Subgroups — C₆₇×#22285 #30701536

Order 17152.

772 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3156	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2957	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1979	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1976	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1912	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1909	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1905	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1902	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1833	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₂²	1	2	yes	—
13	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#568×C₆₇	C₂²	1	2	yes	—
16	#536×C₆₇	C₂²	1	2	yes	—
17	#519×C₆₇	C₂²	1	2	yes	—
18	#497×C₆₇	C₄	1	4	yes	complemented
19	#495×C₆₇	C₄	1	2	yes	complemented
20	#494×C₆₇	C₄	1	2	yes	complemented
21	#444×C₆₇	—	2	2	no	—
22	#443×C₆₇	—	2	2	no	—
23	#441×C₆₇	—	2	2	no	—
24	C₄×Q₁₆×C₆₇	—	2	2	no	—
25	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
26	#427×C₆₇	C₂²	1	1	yes	characteristic
27	#421×C₆₇	C₂²	1	2	yes	—
28	#421×C₆₇	C₂²	1	2	yes	—
29	#419×C₆₇	C₂²	1	2	yes	—
30	#419×C₆₇	C₂²	1	2	yes	—
31	C₂×#101×C₆₇	C₂²	1	2	yes	—
32	#410×C₆₇	C₂²	1	2	yes	—
33	#405×C₆₇	C₂²	1	1	yes	characteristic
34	#405×C₆₇	C₂²	1	1	yes	characteristic
35	#405×C₆₇	C₂²	1	1	yes	characteristic
36	#344×C₆₇	C₂²	1	2	yes	—
37	#338×C₆₇	C₂²	1	1	yes	characteristic
38	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
39	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
40	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
41	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	#137×C₆₇	—	4	2	no	—
45	#137×C₆₇	—	4	2	no	—
46	#135×C₆₇	—	2	1	no	—
47	#135×C₆₇	—	2	1	no	—
48	#135×C₆₇	—	2	1	no	—
49	#135×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	—	2	1	no	—
51	C₂×Q₁₆×C₆₇	—	2	1	no	—
52	C₂×Q₁₆×C₆₇	—	2	1	no	—
53	C₂×Q₁₆×C₆₇	—	2	1	no	—
54	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
60	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	#131×C₆₇	—	2	4	no	—
62	#131×C₆₇	—	4	2	no	—
63	C₂×M₁₆×C₆₇	—	4	2	no	—
64	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
70	#124×C₆₇	C₂×C₄	1	4	yes	—
71	#123×C₆₇	—	4	2	no	—
72	C₂²⋊Q₈×C₆₇	—	4	2	no	—
73	C₄×Q₈×C₆₇	—	2	4	no	—
74	C₄×Q₈×C₆₇	—	4	2	no	—
75	#117×C₆₇	—	4	2	no	—
76	#117×C₆₇	C₂³	1	2	yes	—
77	#117×C₆₇	C₂³	1	1	yes	characteristic
78	#117×C₆₇	C₂³	1	1	yes	characteristic
79	#117×C₆₇	C₂³	1	1	yes	characteristic
80	#108×C₆₇	—	2	1	no	—
81	#108×C₆₇	—	2	1	no	—
82	#107×C₆₇	—	2	1	no	—
83	#106×C₆₇	—	2	1	no	—
84	#104×C₆₇	—	2	4	no	—
85	#104×C₆₇	—	4	2	no	—
86	#103×C₆₇	—	2	4	no	—
87	#103×C₆₇	—	4	2	no	—
88	#101×C₆₇	—	4	2	no	—
89	#99×C₆₇	—	4	2	no	—
90	#97×C₆₇	—	2	1	no	—
91	#97×C₆₇	C₂×C₄	1	2	yes	—
92	#97×C₆₇	D₄	1	2	yes	—
93	C₄×C₈×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
95	C₄×C₈×C₆₇	D₄	1	2	yes	—
96	#41×C₆₇	—	2	2	no	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
106	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
112	Q₁₆×C₆₇	—	2	2	no	—
113	Q₁₆×C₆₇	—	2	2	no	—
114	Q₁₆×C₆₇	—	2	2	no	—
115	Q₁₆×C₆₇	—	2	2	no	—
116	Q₁₆×C₆₇	—	2	1	no	—
117	Q₁₆×C₆₇	—	2	1	no	—
118	Q₁₆×C₆₇	—	4	4	no	—
119	SD₁₆×C₆₇	—	2	2	no	—
120	SD₁₆×C₆₇	—	2	2	no	—
121	SD₁₆×C₆₇	—	4	4	no	—
122	D₈×C₆₇	—	2	1	no	—
123	D₈×C₆₇	—	2	1	no	—
124	M₁₆×C₆₇	—	2	1	no	—
125	M₁₆×C₆₇	—	2	1	no	—
126	M₁₆×C₆₇	—	2	1	no	—
127	M₁₆×C₆₇	—	2	1	no	—
128	M₁₆×C₆₇	—	2	2	no	—
129	M₁₆×C₆₇	—	2	1	no	—
130	M₁₆×C₆₇	—	2	2	no	—
131	M₁₆×C₆₇	—	2	1	no	—
132	M₁₆×C₆₇	—	4	4	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	4	4	no	—
136	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
143	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
144	C₄⋊C₄×C₆₇	—	2	1	no	—
145	C₄⋊C₄×C₆₇	—	2	1	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	4	4	no	—
149	C₄⋊C₄×C₆₇	—	4	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	1	no	—
151	C₂²⋊C₄×C₆₇	—	2	1	no	—
152	C₂²⋊C₄×C₆₇	—	2	2	no	—
153	C₂²⋊C₄×C₆₇	—	2	1	no	—
154	C₂²⋊C₄×C₆₇	—	4	4	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	—	2	2	no	—
157	C₄²×C₆₇	—	4	4	no	—
158	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
159	C₄²×C₆₇	C₂×D₄	1	2	yes	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
162	Q₈×C₆₇	—	2	2	no	—
163	Q₈×C₆₇	—	2	4	no	—
164	Q₈×C₆₇	—	2	4	no	—
165	Q₈×C₆₇	—	2	2	no	—
166	Q₈×C₆₇	—	4	2	no	—
167	Q₈×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	2	2	no	—
169	D₄×C₆₇	—	2	2	no	—
170	D₄×C₆₇	—	4	2	no	—
171	D₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
188	C₈×C₆₇	—	2	2	no	—
189	C₈×C₆₇	—	2	2	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	—	4	1	no	—
192	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
193	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
194	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
195	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂²×C₆₇	—	4	1	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	#518	1	1	yes	characteristic
200	C₂²×C₆₇	#535	1	1	yes	characteristic
201	C₂²×C₆₇	#540	1	1	yes	characteristic
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	4	no	—
206	C₄×C₆₇	—	8	2	no	—
207	C₄×C₆₇	—	8	1	no	—
208	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
209	C₄×C₆₇	#518	1	1	yes	characteristic
210	C₄×C₆₇	#535	1	1	yes	characteristic
211	C₄×C₆₇	#540	1	1	yes	characteristic
212	#22285	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₆₇	—	2	1	no	—
214	C₂×C₆₇	—	2	1	no	—
215	C₂×C₆₇	—	2	1	no	—
216	C₂×C₆₇	—	8	2	no	—
217	C₂×C₆₇	#2328	1	1	yes	characteristic
218	#3156	C₂×C₆₇	1	1	yes	characteristic
219	#2957	C₂×C₆₇	1	1	yes	characteristic
220	#2950	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1979	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#1976	C₂×C₆₇	1	1	yes	characteristic
223	#1912	C₂×C₆₇	1	2	yes	complemented
224	#1909	C₂×C₆₇	1	2	yes	complemented
225	#1905	C₂×C₆₇	1	2	yes	complemented
226	#1902	C₂×C₆₇	1	2	yes	complemented
227	#1833	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1747	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	#22285	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×#137	C₂²×C₆₇	1	2	yes	—
231	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
232	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
233	#568	C₂²×C₆₇	1	2	yes	—
234	#536	C₂²×C₆₇	1	2	yes	—
235	#519	C₂²×C₆₇	1	2	yes	—
236	#497	C₄×C₆₇	1	4	yes	complemented
237	#495	C₄×C₆₇	1	2	yes	complemented
238	#494	C₄×C₆₇	1	2	yes	complemented
239	#444	—	2	2	no	—
240	#443	—	2	2	no	—
241	#441	—	2	2	no	—
242	C₄×Q₁₆	—	2	2	no	—
243	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
244	#427	C₂²×C₆₇	1	1	yes	characteristic
245	#421	C₂²×C₆₇	1	2	yes	—
246	#421	C₂²×C₆₇	1	2	yes	—
247	#419	C₂²×C₆₇	1	2	yes	—
248	#419	C₂²×C₆₇	1	2	yes	—
249	C₂×#101	C₂²×C₆₇	1	2	yes	—
250	#410	C₂²×C₆₇	1	2	yes	—
251	#405	C₂²×C₆₇	1	1	yes	characteristic
252	#405	C₂²×C₆₇	1	1	yes	characteristic
253	#405	C₂²×C₆₇	1	1	yes	characteristic
254	#344	C₂²×C₆₇	1	2	yes	—
255	#338	C₂²×C₆₇	1	1	yes	characteristic
256	H(3,ℤ/4)	C₂²×C₆₇	1	2	yes	—
257	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
258	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
259	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#41	C₂³×C₆₇	1	2	yes	—
261	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
262	#137	—	4	2	no	—
263	#137	—	4	2	no	—
264	#135	—	2	1	no	—
265	#135	—	2	1	no	—
266	#135	—	2	1	no	—
267	#135	—	2	1	no	—
268	C₂×Q₁₆	—	2	1	no	—
269	C₂×Q₁₆	—	2	1	no	—
270	C₂×Q₁₆	—	2	1	no	—
271	C₂×Q₁₆	—	2	1	no	—
272	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
274	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
276	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
277	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
278	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
279	#131	—	2	4	no	—
280	#131	—	4	2	no	—
281	C₂×M₁₆	—	4	2	no	—
282	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
283	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
284	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
285	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
287	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
288	#124	C₂×C₄×C₆₇	1	4	yes	—
289	#123	—	4	2	no	—
290	C₂²⋊Q₈	—	4	2	no	—
291	C₄×Q₈	—	2	4	no	—
292	C₄×Q₈	—	4	2	no	—
293	#117	—	4	2	no	—
294	#117	C₂³×C₆₇	1	2	yes	—
295	#117	C₂³×C₆₇	1	1	yes	characteristic
296	#117	C₂³×C₆₇	1	1	yes	characteristic
297	#117	C₂³×C₆₇	1	1	yes	characteristic
298	#108	—	2	1	no	—
299	#108	—	2	1	no	—
300	#107	—	2	1	no	—
301	#106	—	2	1	no	—
302	#104	—	2	4	no	—
303	#104	—	4	2	no	—
304	#103	—	2	4	no	—
305	#103	—	4	2	no	—
306	#101	—	4	2	no	—
307	#99	—	4	2	no	—
308	#97	—	2	1	no	—
309	#97	C₂×C₄×C₆₇	1	2	yes	—
310	#97	D₄×C₆₇	1	2	yes	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
313	C₄×C₈	D₄×C₆₇	1	2	yes	—
314	#41	—	2	2	no	—
315	#41	—	2	2	no	—
316	#41	—	2	2	no	—
317	#41	—	2	2	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
324	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
330	Q₁₆	—	2	2	no	—
331	Q₁₆	—	2	2	no	—
332	Q₁₆	—	2	2	no	—
333	Q₁₆	—	2	2	no	—
334	Q₁₆	—	2	1	no	—
335	Q₁₆	—	2	1	no	—
336	Q₁₆	—	4	4	no	—
337	SD₁₆	—	2	2	no	—
338	SD₁₆	—	2	2	no	—
339	SD₁₆	—	4	4	no	—
340	D₈	—	2	1	no	—
341	D₈	—	2	1	no	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	2	1	no	—
344	M₁₆	—	2	1	no	—
345	M₁₆	—	2	1	no	—
346	M₁₆	—	2	2	no	—
347	M₁₆	—	2	1	no	—
348	M₁₆	—	2	2	no	—
349	M₁₆	—	2	1	no	—
350	M₁₆	—	4	4	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	4	4	no	—
354	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
355	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
356	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
357	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
358	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
361	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	2	no	—
372	C₂²⋊C₄	—	4	4	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	4	4	no	—
376	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
377	C₄²	C₂×D₄×C₆₇	1	2	yes	—
378	C₂³	—	2	2	no	—
379	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
380	Q₈	—	2	2	no	—
381	Q₈	—	2	4	no	—
382	Q₈	—	2	4	no	—
383	Q₈	—	2	2	no	—
384	Q₈	—	4	2	no	—
385	Q₈	—	4	2	no	—
386	D₄	—	2	2	no	—
387	D₄	—	2	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
403	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
404	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
405	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	1	no	—
410	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
411	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
412	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
413	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
414	C₂²	—	4	1	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	#518×C₆₇	1	1	yes	characteristic
418	C₂²	#535×C₆₇	1	1	yes	characteristic
419	C₂²	#540×C₆₇	1	1	yes	characteristic
420	C₄	—	4	2	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	4	no	—
424	C₄	—	8	2	no	—
425	C₄	—	8	1	no	—
426	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
427	C₄	#518×C₆₇	1	1	yes	characteristic
428	C₄	#535×C₆₇	1	1	yes	characteristic
429	C₄	#540×C₆₇	1	1	yes	characteristic
430	C₂	—	2	1	no	—
431	C₂	—	2	1	no	—
432	C₂	—	2	1	no	—
433	C₂	—	8	2	no	—
434	C₂	C₆₇×#2328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2328
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×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
3. C₆₇×#2950	C₂	semidirect
4. C₆₇×#1979	C₂	semidirect
6. C₆₇×#1912	C₂	semidirect
7. C₆₇×#1909	C₂	semidirect
8. C₆₇×#1905	C₂	semidirect
9. C₆₇×#1902	C₂	semidirect
10. C₆₇×#1833	C₂	semidirect
11. C₆₇×#1747	C₂	semidirect
18. #497×C₆₇	C₄	semidirect
19. #495×C₆₇	C₄	semidirect
20. #494×C₆₇	C₄	semidirect
212. #22285	C₆₇	direct (N × H)
220. #2950	C₂×C₆₇	semidirect
221. #1979	C₂×C₆₇	semidirect
223. #1912	C₂×C₆₇	semidirect
224. #1909	C₂×C₆₇	semidirect
225. #1905	C₂×C₆₇	semidirect
226. #1902	C₂×C₆₇	semidirect
227. #1833	C₂×C₆₇	semidirect
228. #1747	C₂×C₆₇	semidirect
229. C₆₇	#22285	direct (N × H)
236. #497	C₄×C₆₇	semidirect
237. #495	C₄×C₆₇	semidirect
238. #494	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3156	2
C₆₇×#2957	2
C₆₇×#2950	2
C₆₇×#1979	2
C₆₇×#1976	2
C₆₇×#1912	2
C₆₇×#1909	2
C₆₇×#1905	2
C₆₇×#1902	2
C₆₇×#1833	2
C₆₇×#1747	2
#22285	67

Subgroups — C67×#22285 #30701536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22285 #30701536