Subgroups — C₆₇×#25925 #30705176

Order 17152.

1004 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3140	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3040	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2861	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2028	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2027	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1994	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1883	C₂	1	2	yes	maximal
10	C₆₇×#1882	C₂	1	2	yes	maximal, complemented
11	C₄×#104×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
14	#557×C₆₇	—	2	2	no	—
15	#554×C₆₇	—	2	2	no	—
16	#554×C₆₇	—	2	4	no	—
17	#544×C₆₇	—	2	2	no	—
18	#544×C₆₇	C₂²	1	2	yes	—
19	#533×C₆₇	—	2	1	no	—
20	#533×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#125×C₆₇	—	2	1	no	—
22	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
24	#520×C₆₇	C₂²	1	2	yes	—
25	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
26	#489×C₆₇	—	2	2	no	—
27	#488×C₆₇	—	2	1	no	—
28	#488×C₆₇	—	2	1	no	—
29	#464×C₆₇	—	2	2	no	—
30	#463×C₆₇	—	2	2	no	—
31	#423×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#104×C₆₇	C₂²	1	2	yes	—
33	#419×C₆₇	C₂²	1	1	yes	characteristic
34	#419×C₆₇	C₂²	1	2	yes	—
35	#418×C₆₇	C₂²	1	1	yes	characteristic
36	#418×C₆₇	C₂²	1	2	yes	—
37	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	#391×C₆₇	C₂²	1	2	yes	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#388×C₆₇	—	2	4	no	—
41	C₄³×C₆₇	C₂²	1	1	yes	characteristic
42	#343×C₆₇	C₂²	1	2	yes	—
43	#343×C₆₇	C₂²	1	2	yes	—
44	#339×C₆₇	C₂²	1	2	yes	—
45	#339×C₆₇	C₂²	1	2	yes	—
46	#339×C₆₇	C₂²	1	2	yes	—
47	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
50	#137×C₆₇	—	2	4	no	—
51	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
52	C₂×Q₁₆×C₆₇	—	2	1	no	—
53	C₂×SD₁₆×C₆₇	—	2	1	no	—
54	C₂×SD₁₆×C₆₇	—	2	1	no	—
55	C₂×D₈×C₆₇	—	2	1	no	—
56	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
59	C₄⋊Q₈×C₆₇	—	2	4	no	—
60	C₄⋊Q₈×C₆₇	—	2	2	no	—
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
63	#125×C₆₇	—	2	4	no	—
64	#125×C₆₇	—	2	4	no	—
65	#125×C₆₇	—	2	2	no	—
66	#125×C₆₇	—	2	2	no	—
67	#125×C₆₇	—	2	1	no	—
68	#125×C₆₇	—	2	1	no	—
69	#125×C₆₇	—	2	2	no	—
70	#125×C₆₇	D₄	1	1	yes	characteristic
71	#125×C₆₇	D₄	1	1	yes	characteristic
72	#124×C₆₇	—	2	2	no	—
73	#124×C₆₇	—	2	1	no	—
74	#124×C₆₇	—	2	1	no	—
75	C₂²⋊Q₈×C₆₇	—	2	4	no	—
76	C₂²⋊Q₈×C₆₇	—	2	4	no	—
77	C₂²⋊Q₈×C₆₇	—	2	4	no	—
78	C₂²⋊Q₈×C₆₇	—	2	2	no	—
79	C₂²⋊Q₈×C₆₇	—	2	4	no	—
80	C₄×Q₈×C₆₇	—	2	4	no	—
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	—	2	2	no	—
83	C₄×Q₈×C₆₇	D₄	1	2	yes	—
84	C₄×D₄×C₆₇	—	2	2	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	2	4	no	—
87	C₄×D₄×C₆₇	D₄	1	2	yes	—
88	#117×C₆₇	—	2	2	no	—
89	#117×C₆₇	—	2	1	no	—
90	#117×C₆₇	C₂³	1	1	yes	characteristic
91	#117×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
98	C₂×C₄²×C₆₇	—	2	1	no	—
99	C₂×C₄²×C₆₇	—	2	2	no	—
100	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₈×C₆₇	—	2	2	no	—
103	#104×C₆₇	—	2	4	no	—
104	#103×C₆₇	—	2	2	no	—
105	#103×C₆₇	—	2	2	no	—
106	#103×C₆₇	—	2	2	no	—
107	#102×C₆₇	—	2	2	no	—
108	#102×C₆₇	—	2	2	no	—
109	#102×C₆₇	—	2	2	no	—
110	#97×C₆₇	—	2	1	no	—
111	C₄×C₈×C₆₇	—	2	1	no	—
112	#95×C₆₇	—	2	4	no	—
113	C₂⁴×C₆₇	—	2	1	no	—
114	#41×C₆₇	—	2	2	no	—
115	#41×C₆₇	#41	1	4	yes	—
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	—	2	1	no	—
119	C₂×Q₈×C₆₇	—	2	1	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	—	2	2	no	—
122	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	1	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₆₇	—	4	2	no	—
138	SD₁₆×C₆₇	—	4	2	no	—
139	SD₁₆×C₆₇	—	4	2	no	—
140	D₈×C₆₇	—	4	2	no	—
141	M₁₆×C₆₇	—	2	2	no	—
142	M₁₆×C₆₇	—	2	2	no	—
143	M₁₆×C₆₇	—	2	2	no	—
144	M₁₆×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	1	no	—
147	C₂×C₈×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	2	4	no	—
149	C₄⋊C₄×C₆₇	—	2	4	no	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	4	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
162	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₆₇	—	2	4	no	—
164	C₂²⋊C₄×C₆₇	—	2	2	no	—
165	C₂²⋊C₄×C₆₇	—	2	1	no	—
166	C₂²⋊C₄×C₆₇	—	2	2	no	—
167	C₂²⋊C₄×C₆₇	—	4	8	no	—
168	C₂²⋊C₄×C₆₇	—	4	2	no	—
169	C₂²⋊C₄×C₆₇	—	4	2	no	—
170	C₄²×C₆₇	—	2	4	no	—
171	C₄²×C₆₇	—	2	2	no	—
172	C₄²×C₆₇	—	2	1	no	—
173	C₄²×C₆₇	—	2	2	no	—
174	C₄²×C₆₇	—	2	2	no	—
175	C₄²×C₆₇	—	2	2	no	—
176	C₄²×C₆₇	—	4	4	no	—
177	C₄²×C₆₇	C₂×D₄	1	2	yes	—
178	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	—	4	4	no	—
182	C₂³×C₆₇	—	4	1	no	—
183	C₂³×C₆₇	—	4	2	no	—
184	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₆₇	—	2	1	no	—
186	Q₈×C₆₇	—	4	1	no	—
187	Q₈×C₆₇	—	4	1	no	—
188	Q₈×C₆₇	—	4	2	no	—
189	Q₈×C₆₇	—	8	2	no	—
190	Q₈×C₆₇	—	8	4	no	—
191	Q₈×C₆₇	#124	1	2	yes	—
192	D₄×C₆₇	—	2	4	no	—
193	D₄×C₆₇	—	2	1	no	—
194	D₄×C₆₇	—	4	1	no	—
195	D₄×C₆₇	—	4	1	no	—
196	D₄×C₆₇	—	4	2	no	—
197	D₄×C₆₇	#124	1	2	yes	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	4	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	8	2	no	—
212	C₂×C₄×C₆₇	—	8	4	no	—
213	C₂×C₄×C₆₇	#120	1	2	yes	—
214	C₂×C₄×C₆₇	#120	1	2	yes	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
218	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
219	C₈×C₆₇	—	4	1	no	—
220	C₈×C₆₇	—	4	1	no	—
221	C₈×C₆₇	—	4	2	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	8	4	no	—
226	C₂²×C₆₇	—	8	1	no	—
227	C₂²×C₆₇	—	8	2	no	—
228	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
229	C₂²×C₆₇	#548	1	2	yes	—
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	2	no	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	4	4	no	—
234	C₄×C₆₇	—	4	4	no	—
235	C₄×C₆₇	—	4	2	no	—
236	C₄×C₆₇	—	4	2	no	—
237	C₄×C₆₇	—	4	2	no	—
238	C₄×C₆₇	—	8	1	no	—
239	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
240	C₄×C₆₇	#536	1	1	yes	characteristic
241	C₄×C₆₇	#548	1	2	yes	—
242	#25925	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₆₇	—	2	2	no	—
244	C₂×C₆₇	—	4	2	no	—
245	C₂×C₆₇	—	16	1	no	—
246	C₂×C₆₇	#2436	1	1	yes	characteristic
247	C₂×C₆₇	#3018	1	2	yes	—
248	#3504	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#3140	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#3040	C₂×C₆₇	1	1	yes	characteristic
251	#2861	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#2028	C₂×C₆₇	1	2	yes	complemented
253	#2027	C₂×C₆₇	1	2	yes	complemented
254	#1998	C₂×C₆₇	1	1	yes	characteristic, complemented
255	#1994	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#1883	C₂×C₆₇	1	2	yes	—
257	#1882	C₂×C₆₇	1	2	yes	complemented
258	C₄×#104	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₆₇	#25925	1	1	yes	characteristic, direct product factor, Sylow
260	C₂×#137	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
262	#557	—	2	2	no	—
263	#554	—	2	2	no	—
264	#554	—	2	4	no	—
265	#544	—	2	2	no	—
266	#544	C₂²×C₆₇	1	2	yes	—
267	#533	—	2	1	no	—
268	#533	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×#125	—	2	1	no	—
270	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
272	#520	C₂²×C₆₇	1	2	yes	—
273	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
274	#489	—	2	2	no	—
275	#488	—	2	1	no	—
276	#488	—	2	1	no	—
277	#464	—	2	2	no	—
278	#463	—	2	2	no	—
279	#423	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#104	C₂²×C₆₇	1	2	yes	—
281	#419	C₂²×C₆₇	1	2	yes	—
282	#419	C₂²×C₆₇	1	1	yes	characteristic
283	#418	C₂²×C₆₇	1	2	yes	—
284	#418	C₂²×C₆₇	1	1	yes	characteristic
285	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
286	#391	C₂²×C₆₇	1	2	yes	—
287	#390	C₂²×C₆₇	1	2	yes	—
288	#388	—	2	4	no	—
289	C₄³	C₂²×C₆₇	1	1	yes	characteristic
290	#343	C₂²×C₆₇	1	2	yes	—
291	#343	C₂²×C₆₇	1	2	yes	—
292	#339	C₂²×C₆₇	1	2	yes	—
293	#339	C₂²×C₆₇	1	2	yes	—
294	#339	C₂²×C₆₇	1	2	yes	—
295	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
296	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
298	#137	—	2	4	no	—
299	AGL(1,ℤ/8)	—	2	4	no	—
300	C₂×Q₁₆	—	2	1	no	—
301	C₂×SD₁₆	—	2	1	no	—
302	C₂×SD₁₆	—	2	1	no	—
303	C₂×D₈	—	2	1	no	—
304	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
305	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
306	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
307	C₄⋊Q₈	—	2	4	no	—
308	C₄⋊Q₈	—	2	2	no	—
309	C₄⋊Q₈	—	2	2	no	—
310	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
311	#125	—	2	4	no	—
312	#125	—	2	4	no	—
313	#125	—	2	2	no	—
314	#125	—	2	2	no	—
315	#125	—	2	1	no	—
316	#125	—	2	1	no	—
317	#125	—	2	2	no	—
318	#125	D₄×C₆₇	1	1	yes	characteristic
319	#125	D₄×C₆₇	1	1	yes	characteristic
320	#124	—	2	2	no	—
321	#124	—	2	1	no	—
322	#124	—	2	1	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	4	no	—
325	C₂²⋊Q₈	—	2	4	no	—
326	C₂²⋊Q₈	—	2	2	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₄×Q₈	—	2	4	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	D₄×C₆₇	1	2	yes	—
332	C₄×D₄	—	2	2	no	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	—	2	4	no	—
335	C₄×D₄	D₄×C₆₇	1	2	yes	—
336	#117	—	2	2	no	—
337	#117	—	2	1	no	—
338	#117	C₂³×C₆₇	1	1	yes	characteristic
339	#117	C₂³×C₆₇	1	2	yes	—
340	C₂×C₄⋊C₄	—	2	4	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₂²⋊C₄	—	2	4	no	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	—	2	2	no	—
348	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
349	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
350	C₄⋊C₈	—	2	2	no	—
351	#104	—	2	4	no	—
352	#103	—	2	2	no	—
353	#103	—	2	2	no	—
354	#103	—	2	2	no	—
355	#102	—	2	2	no	—
356	#102	—	2	2	no	—
357	#102	—	2	2	no	—
358	#97	—	2	1	no	—
359	C₄×C₈	—	2	1	no	—
360	#95	—	2	4	no	—
361	C₂⁴	—	2	1	no	—
362	#41	—	2	2	no	—
363	#41	#41×C₆₇	1	4	yes	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
383	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
384	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
385	Q₁₆	—	4	2	no	—
386	SD₁₆	—	4	2	no	—
387	SD₁₆	—	4	2	no	—
388	D₈	—	4	2	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	—	2	2	no	—
391	M₁₆	—	2	2	no	—
392	M₁₆	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
409	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
410	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	1	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	4	8	no	—
416	C₂²⋊C₄	—	4	2	no	—
417	C₂²⋊C₄	—	4	2	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	4	4	no	—
425	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
426	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
427	C₄²	C₂×D₄×C₆₇	1	2	yes	—
428	C₂³	—	2	1	no	—
429	C₂³	—	4	4	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	2	no	—
432	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
433	Q₈	—	2	1	no	—
434	Q₈	—	4	1	no	—
435	Q₈	—	4	1	no	—
436	Q₈	—	4	2	no	—
437	Q₈	—	8	2	no	—
438	Q₈	—	8	4	no	—
439	Q₈	#124×C₆₇	1	2	yes	—
440	D₄	—	2	4	no	—
441	D₄	—	2	1	no	—
442	D₄	—	4	1	no	—
443	D₄	—	4	1	no	—
444	D₄	—	4	2	no	—
445	D₄	#124×C₆₇	1	2	yes	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	8	2	no	—
460	C₂×C₄	—	8	4	no	—
461	C₂×C₄	#120×C₆₇	1	2	yes	—
462	C₂×C₄	#120×C₆₇	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
466	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
467	C₈	—	4	1	no	—
468	C₈	—	4	1	no	—
469	C₈	—	4	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	8	4	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	1	no	—
476	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
477	C₂²	#548×C₆₇	1	2	yes	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	4	no	—
482	C₄	—	4	4	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	1	no	—
487	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
488	C₄	#536×C₆₇	1	1	yes	characteristic
489	C₄	#548×C₆₇	1	2	yes	—
490	C₂	—	2	2	no	—
491	C₂	—	4	2	no	—
492	C₂	—	16	1	no	—
493	C₂	C₆₇×#2436	1	1	yes	characteristic
494	C₂	C₆₇×#3018	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₂²
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
1. C₆₇×#3504	C₂	semidirect
2. C₆₇×#3140	C₂	semidirect
4. C₆₇×#2861	C₂	semidirect
5. C₆₇×#2028	C₂	semidirect
6. C₆₇×#2027	C₂	semidirect
7. C₆₇×#1998	C₂	semidirect
8. C₆₇×#1994	C₂	semidirect
10. C₆₇×#1882	C₂	semidirect
11. C₄×#104×C₆₇	C₂	semidirect
242. #25925	C₆₇	direct (N × H)
248. #3504	C₂×C₆₇	semidirect
249. #3140	C₂×C₆₇	semidirect
251. #2861	C₂×C₆₇	semidirect
252. #2028	C₂×C₆₇	semidirect
253. #2027	C₂×C₆₇	semidirect
254. #1998	C₂×C₆₇	semidirect
255. #1994	C₂×C₆₇	semidirect
257. #1882	C₂×C₆₇	semidirect
258. C₄×#104	C₂×C₆₇	semidirect
259. C₆₇	#25925	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×#3504	2
C₆₇×#3140	2
C₆₇×#3040	2
C₆₇×#2861	2
C₆₇×#2028	2
C₆₇×#2027	2
C₆₇×#1998	2
C₆₇×#1994	2
C₆₇×#1883	2
C₆₇×#1882	2
C₄×#104×C₆₇	2
#25925	67

Subgroups — C67×#25925 #30705176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25925 #30705176