈	#120×C₆₇	1	4	yes	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	4	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	8	2	no	—
450	C₂²	—	16	4	no	—
451	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
452	C₂²	#525×C₆₇	1	2	yes	—
453	C₄	—	2	1	no	—
454	C₄	—	2	4	no	—
455	C₄	—	2	1	no	—
456	C₄	—	8	2	no	—
457	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
458	C₄	C₂×#121×C₆₇	1	2	yes	—
459	C₄	#525×C₆₇	1	1	yes	characteristic
460	C₂	—	2	2	no	—
461	C₂	—	4	1	no	—
462	C₂	—	4	1	no	—
463	C₂	—	4	4	no	—
464	C₂	—	16	2	no	—
465	C₂	C₆₇×#2847	1	1	yes	characteristic
466	C₂	C₆₇×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#568×C₆₇	C₂	semidirect
2. C₂×#472×C₆₇	C₂	semidirect
3. C₂×#469×C₆₇	C₂	semidirect
4. C₆₇×#3006	C₂	semidirect
5. C₆₇×#1907	C₂	semidirect
6. C₆₇×#1900	C₂	semidirect
7. C₆₇×#1815	C₂	semidirect
8. C₆₇×#1811	C₂	semidirect
11. C₂²×D₈×C₆₇	C₂²	semidirect
26. C₂×#108×C₆₇	C₂²	semidirect
27. C₂×#106×C₆₇	C₂²	semidirect
29. C₂×#102×C₆₇	C₂²	semidirect
30. C₂×#100×C₆₇	C₂²	semidirect
32. #343×C₆₇	C₂²	semidirect
33. #341×C₆₇	C₂²	semidirect
57. #131×C₆₇	D₄	semidirect
226. #32945	C₆₇	direct (N × H)
234. C₂×#568	C₂×C₆₇	semidirect
235. C₂×#472	C₂×C₆₇	semidirect
236. C₂×#469	C₂×C₆₇	semidirect
237. #3006	C₂×C₆₇	semidirect
238. #1907	C₂×C₆₇	semidirect
239. #1900	C₂×C₆₇	semidirect
240. #1815	C₂×C₆₇	semidirect
241. #1811	C₂×C₆₇	semidirect
242. C₆₇	#32945	direct (N × H)
245. C₂²×D₈	C₂²×C₆₇	semidirect
260. C₂×#108	C₂²×C₆₇	semidirect
261. C₂×#106	C₂²×C₆₇	semidirect
263. C₂×#102	C₂²×C₆₇	semidirect
264. C₂×#100	C₂²×C₆₇	semidirect
266. #343	C₂²×C₆₇	semidirect
267. #341	C₂²×C₆₇	semidirect
291. #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
C₂×#568×C₆₇	2
C₂×#472×C₆₇	2
C₂×#469×C₆₇	2
C₆₇×#3006	2
C₆₇×#1907	2
C₆₇×#1900	2
C₆₇×#1815	2
C₆₇×#1811	2
#32945	67

Subgroups — C67×#32945 #30712196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32945 #30712196