Subgroups — C₆₇×#33722 #30712973

Order 17152.

826 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#509×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3156	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#3155	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2251	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2250	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2249	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2248	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₆₇	—	2	1	no	—
14	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
18	#536×C₆₇	—	2	1	no	—
19	#535×C₆₇	—	2	1	no	—
20	#510×C₆₇	—	2	4	no	—
21	#509×C₆₇	—	2	4	no	—
22	#508×C₆₇	—	2	4	no	—
23	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	complemented
24	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×SD₃₂×C₆₇	—	2	4	no	—
26	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
27	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
28	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
29	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	complemented
31	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
33	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#497×C₆₇	—	2	2	no	—
37	#496×C₆₇	—	2	2	no	—
38	#495×C₆₇	C₂²	1	2	yes	—
39	#495×C₆₇	C₂²	1	2	yes	complemented
40	#494×C₆₇	C₂²	1	2	yes	complemented
41	#493×C₆₇	C₂²	1	2	yes	—
42	#405×C₆₇	C₂²	1	1	yes	characteristic
43	#346×C₆₇	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
45	C₂×#41×C₆₇	—	2	1	no	—
46	C₂×#41×C₆₇	—	2	1	no	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	#137×C₆₇	—	4	4	no	—
50	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
51	#135×C₆₇	—	2	2	no	—
52	#135×C₆₇	—	2	2	no	—
53	#135×C₆₇	—	2	2	no	—
54	#135×C₆₇	—	2	2	no	—
55	C₂×Q₁₆×C₆₇	—	2	2	no	—
56	C₂×Q₁₆×C₆₇	—	2	2	no	—
57	C₂×Q₁₆×C₆₇	—	2	2	no	—
58	C₂×Q₁₆×C₆₇	—	2	1	no	—
59	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×SD₁₆×C₆₇	—	2	2	no	—
63	C₂×SD₁₆×C₆₇	—	2	2	no	—
64	C₂×SD₁₆×C₆₇	—	2	1	no	—
65	C₂×SD₁₆×C₆₇	—	2	1	no	—
66	C₂×D₈×C₆₇	—	2	2	no	—
67	C₂×D₈×C₆₇	—	2	2	no	—
68	C₂×D₈×C₆₇	—	2	2	no	—
69	C₂×D₈×C₆₇	—	2	1	no	—
70	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₆₇	C₂³	1	2	yes	—
72	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₇	—	2	2	no	—
76	#127×C₆₇	—	2	2	no	—
77	#124×C₆₇	—	2	2	no	—
78	#124×C₆₇	—	2	2	no	—
79	#123×C₆₇	—	4	2	no	—
80	C₂²⋊Q₈×C₆₇	—	4	2	no	—
81	#121×C₆₇	—	4	2	no	—
82	#120×C₆₇	—	4	2	no	—
83	#117×C₆₇	D₄	1	1	yes	characteristic
84	Q₃₂×C₆₇	—	2	2	no	—
85	Q₃₂×C₆₇	—	2	4	no	—
86	SD₃₂×C₆₇	—	2	4	no	—
87	SD₃₂×C₆₇	—	2	4	no	—
88	SD₃₂×C₆₇	—	2	4	no	—
89	SD₃₂×C₆₇	—	2	2	no	—
90	SD₃₂×C₆₇	—	2	2	no	—
91	SD₃₂×C₆₇	—	2	4	no	—
92	D₁₆×C₆₇	—	2	4	no	—
93	D₁₆×C₆₇	—	2	2	no	—
94	M₃₂×C₆₇	D₄	1	4	yes	complemented
95	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
96	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
99	#97×C₆₇	D₄	1	2	yes	—
100	C₄×C₈×C₆₇	C₂³	1	2	yes	—
101	C₂⁴×C₆₇	—	4	1	no	—
102	#41×C₆₇	—	4	2	no	—
103	#41×C₆₇	—	4	2	no	—
104	#41×C₆₇	—	4	2	no	—
105	#41×C₆₇	—	4	2	no	—
106	C₂×Q₈×C₆₇	—	2	1	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	1	no	—
109	C₂×Q₈×C₆₇	—	4	1	no	—
110	C₂×Q₈×C₆₇	—	4	2	no	—
111	C₂×Q₈×C₆₇	—	4	2	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	—	2	1	no	—
115	C₂×D₄×C₆₇	—	4	1	no	—
116	C₂×D₄×C₆₇	—	4	1	no	—
117	C₂×D₄×C₆₇	—	4	2	no	—
118	C₂×D₄×C₆₇	—	4	2	no	—
119	C₂×D₄×C₆₇	—	4	1	no	—
120	C₂×D₄×C₆₇	—	4	4	no	—
121	C₂²×C₄×C₆₇	—	4	1	no	—
122	C₂²×C₄×C₆₇	—	4	1	no	—
123	C₂²×C₄×C₆₇	—	4	1	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	Q₁₆×C₆₇	—	2	2	no	—
126	Q₁₆×C₆₇	—	2	1	no	—
127	Q₁₆×C₆₇	—	2	2	no	—
128	Q₁₆×C₆₇	—	2	1	no	—
129	Q₁₆×C₆₇	—	2	1	no	—
130	Q₁₆×C₆₇	—	2	1	no	—
131	Q₁₆×C₆₇	—	4	2	no	—
132	Q₁₆×C₆₇	—	4	4	no	—
133	SD₁₆×C₆₇	—	4	2	no	—
134	SD₁₆×C₆₇	—	4	2	no	—
135	SD₁₆×C₆₇	—	4	4	no	—
136	SD₁₆×C₆₇	—	4	4	no	—
137	D₈×C₆₇	—	2	1	no	—
138	D₈×C₆₇	—	2	2	no	—
139	D₈×C₆₇	—	2	1	no	—
140	D₈×C₆₇	—	2	1	no	—
141	D₈×C₆₇	—	2	2	no	—
142	D₈×C₆₇	—	2	1	no	—
143	D₈×C₆₇	—	4	2	no	—
144	D₈×C₆₇	—	4	4	no	—
145	M₁₆×C₆₇	D₈	1	4	yes	—
146	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
150	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
151	C₄⋊C₄×C₆₇	—	4	4	no	—
152	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
153	C₂²⋊C₄×C₆₇	—	4	4	no	—
154	C₂²⋊C₄×C₆₇	—	4	4	no	—
155	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
156	C₄²×C₆₇	C₂×D₄	1	2	yes	—
157	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
158	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
159	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
160	C₂³×C₆₇	—	4	1	no	—
161	C₂³×C₆₇	—	4	2	no	—
162	C₂³×C₆₇	—	4	1	no	—
163	C₂³×C₆₇	—	8	2	no	—
164	C₂³×C₆₇	—	16	2	no	—
165	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
166	Q₈×C₆₇	—	4	1	no	—
167	Q₈×C₆₇	—	4	1	no	—
168	Q₈×C₆₇	—	4	2	no	—
169	Q₈×C₆₇	—	4	1	no	—
170	Q₈×C₆₇	—	4	2	no	—
171	Q₈×C₆₇	—	4	1	no	—
172	Q₈×C₆₇	—	8	2	no	—
173	D₄×C₆₇	—	4	2	no	—
174	D₄×C₆₇	—	4	1	no	—
175	D₄×C₆₇	—	4	1	no	—
176	D₄×C₆₇	—	4	1	no	—
177	D₄×C₆₇	—	4	2	no	—
178	D₄×C₆₇	—	4	1	no	—
179	D₄×C₆₇	—	8	2	no	—
180	D₄×C₆₇	—	8	2	no	—
181	D₄×C₆₇	—	8	2	no	—
182	D₄×C₆₇	—	16	4	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	8	2	no	—
187	C₂×C₄×C₆₇	—	8	2	no	—
188	C₂×C₄×C₆₇	—	8	2	no	—
189	C₂×C₄×C₆₇	#127	1	2	yes	—
190	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
191	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
192	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₈×C₆₇	#127	1	2	yes	—
195	C₈×C₆₇	C₂×D₈	1	4	yes	—
196	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
197	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	8	1	no	—
200	C₂²×C₆₇	—	8	1	no	—
201	C₂²×C₆₇	—	8	2	no	—
202	C₂²×C₆₇	—	8	2	no	—
203	C₂²×C₆₇	—	8	4	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	16	4	no	—
206	C₂²×C₆₇	#493	1	2	yes	—
207	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
208	C₄×C₆₇	—	2	4	no	—
209	C₄×C₆₇	—	8	1	no	—
210	C₄×C₆₇	—	8	1	no	—
211	C₄×C₆₇	—	8	2	no	—
212	C₄×C₆₇	#493	1	2	yes	—
213	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
214	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
215	#33722	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂×C₆₇	—	2	2	no	—
217	C₂×C₆₇	—	16	1	no	—
218	C₂×C₆₇	—	16	1	no	—
219	C₂×C₆₇	—	16	2	no	—
220	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
221	C₂×C₆₇	#3416	1	2	yes	—
222	C₂×#510	C₂×C₆₇	1	1	yes	characteristic, complemented
223	C₂×#509	C₂×C₆₇	1	1	yes	characteristic, complemented
224	C₂×#508	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₂²×SD₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#3156	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#3155	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#2251	C₂×C₆₇	1	2	yes	complemented
229	#2250	C₂×C₆₇	1	2	yes	complemented
230	#2249	C₂×C₆₇	1	2	yes	complemented
231	#2248	C₂×C₆₇	1	2	yes	complemented
232	#2108	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₆₇	#33722	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×#137	—	2	1	no	—
235	C₂×AGL(1,ℤ/8)	—	2	1	no	—
236	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
238	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
239	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
240	#536	—	2	1	no	—
241	#535	—	2	1	no	—
242	#510	—	2	4	no	—
243	#509	—	2	4	no	—
244	#508	—	2	4	no	—
245	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
246	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	complemented
247	C₂×SD₃₂	—	2	4	no	—
248	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
249	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
250	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
251	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
252	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
253	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	complemented
254	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
255	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
256	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
257	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
258	#497	—	2	2	no	—
259	#496	—	2	2	no	—
260	#495	C₂²×C₆₇	1	2	yes	—
261	#495	C₂²×C₆₇	1	2	yes	complemented
262	#494	C₂²×C₆₇	1	2	yes	complemented
263	#493	C₂²×C₆₇	1	2	yes	—
264	#405	C₂²×C₆₇	1	1	yes	characteristic
265	#346	C₂²×C₆₇	1	2	yes	complemented
266	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
267	C₂×#41	—	2	1	no	—
268	C₂×#41	—	2	1	no	—
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×D₄	—	2	1	no	—
271	#137	—	4	4	no	—
272	AGL(1,ℤ/8)	—	4	4	no	—
273	#135	—	2	2	no	—
274	#135	—	2	2	no	—
275	#135	—	2	2	no	—
276	#135	—	2	2	no	—
277	C₂×Q₁₆	—	2	2	no	—
278	C₂×Q₁₆	—	2	2	no	—
279	C₂×Q₁₆	—	2	1	no	—
280	C₂×Q₁₆	—	2	2	no	—
281	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
282	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
283	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
284	C₂×SD₁₆	—	2	1	no	—
285	C₂×SD₁₆	—	2	1	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×SD₁₆	—	2	2	no	—
288	C₂×D₈	—	2	2	no	—
289	C₂×D₈	—	2	2	no	—
290	C₂×D₈	—	2	1	no	—
291	C₂×D₈	—	2	2	no	—
292	C₂×D₈	C₂³×C₆₇	1	2	yes	—
293	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄⋊Q₈	—	2	2	no	—
298	#127	—	2	2	no	—
299	#124	—	2	2	no	—
300	#124	—	2	2	no	—
301	#123	—	4	2	no	—
302	C₂²⋊Q₈	—	4	2	no	—
303	#121	—	4	2	no	—
304	#120	—	4	2	no	—
305	#117	D₄×C₆₇	1	1	yes	characteristic
306	Q₃₂	—	2	2	no	—
307	Q₃₂	—	2	4	no	—
308	SD₃₂	—	2	4	no	—
309	SD₃₂	—	2	4	no	—
310	SD₃₂	—	2	4	no	—
311	SD₃₂	—	2	2	no	—
312	SD₃₂	—	2	2	no	—
313	SD₃₂	—	2	4	no	—
314	D₁₆	—	2	4	no	—
315	D₁₆	—	2	2	no	—
316	M₃₂	D₄×C₆₇	1	4	yes	complemented
317	C₂×C₁₆	D₄×C₆₇	1	4	yes	complemented
318	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
319	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
320	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
321	#97	D₄×C₆₇	1	2	yes	—
322	C₄×C₈	C₂³×C₆₇	1	2	yes	—
323	C₂⁴	—	4	1	no	—
324	#41	—	4	2	no	—
325	#41	—	4	2	no	—
326	#41	—	4	2	no	—
327	#41	—	4	2	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	4	1	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×Q₈	—	4	2	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	4	1	no	—
338	C₂×D₄	—	4	1	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	4	1	no	—
342	C₂×D₄	—	4	4	no	—
343	C₂²×C₄	—	4	1	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
347	Q₁₆	—	2	2	no	—
348	Q₁₆	—	2	1	no	—
349	Q₁₆	—	2	2	no	—
350	Q₁₆	—	2	1	no	—
351	Q₁₆	—	2	1	no	—
352	Q₁₆	—	2	1	no	—
353	Q₁₆	—	4	2	no	—
354	Q₁₆	—	4	4	no	—
355	SD₁₆	—	4	2	no	—
356	SD₁₆	—	4	2	no	—
357	SD₁₆	—	4	4	no	—
358	SD₁₆	—	4	4	no	—
359	D₈	—	2	1	no	—
360	D₈	—	2	2	no	—
361	D₈	—	2	1	no	—
362	D₈	—	2	1	no	—
363	D₈	—	2	2	no	—
364	D₈	—	2	1	no	—
365	D₈	—	4	2	no	—
366	D₈	—	4	4	no	—
367	M₁₆	D₈×C₆₇	1	4	yes	—
368	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
370	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
371	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
372	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
373	C₄⋊C₄	—	4	4	no	—
374	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
378	C₄²	C₂×D₄×C₆₇	1	2	yes	—
379	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
380	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
381	C₁₆	C₂×D₄×C₆₇	1	4	yes	—
382	C₂³	—	4	1	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	8	2	no	—
386	C₂³	—	16	2	no	—
387	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
388	Q₈	—	4	1	no	—
389	Q₈	—	4	1	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	4	1	no	—
392	Q₈	—	4	2	no	—
393	Q₈	—	4	1	no	—
394	Q₈	—	8	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	1	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	1	no	—
401	D₄	—	8	2	no	—
402	D₄	—	8	2	no	—
403	D₄	—	8	2	no	—
404	D₄	—	16	4	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	8	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	8	2	no	—
411	C₂×C₄	#127×C₆₇	1	2	yes	—
412	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
413	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
416	C₈	#127×C₆₇	1	2	yes	—
417	C₈	C₂×D₈×C₆₇	1	4	yes	—
418	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
419	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
420	C₂²	—	2	2	no	—
421	C₂²	—	8	1	no	—
422	C₂²	—	8	1	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	4	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	16	4	no	—
428	C₂²	#493×C₆₇	1	2	yes	—
429	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
430	C₄	—	2	4	no	—
431	C₄	—	8	1	no	—
432	C₄	—	8	1	no	—
433	C₄	—	8	2	no	—
434	C₄	#493×C₆₇	1	2	yes	—
435	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
436	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
437	C₂	—	2	2	no	—
438	C₂	—	16	2	no	—
439	C₂	—	16	1	no	—
440	C₂	—	16	1	no	—
441	C₂	C₂×#493×C₆₇	1	1	yes	characteristic
442	C₂	C₆₇×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #117×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#510×C₆₇	C₂	semidirect
2. C₂×#509×C₆₇	C₂	semidirect
3. C₂×#508×C₆₇	C₂	semidirect
4. C₂²×SD₃₂×C₆₇	C₂	semidirect
5. C₆₇×#3156	C₂	semidirect
6. C₆₇×#3155	C₂	semidirect
7. C₆₇×#2251	C₂	semidirect
8. C₆₇×#2250	C₂	semidirect
9. C₆₇×#2249	C₂	semidirect
10. C₆₇×#2248	C₂	semidirect
11. C₆₇×#2108	C₂	semidirect
23. C₂×Q₃₂×C₆₇	C₂²	semidirect
24. C₂×Q₃₂×C₆₇	C₂²	semidirect
29. C₂×SD₃₂×C₆₇	C₂²	semidirect
30. C₂×SD₃₂×C₆₇	C₂²	semidirect
31. C₂×SD₃₂×C₆₇	C₂²	semidirect
33. C₂×D₁₆×C₆₇	C₂²	semidirect
39. #495×C₆₇	C₂²	semidirect
40. #494×C₆₇	C₂²	semidirect
43. #346×C₆₇	C₂²	semidirect
44. C₄×C₁₆×C₆₇	C₂²	semidirect
94. M₃₂×C₆₇	D₄	semidirect
95. C₂×C₁₆×C₆₇	D₄	semidirect
215. #33722	C₆₇	direct (N × H)
222. C₂×#510	C₂×C₆₇	semidirect
223. C₂×#509	C₂×C₆₇	semidirect
224. C₂×#508	C₂×C₆₇	semidirect
225. C₂²×SD₃₂	C₂×C₆₇	semidirect
226. #3156	C₂×C₆₇	semidirect
227. #3155	C₂×C₆₇	semidirect
228. #2251	C₂×C₆₇	semidirect
229. #2250	C₂×C₆₇	semidirect
230. #2249	C₂×C₆₇	semidirect
231. #2248	C₂×C₆₇	semidirect
232. #2108	C₂×C₆₇	semidirect
233. C₆₇	#33722	direct (N × H)
245. C₂×Q₃₂	C₂²×C₆₇	semidirect
246. C₂×Q₃₂	C₂²×C₆₇	semidirect
251. C₂×SD₃₂	C₂²×C₆₇	semidirect
252. C₂×SD₃₂	C₂²×C₆₇	semidirect
253. C₂×SD₃₂	C₂²×C₆₇	semidirect
255. C₂×D₁₆	C₂²×C₆₇	semidirect
261. #495	C₂²×C₆₇	semidirect
262. #494	C₂²×C₆₇	semidirect
265. #346	C₂²×C₆₇	semidirect
266. C₄×C₁₆	C₂²×C₆₇	semidirect
316. M₃₂	D₄×C₆₇	semidirect
317. C₂×C₁₆	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×#510×C₆₇	2
C₂×#509×C₆₇	2
C₂×#508×C₆₇	2
C₂²×SD₃₂×C₆₇	2
C₆₇×#3156	2
C₆₇×#3155	2
C₆₇×#2251	2
C₆₇×#2250	2
C₆₇×#2249	2
C₆₇×#2248	2
C₆₇×#2108	2
#33722	67

Subgroups — C67×#33722 #30712973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33722 #30712973