Subgroups — C₆₇×#33194 #30712445

Order 17152.

980 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#497×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#473×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#471×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#3040	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#3039	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1979	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×#1966	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₂²	1	2	yes	—
13	C₂×#137×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂×#131×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×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	#497×C₆₇	—	2	1	no	—
22	#497×C₆₇	—	2	1	no	—
23	#497×C₆₇	—	2	2	no	—
24	#473×C₆₇	—	2	8	no	—
25	#471×C₆₇	—	2	8	no	—
26	#464×C₆₇	—	2	2	no	—
27	#463×C₆₇	—	2	2	no	—
28	#462×C₆₇	—	2	2	no	—
29	#461×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	2	yes	—
35	#419×C₆₇	C₂²	1	1	yes	characteristic
36	#418×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#101×C₆₇	C₂²	1	2	yes	—
38	C₂×#101×C₆₇	C₂²	1	2	yes	—
39	C₂×#101×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×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
51	#137×C₆₇	—	2	4	no	—
52	#137×C₆₇	—	2	4	no	—
53	#137×C₆₇	—	2	4	no	—
54	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
55	C₂×Q₁₆×C₆₇	—	2	2	no	—
56	C₂×Q₁₆×C₆₇	—	2	1	no	—
57	C₂×Q₁₆×C₆₇	—	2	4	no	—
58	C₂×Q₁₆×C₆₇	—	2	2	no	—
59	C₂×Q₁₆×C₆₇	—	2	2	no	—
60	C₂×Q₁₆×C₆₇	—	2	4	no	—
61	C₂×SD₁₆×C₆₇	—	2	1	no	—
62	C₂×SD₁₆×C₆₇	—	2	2	no	—
63	C₂×SD₁₆×C₆₇	—	2	1	no	—
64	C₂×SD₁₆×C₆₇	—	2	4	no	—
65	C₂×SD₁₆×C₆₇	—	2	2	no	—
66	C₂×SD₁₆×C₆₇	—	2	2	no	—
67	C₂×SD₁₆×C₆₇	—	2	4	no	—
68	C₂×D₈×C₆₇	—	2	1	no	—
69	#131×C₆₇	—	2	4	no	—
70	#131×C₆₇	D₄	1	8	yes	—
71	C₂×M₁₆×C₆₇	—	2	2	no	—
72	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
74	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	—	2	2	no	—
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₆₇	—	2	2	no	—
79	C₄⋊Q₈×C₆₇	—	2	1	no	—
80	C₄⋊Q₈×C₆₇	—	2	1	no	—
81	#124×C₆₇	—	2	2	no	—
82	#124×C₆₇	—	2	1	no	—
83	#124×C₆₇	—	2	1	no	—
84	C₄×Q₈×C₆₇	D₄	1	2	yes	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	D₄	1	2	yes	—
87	#117×C₆₇	—	2	1	no	—
88	#117×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
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	4	no	—
98	#103×C₆₇	—	2	2	no	—
99	#102×C₆₇	—	2	2	no	—
100	#101×C₆₇	—	2	4	no	—
101	#101×C₆₇	—	2	4	no	—
102	#101×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	#41×C₆₇	—	2	2	no	—
113	#41×C₆₇	C₂×D₄	1	4	yes	—
114	C₂×Q₈×C₆₇	—	2	2	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	—	2	2	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	—	2	2	no	—
123	C₂×Q₈×C₆₇	—	2	2	no	—
124	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×D₄×C₆₇	—	2	1	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	1	no	—
128	C₂×D₄×C₆₇	—	2	1	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	#41	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₆₇	—	4	4	no	—
138	Q₁₆×C₆₇	—	4	2	no	—
139	Q₁₆×C₆₇	—	4	4	no	—
140	Q₁₆×C₆₇	—	4	4	no	—
141	SD₁₆×C₆₇	—	4	2	no	—
142	SD₁₆×C₆₇	—	4	4	no	—
143	SD₁₆×C₆₇	—	4	2	no	—
144	SD₁₆×C₆₇	—	4	4	no	—
145	SD₁₆×C₆₇	—	4	4	no	—
146	D₈×C₆₇	—	4	2	no	—
147	M₁₆×C₆₇	—	2	2	no	—
148	M₁₆×C₆₇	—	2	8	no	—
149	M₁₆×C₆₇	—	2	2	no	—
150	M₁₆×C₆₇	—	2	2	no	—
151	M₁₆×C₆₇	—	2	2	no	—
152	M₁₆×C₆₇	—	2	2	no	—
153	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	8	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
160	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₆₇	#41	1	4	yes	—
163	C₄⋊C₄×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	4	2	no	—
165	C₄⋊C₄×C₆₇	—	4	2	no	—
166	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²⋊C₄×C₆₇	—	2	1	no	—
168	C₂²⋊C₄×C₆₇	—	2	2	no	—
169	C₂²⋊C₄×C₆₇	—	4	2	no	—
170	C₂²⋊C₄×C₆₇	—	4	2	no	—
171	C₄²×C₆₇	—	2	2	no	—
172	C₄²×C₆₇	—	2	1	no	—
173	C₄²×C₆₇	C₂×D₄	1	2	yes	—
174	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂³×C₆₇	—	2	1	no	—
177	C₂³×C₆₇	—	4	1	no	—
178	C₂³×C₆₇	—	4	2	no	—
179	C₂³×C₆₇	—	8	2	no	—
180	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
181	Q₈×C₆₇	—	2	1	no	—
182	Q₈×C₆₇	—	4	2	no	—
183	Q₈×C₆₇	—	4	2	no	—
184	Q₈×C₆₇	—	4	2	no	—
185	Q₈×C₆₇	—	4	2	no	—
186	Q₈×C₆₇	—	4	1	no	—
187	Q₈×C₆₇	—	4	1	no	—
188	Q₈×C₆₇	—	4	2	no	—
189	Q₈×C₆₇	#127	1	2	yes	—
190	D₄×C₆₇	—	2	4	no	—
191	D₄×C₆₇	—	2	1	no	—
192	D₄×C₆₇	—	4	2	no	—
193	D₄×C₆₇	—	4	1	no	—
194	D₄×C₆₇	—	4	1	no	—
195	D₄×C₆₇	#127	1	2	yes	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	1	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	8	2	no	—
206	C₂×C₄×C₆₇	#121	1	2	yes	—
207	C₂×C₄×C₆₇	#121	1	2	yes	—
208	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
212	C₈×C₆₇	—	2	4	no	—
213	C₈×C₆₇	—	4	1	no	—
214	C₈×C₆₇	—	4	1	no	—
215	C₈×C₆₇	—	4	2	no	—
216	C₈×C₆₇	#121	1	4	yes	—
217	C₈×C₆₇	#121	1	4	yes	—
218	C₂²×C₆₇	—	2	2	no	—
219	C₂²×C₆₇	—	2	2	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	8	1	no	—
222	C₂²×C₆₇	—	8	2	no	—
223	C₂²×C₆₇	—	16	2	no	—
224	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
225	C₂²×C₆₇	#532	1	2	yes	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	4	2	no	—
230	C₄×C₆₇	—	8	1	no	—
231	C₄×C₆₇	—	8	2	no	—
232	C₄×C₆₇	C₂×#121	1	2	yes	—
233	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
234	C₄×C₆₇	#532	1	1	yes	characteristic
235	#33194	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂×C₆₇	—	2	2	no	—
237	C₂×C₆₇	—	4	2	no	—
238	C₂×C₆₇	—	16	1	no	—
239	C₂×C₆₇	#2856	1	1	yes	characteristic
240	C₂×C₆₇	#3069	1	2	yes	—
241	C₂×#497	C₂×C₆₇	1	1	yes	characteristic, complemented
242	C₂×#473	C₂×C₆₇	1	2	yes	complemented
243	C₂×#471	C₂×C₆₇	1	2	yes	—
244	#3040	C₂×C₆₇	1	1	yes	characteristic
245	#3039	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#2875	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#1979	C₂×C₆₇	1	2	yes	complemented
248	#1977	C₂×C₆₇	1	2	yes	complemented
249	#1967	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1966	C₂×C₆₇	1	1	yes	characteristic, complemented
251	#1849	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₆₇	#33194	1	1	yes	characteristic, direct product factor, Sylow
253	C₂×#137	C₂²×C₆₇	1	2	yes	—
254	C₂×#137	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
256	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
257	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
258	C₂×#131	C₂²×C₆₇	1	2	yes	—
259	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
261	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
262	#497	—	2	1	no	—
263	#497	—	2	1	no	—
264	#497	—	2	2	no	—
265	#473	—	2	8	no	—
266	#471	—	2	8	no	—
267	#464	—	2	2	no	—
268	#463	—	2	2	no	—
269	#462	—	2	2	no	—
270	#461	—	2	2	no	—
271	C₂×#108	C₂²×C₆₇	1	2	yes	—
272	C₂×#108	C₂²×C₆₇	1	2	yes	—
273	#423	C₂²×C₆₇	1	1	yes	characteristic
274	#423	C₂²×C₆₇	1	1	yes	characteristic
275	#419	C₂²×C₆₇	1	2	yes	—
276	#419	C₂²×C₆₇	1	1	yes	characteristic
277	#418	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×#101	C₂²×C₆₇	1	2	yes	—
279	C₂×#101	C₂²×C₆₇	1	2	yes	—
280	C₂×#101	C₂²×C₆₇	1	2	yes	—
281	C₂×#100	C₂²×C₆₇	1	2	yes	—
282	#405	—	2	2	no	—
283	#405	C₂²×C₆₇	1	2	yes	—
284	C₄×M₁₆	—	2	1	no	—
285	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×#97	—	2	1	no	—
287	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
289	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
291	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
292	#137	—	2	4	no	—
293	#137	—	2	4	no	—
294	#137	—	2	4	no	—
295	AGL(1,ℤ/8)	—	2	4	no	—
296	C₂×Q₁₆	—	2	2	no	—
297	C₂×Q₁₆	—	2	1	no	—
298	C₂×Q₁₆	—	2	4	no	—
299	C₂×Q₁₆	—	2	2	no	—
300	C₂×Q₁₆	—	2	2	no	—
301	C₂×Q₁₆	—	2	4	no	—
302	C₂×SD₁₆	—	2	1	no	—
303	C₂×SD₁₆	—	2	2	no	—
304	C₂×SD₁₆	—	2	1	no	—
305	C₂×SD₁₆	—	2	4	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×SD₁₆	—	2	4	no	—
309	C₂×D₈	—	2	1	no	—
310	#131	—	2	4	no	—
311	#131	D₄×C₆₇	1	8	yes	—
312	C₂×M₁₆	—	2	2	no	—
313	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
315	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
316	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
317	C₂²×C₈	—	2	2	no	—
318	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	—	2	1	no	—
321	C₄⋊Q₈	—	2	1	no	—
322	#124	—	2	2	no	—
323	#124	—	2	1	no	—
324	#124	—	2	1	no	—
325	C₄×Q₈	D₄×C₆₇	1	2	yes	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	D₄×C₆₇	1	2	yes	—
328	#117	—	2	1	no	—
329	#117	C₂³×C₆₇	1	1	yes	characteristic
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
334	#108	—	2	4	no	—
335	#108	—	2	4	no	—
336	C₄⋊C₈	—	2	2	no	—
337	C₄⋊C₈	—	2	2	no	—
338	#103	—	2	4	no	—
339	#103	—	2	2	no	—
340	#102	—	2	2	no	—
341	#101	—	2	4	no	—
342	#101	—	2	4	no	—
343	#101	—	2	4	no	—
344	#100	—	2	4	no	—
345	#97	—	2	2	no	—
346	#97	—	2	1	no	—
347	#97	D₄×C₆₇	1	1	yes	characteristic
348	#97	D₄×C₆₇	1	1	yes	characteristic
349	#97	D₄×C₆₇	1	2	yes	—
350	C₄×C₈	—	2	2	no	—
351	C₄×C₈	—	2	1	no	—
352	C₂⁴	—	2	1	no	—
353	#41	—	2	2	no	—
354	#41	C₂×D₄×C₆₇	1	4	yes	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	#41×C₆₇	1	2	yes	—
377	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
378	Q₁₆	—	4	4	no	—
379	Q₁₆	—	4	2	no	—
380	Q₁₆	—	4	4	no	—
381	Q₁₆	—	4	4	no	—
382	SD₁₆	—	4	2	no	—
383	SD₁₆	—	4	4	no	—
384	SD₁₆	—	4	2	no	—
385	SD₁₆	—	4	4	no	—
386	SD₁₆	—	4	4	no	—
387	D₈	—	4	2	no	—
388	M₁₆	—	2	2	no	—
389	M₁₆	—	2	8	no	—
390	M₁₆	—	2	2	no	—
391	M₁₆	—	2	2	no	—
392	M₁₆	—	2	2	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	8	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
401	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
402	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
403	C₂×C₈	#41×C₆₇	1	4	yes	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	—	4	2	no	—
407	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
408	C₂²⋊C₄	—	2	1	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	4	2	no	—
411	C₂²⋊C₄	—	4	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	1	no	—
414	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
415	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
416	C₄²	C₂×D₄×C₆₇	1	2	yes	—
417	C₂³	—	2	1	no	—
418	C₂³	—	4	1	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	8	2	no	—
421	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
422	Q₈	—	2	1	no	—
423	Q₈	—	4	2	no	—
424	Q₈	—	4	2	no	—
425	Q₈	—	4	2	no	—
426	Q₈	—	4	2	no	—
427	Q₈	—	4	1	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	2	no	—
430	Q₈	#127×C₆₇	1	2	yes	—
431	D₄	—	2	4	no	—
432	D₄	—	2	1	no	—
433	D₄	—	4	2	no	—
434	D₄	—	4	1	no	—
435	D₄	—	4	1	no	—
436	D₄	#127×C₆₇	1	2	yes	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	8	2	no	—
447	C₂×C₄	#121×C₆₇	1	2	yes	—
448	C₂×C₄	#121×C₆₇	1	2	yes	—
449	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
451	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
453	C₈	—	2	4	no	—
454	C₈	—	4	1	no	—
455	C₈	—	4	1	no	—
456	C₈	—	4	2	no	—
457	C₈	#121×C₆₇	1	4	yes	—
458	C₈	#121×C₆₇	1	4	yes	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	16	2	no	—
465	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
466	C₂²	#532×C₆₇	1	2	yes	—
467	C₄	—	2	2	no	—
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	4	2	no	—
471	C₄	—	8	1	no	—
472	C₄	—	8	2	no	—
473	C₄	C₂×#121×C₆₇	1	2	yes	—
474	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
475	C₄	#532×C₆₇	1	1	yes	characteristic
476	C₂	—	2	2	no	—
477	C₂	—	4	2	no	—
478	C₂	—	16	1	no	—
479	C₂	C₆₇×#2856	1	1	yes	characteristic
480	C₂	C₆₇×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
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. C₂×#497×C₆₇	C₂	semidirect
2. C₂×#473×C₆₇	C₂	semidirect
5. C₆₇×#3039	C₂	semidirect
6. C₆₇×#2875	C₂	semidirect
7. C₆₇×#1979	C₂	semidirect
8. C₆₇×#1977	C₂	semidirect
9. C₆₇×#1967	C₂	semidirect
10. C₆₇×#1966	C₂	semidirect
11. C₆₇×#1849	C₂	semidirect
235. #33194	C₆₇	direct (N × H)
241. C₂×#497	C₂×C₆₇	semidirect
242. C₂×#473	C₂×C₆₇	semidirect
245. #3039	C₂×C₆₇	semidirect
246. #2875	C₂×C₆₇	semidirect
247. #1979	C₂×C₆₇	semidirect
248. #1977	C₂×C₆₇	semidirect
249. #1967	C₂×C₆₇	semidirect
250. #1966	C₂×C₆₇	semidirect
251. #1849	C₂×C₆₇	semidirect
252. C₆₇	#33194	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#497×C₆₇	2
C₂×#473×C₆₇	2
C₂×#471×C₆₇	2
C₆₇×#3040	2
C₆₇×#3039	2
C₆₇×#2875	2
C₆₇×#1979	2
C₆₇×#1977	2
C₆₇×#1967	2
C₆₇×#1966	2
C₆₇×#1849	2
#33194	67

Subgroups — C67×#33194 #30712445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33194 #30712445