Subgroups — C₆₇×#32950 #30712201

Order 17152.

1282 conjugacy classes of subgroups.

The subgroup lattice has 552 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×#471×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3057	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3022	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3021	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1915	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1914	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1903	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1902	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1814	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1813	C₂	1	1	yes	maximal, characteristic, complemented
12	#585×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#137×C₆₇	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	#568×C₆₇	—	2	4	no	—
17	#568×C₆₇	—	2	2	no	—
18	#568×C₆₇	C₂²	1	2	yes	—
19	#568×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#131×C₆₇	—	2	4	no	—
21	C₂×#131×C₆₇	—	2	2	no	—
22	C₂×#131×C₆₇	C₂²	1	2	yes	—
23	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
24	#536×C₆₇	C₂²	1	1	yes	characteristic
25	#535×C₆₇	C₂²	1	1	yes	characteristic
26	#471×C₆₇	—	2	4	no	—
27	#471×C₆₇	—	2	4	no	—
28	#468×C₆₇	—	2	2	no	—
29	#465×C₆₇	—	2	2	no	—
30	#456×C₆₇	—	2	2	no	—
31	#455×C₆₇	—	2	2	no	—
32	#454×C₆₇	—	2	2	no	—
33	#453×C₆₇	—	2	2	no	—
34	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
35	#427×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#421×C₆₇	C₂²	1	2	yes	—
37	#421×C₆₇	C₂²	1	2	yes	—
38	#419×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#418×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#413×C₆₇	C₂²	1	1	yes	characteristic
41	#413×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
43	C₂×#100×C₆₇	C₂²	1	2	yes	complemented
44	#344×C₆₇	C₂²	1	2	yes	complemented
45	#344×C₆₇	C₂²	1	2	yes	complemented
46	#338×C₆₇	C₂²	1	2	yes	complemented
47	2₋¹⁺⁴×C₆₇	—	2	2	no	—
48	2₋¹⁺⁴×C₆₇	D₄	1	2	yes	—
49	H(5,𝔽₂)×C₆₇	—	2	2	no	—
50	H(5,𝔽₂)×C₆₇	—	2	2	no	—
51	H(5,𝔽₂)×C₆₇	D₄	1	2	yes	—
52	C₂×#41×C₆₇	—	2	4	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	C₂³	1	2	yes	—
55	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	#137×C₆₇	—	2	4	no	—
59	#137×C₆₇	—	2	4	no	—
60	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
61	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
62	C₂×Q₁₆×C₆₇	—	2	2	no	—
63	C₂×SD₁₆×C₆₇	—	2	2	no	—
64	C₂×SD₁₆×C₆₇	—	2	2	no	—
65	C₂×SD₁₆×C₆₇	—	2	1	no	—
66	C₂×SD₁₆×C₆₇	—	2	1	no	—
67	C₂×SD₁₆×C₆₇	—	2	2	no	—
68	C₂×SD₁₆×C₆₇	—	2	2	no	—
69	C₂×D₈×C₆₇	—	2	2	no	—
70	#131×C₆₇	—	2	4	no	—
71	#131×C₆₇	—	2	8	no	—
72	#131×C₆₇	—	2	4	no	—
73	#131×C₆₇	—	2	4	no	—
74	#131×C₆₇	—	2	4	no	—
75	#131×C₆₇	—	2	2	no	—
76	#131×C₆₇	—	2	2	no	—
77	#131×C₆₇	—	2	4	no	—
78	#131×C₆₇	—	2	2	no	—
79	#131×C₆₇	—	2	2	no	—
80	#131×C₆₇	D₄	1	4	yes	complemented
81	#131×C₆₇	D₄	1	4	yes	complemented
82	C₂×M₁₆×C₆₇	—	2	8	no	—
83	C₂×M₁₆×C₆₇	—	2	4	no	—
84	C₂×M₁₆×C₆₇	—	2	2	no	—
85	C₂×M₁₆×C₆₇	—	2	4	no	—
86	C₂×M₁₆×C₆₇	—	2	2	no	—
87	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
88	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
89	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
90	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₆₇	—	2	4	no	—
93	C₂²×C₈×C₆₇	—	2	2	no	—
94	C₂²×C₈×C₆₇	D₄	1	2	yes	—
95	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊Q₈×C₆₇	—	2	1	no	—
97	#127×C₆₇	—	2	1	no	—
98	#124×C₆₇	—	2	1	no	—
99	#124×C₆₇	—	2	1	no	—
100	#123×C₆₇	—	2	2	no	—
101	C₂²⋊Q₈×C₆₇	—	2	2	no	—
102	#121×C₆₇	—	2	2	no	—
103	#120×C₆₇	—	2	2	no	—
104	#117×C₆₇	C₂³	1	1	yes	characteristic
105	#117×C₆₇	C₂³	1	1	yes	characteristic
106	#108×C₆₇	—	2	2	no	—
107	#108×C₆₇	—	2	2	no	—
108	#107×C₆₇	—	2	1	no	—
109	#106×C₆₇	—	2	1	no	—
110	#104×C₆₇	—	2	4	no	—
111	#104×C₆₇	—	2	4	no	—
112	#103×C₆₇	—	2	2	no	—
113	#102×C₆₇	—	2	2	no	—
114	#101×C₆₇	—	2	1	no	—
115	#101×C₆₇	—	2	1	no	—
116	#101×C₆₇	—	2	4	no	—
117	#100×C₆₇	—	2	1	no	—
118	#100×C₆₇	—	2	1	no	—
119	#100×C₆₇	—	2	4	no	—
120	C₂⁴×C₆₇	—	2	1	no	—
121	#41×C₆₇	—	2	4	no	—
122	#41×C₆₇	—	2	8	no	—
123	#41×C₆₇	—	2	4	no	—
124	#41×C₆₇	—	2	4	no	—
125	#41×C₆₇	—	2	4	no	—
126	#41×C₆₇	—	2	4	no	—
127	#41×C₆₇	—	2	2	no	—
128	#41×C₆₇	—	2	2	no	—
129	#41×C₆₇	—	2	2	no	—
130	#41×C₆₇	—	2	2	no	—
131	#41×C₆₇	C₂×D₄	1	4	yes	—
132	#41×C₆₇	C₂×D₄	1	4	yes	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	2	4	no	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	1	no	—
137	C₂×Q₈×C₆₇	—	2	1	no	—
138	C₂×Q₈×C₆₇	—	2	2	no	—
139	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂×Q₈×C₆₇	#41	1	2	yes	—
141	C₂×D₄×C₆₇	—	2	8	no	—
142	C₂×D₄×C₆₇	—	2	2	no	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	—	2	2	no	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	2	1	no	—
148	C₂×D₄×C₆₇	—	2	1	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	4	2	no	—
151	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂×D₄×C₆₇	#41	1	2	yes	—
153	C₂×D₄×C₆₇	#41	1	2	yes	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	1	no	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
159	Q₁₆×C₆₇	—	4	4	no	—
160	SD₁₆×C₆₇	—	4	4	no	—
161	SD₁₆×C₆₇	—	4	4	no	—
162	SD₁₆×C₆₇	—	4	1	no	—
163	SD₁₆×C₆₇	—	4	1	no	—
164	SD₁₆×C₆₇	—	4	2	no	—
165	D₈×C₆₇	—	4	4	no	—
166	M₁₆×C₆₇	—	2	4	no	—
167	M₁₆×C₆₇	—	2	8	no	—
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	—	2	2	no	—
170	M₁₆×C₆₇	—	2	4	no	—
171	M₁₆×C₆₇	—	2	4	no	—
172	M₁₆×C₆₇	—	2	4	no	—
173	M₁₆×C₆₇	—	2	2	no	—
174	M₁₆×C₆₇	—	2	2	no	—
175	M₁₆×C₆₇	—	2	2	no	—
176	M₁₆×C₆₇	—	2	2	no	—
177	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
178	C₂×C₈×C₆₇	—	2	4	no	—
179	C₂×C₈×C₆₇	—	2	8	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	—	2	4	no	—
182	C₂×C₈×C₆₇	—	2	2	no	—
183	C₂×C₈×C₆₇	—	2	4	no	—
184	C₂×C₈×C₆₇	—	2	2	no	—
185	C₂×C₈×C₆₇	—	2	2	no	—
186	C₂×C₈×C₆₇	—	2	2	no	—
187	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
188	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
191	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₆₇	—	2	1	no	—
193	C₄⋊C₄×C₆₇	—	2	1	no	—
194	C₄⋊C₄×C₆₇	—	4	2	no	—
195	C₂²⋊C₄×C₆₇	—	2	1	no	—
196	C₂²⋊C₄×C₆₇	—	2	1	no	—
197	C₂²⋊C₄×C₆₇	—	4	2	no	—
198	C₂²⋊C₄×C₆₇	—	4	2	no	—
199	C₄²×C₆₇	—	2	1	no	—
200	C₄²×C₆₇	—	2	1	no	—
201	C₂³×C₆₇	—	2	4	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	4	1	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	—	16	1	no	—
206	C₂³×C₆₇	#121	1	2	yes	—
207	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
208	Q₈×C₆₇	—	2	4	no	—
209	Q₈×C₆₇	—	2	2	no	—
210	Q₈×C₆₇	—	2	2	no	—
211	Q₈×C₆₇	—	4	1	no	—
212	Q₈×C₆₇	—	4	2	no	—
213	Q₈×C₆₇	—	4	1	no	—
214	Q₈×C₆₇	#121	1	4	yes	—
215	D₄×C₆₇	—	2	8	no	—
216	D₄×C₆₇	—	2	4	no	—
217	D₄×C₆₇	—	2	4	no	—
218	D₄×C₆₇	—	2	4	no	—
219	D₄×C₆₇	—	2	4	no	—
220	D₄×C₆₇	—	2	2	no	—
221	D₄×C₆₇	—	2	2	no	—
222	D₄×C₆₇	—	4	1	no	—
223	D₄×C₆₇	—	4	2	no	—
224	D₄×C₆₇	—	4	1	no	—
225	D₄×C₆₇	—	16	2	no	—
226	D₄×C₆₇	#121	1	4	yes	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	8	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	1	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	2	no	—
238	C₂×C₄×C₆₇	#121	1	2	yes	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
243	C₈×C₆₇	—	2	4	no	—
244	C₈×C₆₇	—	2	2	no	—
245	C₈×C₆₇	—	4	2	no	—
246	C₈×C₆₇	—	4	2	no	—
247	C₈×C₆₇	#120	1	1	yes	characteristic
248	C₈×C₆₇	#120	1	2	yes	—
249	C₈×C₆₇	#120	1	1	yes	characteristic
250	C₂²×C₆₇	—	2	2	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	—	4	1	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	—	8	2	no	—
255	C₂²×C₆₇	—	8	2	no	—
256	C₂²×C₆₇	—	8	4	no	—
257	C₂²×C₆₇	—	8	1	no	—
258	C₂²×C₆₇	—	16	2	no	—
259	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
260	C₂²×C₆₇	#525	1	2	yes	—
261	C₄×C₆₇	—	2	4	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	8	1	no	—
264	C₄×C₆₇	—	8	1	no	—
265	C₄×C₆₇	—	8	1	no	—
266	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
267	C₄×C₆₇	C₂×#121	1	2	yes	—
268	C₄×C₆₇	#525	1	1	yes	characteristic
269	#32950	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂×C₆₇	—	2	1	no	—
271	C₂×C₆₇	—	2	2	no	—
272	C₂×C₆₇	—	4	2	no	—
273	C₂×C₆₇	—	4	4	no	—
274	C₂×C₆₇	—	16	1	no	—
275	C₂×C₆₇	#2847	1	1	yes	characteristic
276	#3573	C₂×C₆₇	1	1	yes	characteristic, complemented
277	C₂×#471	C₂×C₆₇	1	2	yes	complemented
278	#3057	C₂×C₆₇	1	1	yes	characteristic, complemented
279	#3022	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#3021	C₂×C₆₇	1	1	yes	characteristic
281	#1915	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#1914	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#1903	C₂×C₆₇	1	2	yes	complemented
284	#1902	C₂×C₆₇	1	2	yes	complemented
285	#1814	C₂×C₆₇	1	2	yes	complemented
286	#1813	C₂×C₆₇	1	1	yes	characteristic, complemented
287	C₆₇	#32950	1	1	yes	characteristic, direct product factor, Sylow
288	#585	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×#137	C₂²×C₆₇	1	2	yes	—
290	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
291	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
292	#568	—	2	4	no	—
293	#568	—	2	2	no	—
294	#568	C₂²×C₆₇	1	2	yes	—
295	#568	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×#131	—	2	4	no	—
297	C₂×#131	—	2	2	no	—
298	C₂×#131	C₂²×C₆₇	1	2	yes	—
299	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
300	#536	C₂²×C₆₇	1	1	yes	characteristic
301	#535	C₂²×C₆₇	1	1	yes	characteristic
302	#471	—	2	4	no	—
303	#471	—	2	4	no	—
304	#468	—	2	2	no	—
305	#465	—	2	2	no	—
306	#456	—	2	2	no	—
307	#455	—	2	2	no	—
308	#454	—	2	2	no	—
309	#453	—	2	2	no	—
310	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
311	#427	C₂²×C₆₇	1	1	yes	characteristic, complemented
312	#421	C₂²×C₆₇	1	2	yes	—
313	#421	C₂²×C₆₇	1	2	yes	—
314	#419	C₂²×C₆₇	1	1	yes	characteristic, complemented
315	#418	C₂²×C₆₇	1	1	yes	characteristic, complemented
316	#413	C₂²×C₆₇	1	1	yes	characteristic
317	#413	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
319	C₂×#100	C₂²×C₆₇	1	2	yes	complemented
320	#344	C₂²×C₆₇	1	2	yes	complemented
321	#344	C₂²×C₆₇	1	2	yes	complemented
322	#338	C₂²×C₆₇	1	2	yes	complemented
323	2₋¹⁺⁴	—	2	2	no	—
324	2₋¹⁺⁴	D₄×C₆₇	1	2	yes	—
325	H(5,𝔽₂)	—	2	2	no	—
326	H(5,𝔽₂)	—	2	2	no	—
327	H(5,𝔽₂)	D₄×C₆₇	1	2	yes	—
328	C₂×#41	—	2	4	no	—
329	C₂×#41	—	2	2	no	—
330	C₂×#41	C₂³×C₆₇	1	2	yes	—
331	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
332	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
333	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
334	#137	—	2	4	no	—
335	#137	—	2	4	no	—
336	AGL(1,ℤ/8)	—	2	4	no	—
337	AGL(1,ℤ/8)	—	2	4	no	—
338	C₂×Q₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	1	no	—
342	C₂×SD₁₆	—	2	1	no	—
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×SD₁₆	—	2	2	no	—
345	C₂×D₈	—	2	2	no	—
346	#131	—	2	4	no	—
347	#131	—	2	8	no	—
348	#131	—	2	4	no	—
349	#131	—	2	4	no	—
350	#131	—	2	4	no	—
351	#131	—	2	2	no	—
352	#131	—	2	2	no	—
353	#131	—	2	4	no	—
354	#131	—	2	2	no	—
355	#131	—	2	2	no	—
356	#131	D₄×C₆₇	1	4	yes	complemented
357	#131	D₄×C₆₇	1	4	yes	complemented
358	C₂×M₁₆	—	2	8	no	—
359	C₂×M₁₆	—	2	4	no	—
360	C₂×M₁₆	—	2	2	no	—
361	C₂×M₁₆	—	2	4	no	—
362	C₂×M₁₆	—	2	2	no	—
363	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
364	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
365	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
366	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
367	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
368	C₂²×C₈	—	2	4	no	—
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	D₄×C₆₇	1	2	yes	—
371	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
372	C₄⋊Q₈	—	2	1	no	—
373	#127	—	2	1	no	—
374	#124	—	2	1	no	—
375	#124	—	2	1	no	—
376	#123	—	2	2	no	—
377	C₂²⋊Q₈	—	2	2	no	—
378	#121	—	2	2	no	—
379	#120	—	2	2	no	—
380	#117	C₂³×C₆₇	1	1	yes	characteristic
381	#117	C₂³×C₆₇	1	1	yes	characteristic
382	#108	—	2	2	no	—
383	#108	—	2	2	no	—
384	#107	—	2	1	no	—
385	#106	—	2	1	no	—
386	#104	—	2	4	no	—
387	#104	—	2	4	no	—
388	#103	—	2	2	no	—
389	#102	—	2	2	no	—
390	#101	—	2	1	no	—
391	#101	—	2	1	no	—
392	#101	—	2	4	no	—
393	#100	—	2	1	no	—
394	#100	—	2	1	no	—
395	#100	—	2	4	no	—
396	C₂⁴	—	2	1	no	—
397	#41	—	2	4	no	—
398	#41	—	2	8	no	—
399	#41	—	2	4	no	—
400	#41	—	2	4	no	—
401	#41	—	2	4	no	—
402	#41	—	2	4	no	—
403	#41	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	C₂×D₄×C₆₇	1	4	yes	—
408	#41	C₂×D₄×C₆₇	1	4	yes	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
416	C₂×Q₈	#41×C₆₇	1	2	yes	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
428	C₂×D₄	#41×C₆₇	1	2	yes	—
429	C₂×D₄	#41×C₆₇	1	2	yes	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
435	Q₁₆	—	4	4	no	—
436	SD₁₆	—	4	4	no	—
437	SD₁₆	—	4	4	no	—
438	SD₁₆	—	4	1	no	—
439	SD₁₆	—	4	1	no	—
440	SD₁₆	—	4	2	no	—
441	D₈	—	4	4	no	—
442	M₁₆	—	2	4	no	—
443	M₁₆	—	2	8	no	—
444	M₁₆	—	2	2	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	4	no	—
447	M₁₆	—	2	4	no	—
448	M₁₆	—	2	4	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	2	2	no	—
453	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
454	C₂×C₈	—	2	4	no	—
455	C₂×C₈	—	2	8	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	4	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
464	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
465	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
467	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	4	2	no	—
471	C₂²⋊C₄	—	2	1	no	—
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	4	2	no	—
474	C₂²⋊C₄	—	4	2	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	4	1	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	16	1	no	—
482	C₂³	#121×C₆₇	1	2	yes	—
483	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
484	Q₈	—	2	4	no	—
485	Q₈	—	2	2	no	—
486	Q₈	—	2	2	no	—
487	Q₈	—	4	1	no	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	1	no	—
490	Q₈	#121×C₆₇	1	4	yes	—
491	D₄	—	2	8	no	—
492	D₄	—	2	4	no	—
493	D₄	—	2	4	no	—
494	D₄	—	2	4	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	2	no	—
497	D₄	—	2	2	no	—
498	D₄	—	4	1	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	1	no	—
501	D₄	—	16	2	no	—
502	D₄	#121×C₆₇	1	4	yes	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	#121×C₆₇	1	2	yes	—
515	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
516	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
518	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
519	C₈	—	2	4	no	—
520	C₈	—	2	2	no	—
521	C₈	—	4	2	no	—
522	C₈	—	4	2	no	—
523	C₈	#120×C₆₇	1	1	yes	characteristic
524	C₈	#120×C₆₇	1	2	yes	—
525	C₈	#120×C₆₇	1	1	yes	characteristic
526	C₂²	—	2	2	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	4	1	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	2	no	—
532	C₂²	—	8	4	no	—
533	C₂²	—	8	1	no	—
534	C₂²	—	16	2	no	—
535	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
536	C₂²	#525×C₆₇	1	2	yes	—
537	C₄	—	2	4	no	—
538	C₄	—	2	2	no	—
539	C₄	—	8	1	no	—
540	C₄	—	8	1	no	—
541	C₄	—	8	1	no	—
542	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
543	C₄	C₂×#121×C₆₇	1	2	yes	—
544	C₄	#525×C₆₇	1	1	yes	characteristic
545	C₂	—	2	1	no	—
546	C₂	—	2	2	no	—
547	C₂	—	4	2	no	—
548	C₂	—	4	4	no	—
549	C₂	—	16	1	no	—
550	C₂	C₆₇×#2847	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3021	C₂
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×#471×C₆₇	C₂	semidirect
3. C₆₇×#3057	C₂	semidirect
4. C₆₇×#3022	C₂	semidirect
6. C₆₇×#1915	C₂	semidirect
7. C₆₇×#1914	C₂	semidirect
8. C₆₇×#1903	C₂	semidirect
9. C₆₇×#1902	C₂	semidirect
10. C₆₇×#1814	C₂	semidirect
11. C₆₇×#1813	C₂	semidirect
15. C₂²×SD₁₆×C₆₇	C₂²	semidirect
34. C₂×#108×C₆₇	C₂²	semidirect
35. #427×C₆₇	C₂²	semidirect
38. #419×C₆₇	C₂²	semidirect
39. #418×C₆₇	C₂²	semidirect
42. C₂×#101×C₆₇	C₂²	semidirect
43. C₂×#100×C₆₇	C₂²	semidirect
44. #344×C₆₇	C₂²	semidirect
45. #344×C₆₇	C₂²	semidirect
46. #338×C₆₇	C₂²	semidirect
80. #131×C₆₇	D₄	semidirect
81. #131×C₆₇	D₄	semidirect
269. #32950	C₆₇	direct (N × H)
276. #3573	C₂×C₆₇	semidirect
277. C₂×#471	C₂×C₆₇	semidirect
278. #3057	C₂×C₆₇	semidirect
279. #3022	C₂×C₆₇	semidirect
281. #1915	C₂×C₆₇	semidirect
282. #1914	C₂×C₆₇	semidirect
283. #1903	C₂×C₆₇	semidirect
284. #1902	C₂×C₆₇	semidirect
285. #1814	C₂×C₆₇	semidirect
286. #1813	C₂×C₆₇	semidirect
287. C₆₇	#32950	direct (N × H)
291. C₂²×SD₁₆	C₂²×C₆₇	semidirect
310. C₂×#108	C₂²×C₆₇	semidirect
311. #427	C₂²×C₆₇	semidirect
314. #419	C₂²×C₆₇	semidirect
315. #418	C₂²×C₆₇	semidirect
318. C₂×#101	C₂²×C₆₇	semidirect
319. C₂×#100	C₂²×C₆₇	semidirect
320. #344	C₂²×C₆₇	semidirect
321. #344	C₂²×C₆₇	semidirect
322. #338	C₂²×C₆₇	semidirect
356. #131	D₄×C₆₇	semidirect
357. #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×#471×C₆₇	2
C₆₇×#3057	2
C₆₇×#3022	2
C₆₇×#3021	2
C₆₇×#1915	2
C₆₇×#1914	2
C₆₇×#1903	2
C₆₇×#1902	2
C₆₇×#1814	2
C₆₇×#1813	2
#32950	67

Subgroups — C67×#32950 #30712201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32950 #30712201