Subgroups — C₆₇×#33193 #30712444

Order 17152.

1032 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#496×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#471×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3040	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3038	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1978	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1977	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1967	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1965	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1849	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	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#131×C₆₇	C₂²	1	2	yes	—
18	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
21	#496×C₆₇	—	2	1	no	—
22	#496×C₆₇	—	2	1	no	—
23	#496×C₆₇	—	2	2	no	—
24	#472×C₆₇	—	2	8	no	—
25	#471×C₆₇	—	2	8	no	—
26	#464×C₆₇	—	2	2	no	—
27	#463×C₆₇	—	2	2	no	—
28	#460×C₆₇	—	2	2	no	—
29	#459×C₆₇	—	2	2	no	—
30	C₂×#108×C₆₇	C₂²	1	2	yes	—
31	C₂×#108×C₆₇	C₂²	1	2	yes	—
32	#423×C₆₇	C₂²	1	1	yes	characteristic
33	#423×C₆₇	C₂²	1	1	yes	characteristic
34	#419×C₆₇	C₂²	1	1	yes	characteristic
35	#418×C₆₇	C₂²	1	2	yes	—
36	#418×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#101×C₆₇	C₂²	1	2	yes	—
38	C₂×#100×C₆₇	C₂²	1	2	yes	—
39	C₂×#100×C₆₇	C₂²	1	2	yes	—
40	C₂×#100×C₆₇	C₂²	1	2	yes	—
41	#405×C₆₇	—	2	2	no	—
42	#405×C₆₇	C₂²	1	2	yes	—
43	C₄×M₁₆×C₆₇	—	2	1	no	—
44	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#97×C₆₇	—	2	1	no	—
46	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
52	#137×C₆₇	—	2	4	no	—
53	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
54	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
55	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
56	C₂×Q₁₆×C₆₇	—	2	1	no	—
57	C₂×SD₁₆×C₆₇	—	2	2	no	—
58	C₂×SD₁₆×C₆₇	—	2	1	no	—
59	C₂×SD₁₆×C₆₇	—	2	1	no	—
60	C₂×SD₁₆×C₆₇	—	2	4	no	—
61	C₂×SD₁₆×C₆₇	—	2	2	no	—
62	C₂×SD₁₆×C₆₇	—	2	2	no	—
63	C₂×SD₁₆×C₆₇	—	2	4	no	—
64	C₂×D₈×C₆₇	—	2	2	no	—
65	C₂×D₈×C₆₇	—	2	1	no	—
66	C₂×D₈×C₆₇	—	2	4	no	—
67	C₂×D₈×C₆₇	—	2	2	no	—
68	C₂×D₈×C₆₇	—	2	2	no	—
69	C₂×D₈×C₆₇	—	2	4	no	—
70	#131×C₆₇	—	2	4	no	—
71	#131×C₆₇	D₄	1	8	yes	complemented
72	C₂×M₁₆×C₆₇	—	2	2	no	—
73	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	—	2	2	no	—
78	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
79	#127×C₆₇	—	2	2	no	—
80	#127×C₆₇	—	2	1	no	—
81	#127×C₆₇	—	2	1	no	—
82	#124×C₆₇	—	2	2	no	—
83	#124×C₆₇	—	2	1	no	—
84	#124×C₆₇	—	2	1	no	—
85	C₄×Q₈×C₆₇	D₄	1	2	yes	—
86	C₄×D₄×C₆₇	—	2	2	no	—
87	C₄×D₄×C₆₇	D₄	1	2	yes	—
88	#117×C₆₇	—	2	1	no	—
89	#117×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	#108×C₆₇	—	2	4	no	—
94	#108×C₆₇	—	2	4	no	—
95	C₄⋊C₈×C₆₇	—	2	2	no	—
96	C₄⋊C₈×C₆₇	—	2	2	no	—
97	#103×C₆₇	—	2	2	no	—
98	#102×C₆₇	—	2	4	no	—
99	#102×C₆₇	—	2	2	no	—
100	#101×C₆₇	—	2	4	no	—
101	#100×C₆₇	—	2	4	no	—
102	#100×C₆₇	—	2	4	no	—
103	#100×C₆₇	—	2	4	no	—
104	#97×C₆₇	—	2	2	no	—
105	#97×C₆₇	—	2	1	no	—
106	#97×C₆₇	D₄	1	1	yes	characteristic
107	#97×C₆₇	D₄	1	1	yes	characteristic
108	#97×C₆₇	D₄	1	2	yes	—
109	C₄×C₈×C₆₇	—	2	2	no	—
110	C₄×C₈×C₆₇	—	2	1	no	—
111	C₂⁴×C₆₇	—	2	1	no	—
112	C₂⁴×C₆₇	—	2	2	no	—
113	#41×C₆₇	—	2	2	no	—
114	#41×C₆₇	C₂×D₄	1	4	yes	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	1	no	—
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×D₄×C₆₇	—	2	1	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	1	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	—	4	2	no	—
132	C₂×D₄×C₆₇	—	4	2	no	—
133	C₂×D₄×C₆₇	—	8	4	no	—
134	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	#41	1	2	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₆₇	—	4	2	no	—
141	SD₁₆×C₆₇	—	4	4	no	—
142	SD₁₆×C₆₇	—	4	2	no	—
143	SD₁₆×C₆₇	—	4	2	no	—
144	SD₁₆×C₆₇	—	4	4	no	—
145	SD₁₆×C₆₇	—	4	4	no	—
146	D₈×C₆₇	—	4	4	no	—
147	D₈×C₆₇	—	4	2	no	—
148	D₈×C₆₇	—	4	4	no	—
149	D₈×C₆₇	—	4	4	no	—
150	M₁₆×C₆₇	—	2	2	no	—
151	M₁₆×C₆₇	—	2	8	no	—
152	M₁₆×C₆₇	—	2	2	no	—
153	M₁₆×C₆₇	—	2	2	no	—
154	M₁₆×C₆₇	—	2	2	no	—
155	M₁₆×C₆₇	—	2	2	no	—
156	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	8	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	1	no	—
161	C₂×C₈×C₆₇	—	2	1	no	—
162	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
163	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
164	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₆₇	#41	1	4	yes	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂²⋊C₄×C₆₇	—	2	1	no	—
169	C₂²⋊C₄×C₆₇	—	2	2	no	—
170	C₂²⋊C₄×C₆₇	—	4	2	no	—
171	C₂²⋊C₄×C₆₇	—	4	2	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	1	no	—
174	C₄²×C₆₇	C₂×D₄	1	2	yes	—
175	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	—	4	1	no	—
179	C₂³×C₆₇	—	4	2	no	—
180	C₂³×C₆₇	—	4	2	no	—
181	C₂³×C₆₇	—	4	2	no	—
182	C₂³×C₆₇	—	4	2	no	—
183	C₂³×C₆₇	—	8	4	no	—
184	C₂³×C₆₇	—	8	2	no	—
185	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
186	Q₈×C₆₇	—	2	1	no	—
187	Q₈×C₆₇	—	4	2	no	—
188	Q₈×C₆₇	—	4	1	no	—
189	Q₈×C₆₇	—	4	1	no	—
190	Q₈×C₆₇	#127	1	2	yes	—
191	D₄×C₆₇	—	2	4	no	—
192	D₄×C₆₇	—	2	1	no	—
193	D₄×C₆₇	—	4	2	no	—
194	D₄×C₆₇	—	4	2	no	—
195	D₄×C₆₇	—	4	2	no	—
196	D₄×C₆₇	—	4	1	no	—
197	D₄×C₆₇	—	4	2	no	—
198	D₄×C₆₇	—	4	1	no	—
199	D₄×C₆₇	—	4	2	no	—
200	D₄×C₆₇	—	8	4	no	—
201	D₄×C₆₇	—	8	4	no	—
202	D₄×C₆₇	#127	1	2	yes	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	8	2	no	—
210	C₂×C₄×C₆₇	#121	1	2	yes	—
211	C₂×C₄×C₆₇	#121	1	2	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
216	C₈×C₆₇	—	2	4	no	—
217	C₈×C₆₇	—	4	2	no	—
218	C₈×C₆₇	—	4	1	no	—
219	C₈×C₆₇	—	4	1	no	—
220	C₈×C₆₇	#121	1	4	yes	—
221	C₈×C₆₇	#121	1	4	yes	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	8	1	no	—
226	C₂²×C₆₇	—	8	2	no	—
227	C₂²×C₆₇	—	8	4	no	—
228	C₂²×C₆₇	—	8	2	no	—
229	C₂²×C₆₇	—	16	2	no	—
230	C₂²×C₆₇	—	16	2	no	—
231	C₂²×C₆₇	—	16	2	no	—
232	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
233	C₂²×C₆₇	#532	1	2	yes	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	2	2	no	—
237	C₄×C₆₇	—	4	2	no	—
238	C₄×C₆₇	—	8	1	no	—
239	C₄×C₆₇	C₂×#121	1	2	yes	—
240	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
241	C₄×C₆₇	#532	1	1	yes	characteristic
242	#33193	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	2	no	—
246	C₂×C₆₇	—	16	1	no	—
247	C₂×C₆₇	#2856	1	1	yes	characteristic
248	C₂×C₆₇	#3068	1	2	yes	—
249	C₂×#496	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₂×#472	C₂×C₆₇	1	2	yes	complemented
251	C₂×#471	C₂×C₆₇	1	2	yes	complemented
252	#3040	C₂×C₆₇	1	1	yes	characteristic, complemented
253	#3038	C₂×C₆₇	1	1	yes	characteristic, complemented
254	#2875	C₂×C₆₇	1	1	yes	characteristic, complemented
255	#1978	C₂×C₆₇	1	2	yes	complemented
256	#1977	C₂×C₆₇	1	2	yes	complemented
257	#1967	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#1965	C₂×C₆₇	1	1	yes	characteristic, complemented
259	#1849	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₆₇	#33193	1	1	yes	characteristic, direct product factor, Sylow
261	C₂×#137	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
263	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
264	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
265	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
266	C₂×#131	C₂²×C₆₇	1	2	yes	—
267	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
269	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
270	#496	—	2	1	no	—
271	#496	—	2	1	no	—
272	#496	—	2	2	no	—
273	#472	—	2	8	no	—
274	#471	—	2	8	no	—
275	#464	—	2	2	no	—
276	#463	—	2	2	no	—
277	#460	—	2	2	no	—
278	#459	—	2	2	no	—
279	C₂×#108	C₂²×C₆₇	1	2	yes	—
280	C₂×#108	C₂²×C₆₇	1	2	yes	—
281	#423	C₂²×C₆₇	1	1	yes	characteristic
282	#423	C₂²×C₆₇	1	1	yes	characteristic
283	#419	C₂²×C₆₇	1	1	yes	characteristic
284	#418	C₂²×C₆₇	1	2	yes	—
285	#418	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×#101	C₂²×C₆₇	1	2	yes	—
287	C₂×#100	C₂²×C₆₇	1	2	yes	—
288	C₂×#100	C₂²×C₆₇	1	2	yes	—
289	C₂×#100	C₂²×C₆₇	1	2	yes	—
290	#405	—	2	2	no	—
291	#405	C₂²×C₆₇	1	2	yes	—
292	C₄×M₁₆	—	2	1	no	—
293	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×#97	—	2	1	no	—
295	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
297	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
298	C₂²×D₄	—	2	2	no	—
299	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
300	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
301	#137	—	2	4	no	—
302	AGL(1,ℤ/8)	—	2	4	no	—
303	AGL(1,ℤ/8)	—	2	4	no	—
304	AGL(1,ℤ/8)	—	2	4	no	—
305	C₂×Q₁₆	—	2	1	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	1	no	—
308	C₂×SD₁₆	—	2	1	no	—
309	C₂×SD₁₆	—	2	4	no	—
310	C₂×SD₁₆	—	2	2	no	—
311	C₂×SD₁₆	—	2	2	no	—
312	C₂×SD₁₆	—	2	4	no	—
313	C₂×D₈	—	2	2	no	—
314	C₂×D₈	—	2	1	no	—
315	C₂×D₈	—	2	4	no	—
316	C₂×D₈	—	2	2	no	—
317	C₂×D₈	—	2	2	no	—
318	C₂×D₈	—	2	4	no	—
319	#131	—	2	4	no	—
320	#131	D₄×C₆₇	1	8	yes	complemented
321	C₂×M₁₆	—	2	2	no	—
322	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
323	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
324	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
325	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
326	C₂²×C₈	—	2	2	no	—
327	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
328	#127	—	2	2	no	—
329	#127	—	2	1	no	—
330	#127	—	2	1	no	—
331	#124	—	2	2	no	—
332	#124	—	2	1	no	—
333	#124	—	2	1	no	—
334	C₄×Q₈	D₄×C₆₇	1	2	yes	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	D₄×C₆₇	1	2	yes	—
337	#117	—	2	1	no	—
338	#117	C₂³×C₆₇	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
342	#108	—	2	4	no	—
343	#108	—	2	4	no	—
344	C₄⋊C₈	—	2	2	no	—
345	C₄⋊C₈	—	2	2	no	—
346	#103	—	2	2	no	—
347	#102	—	2	4	no	—
348	#102	—	2	2	no	—
349	#101	—	2	4	no	—
350	#100	—	2	4	no	—
351	#100	—	2	4	no	—
352	#100	—	2	4	no	—
353	#97	—	2	2	no	—
354	#97	—	2	1	no	—
355	#97	D₄×C₆₇	1	1	yes	characteristic
356	#97	D₄×C₆₇	1	1	yes	characteristic
357	#97	D₄×C₆₇	1	2	yes	—
358	C₄×C₈	—	2	2	no	—
359	C₄×C₈	—	2	1	no	—
360	C₂⁴	—	2	1	no	—
361	C₂⁴	—	2	2	no	—
362	#41	—	2	2	no	—
363	#41	C₂×D₄×C₆₇	1	4	yes	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	8	4	no	—
383	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	#41×C₆₇	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
389	Q₁₆	—	4	2	no	—
390	SD₁₆	—	4	4	no	—
391	SD₁₆	—	4	2	no	—
392	SD₁₆	—	4	2	no	—
393	SD₁₆	—	4	4	no	—
394	SD₁₆	—	4	4	no	—
395	D₈	—	4	4	no	—
396	D₈	—	4	2	no	—
397	D₈	—	4	4	no	—
398	D₈	—	4	4	no	—
399	M₁₆	—	2	2	no	—
400	M₁₆	—	2	8	no	—
401	M₁₆	—	2	2	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	8	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
412	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
413	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
414	C₂×C₈	#41×C₆₇	1	4	yes	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	4	2	no	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	1	no	—
423	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
425	C₄²	C₂×D₄×C₆₇	1	2	yes	—
426	C₂³	—	2	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	—	8	4	no	—
433	C₂³	—	8	2	no	—
434	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
435	Q₈	—	2	1	no	—
436	Q₈	—	4	2	no	—
437	Q₈	—	4	1	no	—
438	Q₈	—	4	1	no	—
439	Q₈	#127×C₆₇	1	2	yes	—
440	D₄	—	2	4	no	—
441	D₄	—	2	1	no	—
442	D₄	—	4	2	no	—
443	D₄	—	4	2	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	1	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	1	no	—
448	D₄	—	4	2	no	—
449	D₄	—	8	4	no	—
450	D₄	—	8	4	no	—
451	D₄	#127×C₆₇	1	2	yes	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	8	2	no	—
459	C₂×C₄	#121×C₆₇	1	2	yes	—
460	C₂×C₄	#121×C₆₇	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
464	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
465	C₈	—	2	4	no	—
466	C₈	—	4	2	no	—
467	C₈	—	4	1	no	—
468	C₈	—	4	1	no	—
469	C₈	#121×C₆₇	1	4	yes	—
470	C₈	#121×C₆₇	1	4	yes	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	8	4	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	1	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	16	2	no	—
479	C₂²	—	16	2	no	—
480	C₂²	—	16	2	no	—
481	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
482	C₂²	#532×C₆₇	1	2	yes	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	2	no	—
487	C₄	—	8	1	no	—
488	C₄	C₂×#121×C₆₇	1	2	yes	—
489	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
490	C₄	#532×C₆₇	1	1	yes	characteristic
491	C₂	—	2	2	no	—
492	C₂	—	4	2	no	—
493	C₂	—	16	2	no	—
494	C₂	—	16	1	no	—
495	C₂	C₆₇×#2856	1	1	yes	characteristic
496	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 ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#496×C₆₇	C₂	semidirect
2. C₂×#472×C₆₇	C₂	semidirect
3. C₂×#471×C₆₇	C₂	semidirect
4. C₆₇×#3040	C₂	semidirect
5. C₆₇×#3038	C₂	semidirect
6. C₆₇×#2875	C₂	semidirect
7. C₆₇×#1978	C₂	semidirect
8. C₆₇×#1977	C₂	semidirect
9. C₆₇×#1967	C₂	semidirect
10. C₆₇×#1965	C₂	semidirect
11. C₆₇×#1849	C₂	semidirect
71. #131×C₆₇	D₄	semidirect
242. #33193	C₆₇	direct (N × H)
249. C₂×#496	C₂×C₆₇	semidirect
250. C₂×#472	C₂×C₆₇	semidirect
251. C₂×#471	C₂×C₆₇	semidirect
252. #3040	C₂×C₆₇	semidirect
253. #3038	C₂×C₆₇	semidirect
254. #2875	C₂×C₆₇	semidirect
255. #1978	C₂×C₆₇	semidirect
256. #1977	C₂×C₆₇	semidirect
257. #1967	C₂×C₆₇	semidirect
258. #1965	C₂×C₆₇	semidirect
259. #1849	C₂×C₆₇	semidirect
260. C₆₇	#33193	direct (N × H)
320. #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×#496×C₆₇	2
C₂×#472×C₆₇	2
C₂×#471×C₆₇	2
C₆₇×#3040	2
C₆₇×#3038	2
C₆₇×#2875	2
C₆₇×#1978	2
C₆₇×#1977	2
C₆₇×#1967	2
C₆₇×#1965	2
C₆₇×#1849	2
#33193	67

Subgroups — C67×#33193 #30712444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33193 #30712444