Subgroups — C₆₇×#61105 #30740356

Order 17152.

2620 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3584	C₂	1	4	yes	maximal, complemented
2	C₆₇×#3293	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3292	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3291	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3290	C₂	1	8	yes	maximal, complemented
6	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#3067	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3018	C₂	1	4	yes	maximal, complemented
9	C₂×#443×C₆₇	C₂	1	2	yes	maximal, complemented
10	#585×C₆₇	C₂²	1	4	yes	complemented
11	#578×C₆₇	—	2	4	no	—
12	#578×C₆₇	C₂²	1	8	yes	complemented
13	#577×C₆₇	—	2	8	no	—
14	#577×C₆₇	C₂²	1	16	yes	complemented
15	#576×C₆₇	—	2	4	no	—
16	#576×C₆₇	C₂²	1	8	yes	complemented
17	#575×C₆₇	—	2	8	no	—
18	#575×C₆₇	C₂²	1	8	yes	complemented
19	C₂×#135×C₆₇	—	2	8	no	—
20	C₂×#135×C₆₇	C₂²	1	2	yes	—
21	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
22	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
23	#532×C₆₇	C₂²	1	2	yes	complemented
24	#495×C₆₇	C₂²	1	4	yes	complemented
25	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	#492×C₆₇	C₂²	1	2	yes	complemented
28	#473×C₆₇	C₂²	1	8	yes	complemented
29	#472×C₆₇	C₂²	1	8	yes	complemented
30	#471×C₆₇	C₂²	1	16	yes	complemented
31	#456×C₆₇	C₂²	1	8	yes	complemented
32	#455×C₆₇	C₂²	1	8	yes	complemented
33	#454×C₆₇	C₂²	1	8	yes	complemented
34	#453×C₆₇	C₂²	1	8	yes	complemented
35	#443×C₆₇	C₂²	1	4	yes	complemented
36	#443×C₆₇	C₂²	1	4	yes	complemented
37	#443×C₆₇	C₂²	1	4	yes	complemented
38	#443×C₆₇	C₂²	1	4	yes	complemented
39	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
40	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
41	#413×C₆₇	C₂²	1	4	yes	complemented
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	2₋¹⁺⁴×C₆₇	—	2	8	no	—
44	2₋¹⁺⁴×C₆₇	C₂³	1	8	yes	—
45	H(5,𝔽₂)×C₆₇	—	2	8	no	—
46	H(5,𝔽₂)×C₆₇	—	4	4	no	—
47	H(5,𝔽₂)×C₆₇	C₂³	1	8	yes	—
48	C₂×#41×C₆₇	—	2	16	no	—
49	C₂×#41×C₆₇	—	4	4	no	—
50	C₂×#41×C₆₇	C₂³	1	4	yes	—
51	C₂×#41×C₆₇	C₂³	1	4	yes	—
52	#137×C₆₇	—	2	8	no	—
53	#137×C₆₇	—	2	16	no	—
54	#137×C₆₇	—	2	16	no	—
55	#137×C₆₇	C₂³	1	16	yes	—
56	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
57	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
58	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
59	AGL(1,ℤ/8)×C₆₇	C₂³	1	16	yes	—
60	#135×C₆₇	—	2	16	no	—
61	#135×C₆₇	—	2	8	no	—
62	#135×C₆₇	—	2	16	no	—
63	#135×C₆₇	—	2	8	no	—
64	#135×C₆₇	—	2	4	no	—
65	#135×C₆₇	—	2	4	no	—
66	#135×C₆₇	—	2	4	no	—
67	#135×C₆₇	—	2	4	no	—
68	#135×C₆₇	D₄	1	4	yes	complemented
69	#135×C₆₇	D₄	1	4	yes	complemented
70	#135×C₆₇	C₂³	1	4	yes	—
71	#135×C₆₇	C₂³	1	4	yes	—
72	C₂×Q₁₆×C₆₇	—	2	8	no	—
73	C₂×Q₁₆×C₆₇	D₄	1	2	yes	complemented
74	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₆₇	—	2	16	no	—
76	C₂×SD₁₆×C₆₇	D₄	1	4	yes	complemented
77	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
78	C₂×D₈×C₆₇	—	2	8	no	—
79	C₂×D₈×C₆₇	D₄	1	2	yes	complemented
80	C₂×D₈×C₆₇	C₂³	1	4	yes	—
81	#131×C₆₇	—	2	4	no	—
82	#131×C₆₇	—	2	4	no	—
83	#131×C₆₇	C₂³	1	8	yes	—
84	#131×C₆₇	C₂³	1	8	yes	—
85	C₂×M₁₆×C₆₇	—	2	4	no	—
86	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
87	C₂²×C₈×C₆₇	—	2	4	no	—
88	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
89	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
90	#127×C₆₇	C₂³	1	2	yes	—
91	#124×C₆₇	C₂³	1	4	yes	—
92	#121×C₆₇	—	4	2	no	—
93	C₄×D₄×C₆₇	—	4	2	no	—
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	#108×C₆₇	C₂³	1	4	yes	—
96	#108×C₆₇	C₂³	1	4	yes	—
97	#104×C₆₇	C₂³	1	8	yes	—
98	#104×C₆₇	C₂³	1	8	yes	—
99	#101×C₆₇	C₂³	1	8	yes	—
100	#100×C₆₇	C₂³	1	8	yes	—
101	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₄×C₈×C₆₇	C₂³	1	2	yes	—
104	#41×C₆₇	—	2	4	no	—
105	#41×C₆₇	—	2	4	no	—
106	#41×C₆₇	—	2	16	no	—
107	#41×C₆₇	—	2	16	no	—
108	#41×C₆₇	—	2	8	no	—
109	#41×C₆₇	—	2	8	no	—
110	#41×C₆₇	—	4	8	no	—
111	#41×C₆₇	—	4	8	no	—
112	#41×C₆₇	—	4	16	no	—
113	#41×C₆₇	—	4	8	no	—
114	#41×C₆₇	C₂×D₄	1	8	yes	—
115	#41×C₆₇	C₂⁴	1	8	yes	—
116	C₂×Q₈×C₆₇	—	2	16	no	—
117	C₂×Q₈×C₆₇	—	4	4	no	—
118	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
119	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
120	C₂×D₄×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	2	16	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	4	2	no	—
124	C₂×D₄×C₆₇	—	4	8	no	—
125	C₂×D₄×C₆₇	—	4	4	no	—
126	C₂×D₄×C₆₇	—	4	16	no	—
127	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
128	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	4	8	no	—
133	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
134	Q₁₆×C₆₇	—	2	16	no	—
135	Q₁₆×C₆₇	—	2	4	no	—
136	Q₁₆×C₆₇	—	2	4	no	—
137	Q₁₆×C₆₇	C₂×D₄	1	4	yes	—
138	Q₁₆×C₆₇	C₂×D₄	1	4	yes	—
139	SD₁₆×C₆₇	—	2	16	no	—
140	SD₁₆×C₆₇	—	2	16	no	—
141	SD₁₆×C₆₇	—	2	8	no	—
142	SD₁₆×C₆₇	—	2	8	no	—
143	SD₁₆×C₆₇	C₂×D₄	1	8	yes	—
144	SD₁₆×C₆₇	C₂×D₄	1	8	yes	—
145	D₈×C₆₇	—	2	16	no	—
146	D₈×C₆₇	—	2	4	no	—
147	D₈×C₆₇	—	2	4	no	—
148	D₈×C₆₇	C₂×D₄	1	4	yes	—
149	D₈×C₆₇	C₂×D₄	1	4	yes	—
150	M₁₆×C₆₇	—	2	4	no	—
151	M₁₆×C₆₇	—	2	8	no	—
152	M₁₆×C₆₇	—	2	8	no	—
153	M₁₆×C₆₇	C₂⁴	1	8	yes	—
154	C₂×C₈×C₆₇	—	2	4	no	—
155	C₂×C₈×C₆₇	—	2	8	no	—
156	C₂×C₈×C₆₇	—	2	8	no	—
157	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
162	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
163	C₄⋊C₄×C₆₇	—	4	2	no	—
164	C₂²⋊C₄×C₆₇	—	4	4	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂³×C₆₇	—	2	4	no	—
169	C₂³×C₆₇	—	2	4	no	—
170	C₂³×C₆₇	—	4	8	no	—
171	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
172	Q₈×C₆₇	—	2	4	no	—
173	Q₈×C₆₇	—	2	8	no	—
174	Q₈×C₆₇	—	4	8	no	—
175	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
176	D₄×C₆₇	—	2	8	no	—
177	D₄×C₆₇	—	2	8	no	—
178	D₄×C₆₇	—	2	4	no	—
179	D₄×C₆₇	—	2	8	no	—
180	D₄×C₆₇	—	4	16	no	—
181	D₄×C₆₇	—	4	8	no	—
182	D₄×C₆₇	—	4	8	no	—
183	D₄×C₆₇	—	8	4	no	—
184	D₄×C₆₇	C₂²×D₄	1	8	yes	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	8	no	—
188	C₂×C₄×C₆₇	—	2	8	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	4	8	no	—
192	C₂×C₄×C₆₇	—	4	16	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	4	no	—
195	C₂×C₄×C₆₇	—	4	4	no	—
196	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
198	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
200	C₈×C₆₇	—	2	8	no	—
201	C₈×C₆₇	C₂²×D₄	1	2	yes	—
202	C₈×C₆₇	C₂²×D₄	1	2	yes	—
203	C₈×C₆₇	C₂²×D₄	1	2	yes	—
204	C₈×C₆₇	C₂²×D₄	1	2	yes	—
205	C₂²×C₆₇	—	2	8	no	—
206	C₂²×C₆₇	—	4	1	no	—
207	C₂²×C₆₇	—	4	4	no	—
208	C₂²×C₆₇	—	8	4	no	—
209	C₂²×C₆₇	—	8	4	no	—
210	C₂²×C₆₇	—	16	8	no	—
211	C₂²×C₆₇	D₄²	1	2	yes	—
212	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	—	2	8	no	—
215	C₄×C₆₇	—	4	1	no	—
216	C₄×C₆₇	—	4	4	no	—
217	C₄×C₆₇	D₄²	1	1	yes	characteristic
218	C₄×C₆₇	D₄²	1	1	yes	characteristic
219	C₄×C₆₇	C₂³×D₄	1	2	yes	—
220	#61105	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₆₇	—	2	1	no	—
222	C₂×C₆₇	—	2	2	no	—
223	C₂×C₆₇	—	4	8	no	—
224	C₂×C₆₇	—	8	4	no	—
225	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
226	#3584	C₂×C₆₇	1	4	yes	complemented
227	#3293	C₂×C₆₇	1	2	yes	complemented
228	#3292	C₂×C₆₇	1	2	yes	complemented
229	#3291	C₂×C₆₇	1	4	yes	complemented
230	#3290	C₂×C₆₇	1	8	yes	complemented
231	#3147	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#3067	C₂×C₆₇	1	4	yes	complemented
233	#3018	C₂×C₆₇	1	4	yes	complemented
234	C₂×#443	C₂×C₆₇	1	2	yes	complemented
235	C₆₇	#61105	1	1	yes	characteristic, direct product factor, Sylow
236	#585	C₂²×C₆₇	1	4	yes	complemented
237	#578	—	2	4	no	—
238	#578	C₂²×C₆₇	1	8	yes	complemented
239	#577	—	2	8	no	—
240	#577	C₂²×C₆₇	1	16	yes	complemented
241	#576	—	2	4	no	—
242	#576	C₂²×C₆₇	1	8	yes	complemented
243	#575	—	2	8	no	—
244	#575	C₂²×C₆₇	1	8	yes	complemented
245	C₂×#135	—	2	8	no	—
246	C₂×#135	C₂²×C₆₇	1	2	yes	—
247	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
248	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
249	#532	C₂²×C₆₇	1	2	yes	complemented
250	#495	C₂²×C₆₇	1	4	yes	complemented
251	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
252	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
253	#492	C₂²×C₆₇	1	2	yes	complemented
254	#473	C₂²×C₆₇	1	8	yes	complemented
255	#472	C₂²×C₆₇	1	8	yes	complemented
256	#471	C₂²×C₆₇	1	16	yes	complemented
257	#456	C₂²×C₆₇	1	8	yes	complemented
258	#455	C₂²×C₆₇	1	8	yes	complemented
259	#454	C₂²×C₆₇	1	8	yes	complemented
260	#453	C₂²×C₆₇	1	8	yes	complemented
261	#443	C₂²×C₆₇	1	4	yes	complemented
262	#443	C₂²×C₆₇	1	4	yes	complemented
263	#443	C₂²×C₆₇	1	4	yes	complemented
264	#443	C₂²×C₆₇	1	4	yes	complemented
265	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
266	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
267	#413	C₂²×C₆₇	1	4	yes	complemented
268	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
269	2₋¹⁺⁴	—	2	8	no	—
270	2₋¹⁺⁴	C₂³×C₆₇	1	8	yes	—
271	H(5,𝔽₂)	—	2	8	no	—
272	H(5,𝔽₂)	—	4	4	no	—
273	H(5,𝔽₂)	C₂³×C₆₇	1	8	yes	—
274	C₂×#41	—	2	16	no	—
275	C₂×#41	—	4	4	no	—
276	C₂×#41	C₂³×C₆₇	1	4	yes	—
277	C₂×#41	C₂³×C₆₇	1	4	yes	—
278	#137	—	2	8	no	—
279	#137	—	2	16	no	—
280	#137	—	2	16	no	—
281	#137	C₂³×C₆₇	1	16	yes	—
282	AGL(1,ℤ/8)	—	2	8	no	—
283	AGL(1,ℤ/8)	—	2	16	no	—
284	AGL(1,ℤ/8)	—	2	16	no	—
285	AGL(1,ℤ/8)	C₂³×C₆₇	1	16	yes	—
286	#135	—	2	16	no	—
287	#135	—	2	8	no	—
288	#135	—	2	16	no	—
289	#135	—	2	8	no	—
290	#135	—	2	4	no	—
291	#135	—	2	4	no	—
292	#135	—	2	4	no	—
293	#135	—	2	4	no	—
294	#135	D₄×C₆₇	1	4	yes	complemented
295	#135	D₄×C₆₇	1	4	yes	complemented
296	#135	C₂³×C₆₇	1	4	yes	—
297	#135	C₂³×C₆₇	1	4	yes	—
298	C₂×Q₁₆	—	2	8	no	—
299	C₂×Q₁₆	D₄×C₆₇	1	2	yes	complemented
300	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
301	C₂×SD₁₆	—	2	16	no	—
302	C₂×SD₁₆	D₄×C₆₇	1	4	yes	complemented
303	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
304	C₂×D₈	—	2	8	no	—
305	C₂×D₈	D₄×C₆₇	1	2	yes	complemented
306	C₂×D₈	C₂³×C₆₇	1	4	yes	—
307	#131	—	2	4	no	—
308	#131	—	2	4	no	—
309	#131	C₂³×C₆₇	1	8	yes	—
310	#131	C₂³×C₆₇	1	8	yes	—
311	C₂×M₁₆	—	2	4	no	—
312	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
313	C₂²×C₈	—	2	4	no	—
314	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
315	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
316	#127	C₂³×C₆₇	1	2	yes	—
317	#124	C₂³×C₆₇	1	4	yes	—
318	#121	—	4	2	no	—
319	C₄×D₄	—	4	2	no	—
320	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
321	#108	C₂³×C₆₇	1	4	yes	—
322	#108	C₂³×C₆₇	1	4	yes	—
323	#104	C₂³×C₆₇	1	8	yes	—
324	#104	C₂³×C₆₇	1	8	yes	—
325	#101	C₂³×C₆₇	1	8	yes	—
326	#100	C₂³×C₆₇	1	8	yes	—
327	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
329	C₄×C₈	C₂³×C₆₇	1	2	yes	—
330	#41	—	2	4	no	—
331	#41	—	2	4	no	—
332	#41	—	2	16	no	—
333	#41	—	2	16	no	—
334	#41	—	2	8	no	—
335	#41	—	2	8	no	—
336	#41	—	4	8	no	—
337	#41	—	4	8	no	—
338	#41	—	4	16	no	—
339	#41	—	4	8	no	—
340	#41	C₂×D₄×C₆₇	1	8	yes	—
341	#41	C₂⁴×C₆₇	1	8	yes	—
342	C₂×Q₈	—	2	16	no	—
343	C₂×Q₈	—	4	4	no	—
344	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
345	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	16	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	4	2	no	—
350	C₂×D₄	—	4	8	no	—
351	C₂×D₄	—	4	4	no	—
352	C₂×D₄	—	4	16	no	—
353	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
354	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	4	8	no	—
359	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
360	Q₁₆	—	2	16	no	—
361	Q₁₆	—	2	4	no	—
362	Q₁₆	—	2	4	no	—
363	Q₁₆	C₂×D₄×C₆₇	1	4	yes	—
364	Q₁₆	C₂×D₄×C₆₇	1	4	yes	—
365	SD₁₆	—	2	16	no	—
366	SD₁₆	—	2	16	no	—
367	SD₁₆	—	2	8	no	—
368	SD₁₆	—	2	8	no	—
369	SD₁₆	C₂×D₄×C₆₇	1	8	yes	—
370	SD₁₆	C₂×D₄×C₆₇	1	8	yes	—
371	D₈	—	2	16	no	—
372	D₈	—	2	4	no	—
373	D₈	—	2	4	no	—
374	D₈	C₂×D₄×C₆₇	1	4	yes	—
375	D₈	C₂×D₄×C₆₇	1	4	yes	—
376	M₁₆	—	2	4	no	—
377	M₁₆	—	2	8	no	—
378	M₁₆	—	2	8	no	—
379	M₁₆	C₂⁴×C₆₇	1	8	yes	—
380	C₂×C₈	—	2	4	no	—
381	C₂×C₈	—	2	8	no	—
382	C₂×C₈	—	2	8	no	—
383	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
384	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
385	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
387	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
388	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
389	C₄⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₄²	—	2	1	no	—
392	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
393	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
394	C₂³	—	2	4	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	4	8	no	—
397	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
398	Q₈	—	2	4	no	—
399	Q₈	—	2	8	no	—
400	Q₈	—	4	8	no	—
401	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
402	D₄	—	2	8	no	—
403	D₄	—	2	8	no	—
404	D₄	—	2	4	no	—
405	D₄	—	2	8	no	—
406	D₄	—	4	16	no	—
407	D₄	—	4	8	no	—
408	D₄	—	4	8	no	—
409	D₄	—	8	4	no	—
410	D₄	C₂²×D₄×C₆₇	1	8	yes	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	—	4	16	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
423	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
424	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
425	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
426	C₈	—	2	8	no	—
427	C₈	C₂²×D₄×C₆₇	1	2	yes	—
428	C₈	C₂²×D₄×C₆₇	1	2	yes	—
429	C₈	C₂²×D₄×C₆₇	1	2	yes	—
430	C₈	C₂²×D₄×C₆₇	1	2	yes	—
431	C₂²	—	2	8	no	—
432	C₂²	—	4	1	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	8	4	no	—
435	C₂²	—	8	4	no	—
436	C₂²	—	16	8	no	—
437	C₂²	D₄²×C₆₇	1	2	yes	—
438	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
439	C₄	—	2	2	no	—
440	C₄	—	2	8	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	4	no	—
443	C₄	D₄²×C₆₇	1	1	yes	characteristic
444	C₄	D₄²×C₆₇	1	1	yes	characteristic
445	C₄	C₂³×D₄×C₆₇	1	2	yes	—
446	C₂	—	2	1	no	—
447	C₂	—	2	2	no	—
448	C₂	—	4	8	no	—
449	C₂	—	8	4	no	—
450	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂×C₆₇	C₂×D₄²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3584	C₂	semidirect
2. C₆₇×#3293	C₂	semidirect
3. C₆₇×#3292	C₂	semidirect
4. C₆₇×#3291	C₂	semidirect
5. C₆₇×#3290	C₂	semidirect
6. C₆₇×#3147	C₂	semidirect
7. C₆₇×#3067	C₂	semidirect
8. C₆₇×#3018	C₂	semidirect
9. C₂×#443×C₆₇	C₂	semidirect
10. #585×C₆₇	C₂²	semidirect
12. #578×C₆₇	C₂²	semidirect
14. #577×C₆₇	C₂²	semidirect
16. #576×C₆₇	C₂²	semidirect
18. #575×C₆₇	C₂²	semidirect
21. C₂×#135×C₆₇	C₂²	semidirect
22. C₂×#131×C₆₇	C₂²	semidirect
23. #532×C₆₇	C₂²	semidirect
24. #495×C₆₇	C₂²	semidirect
25. #494×C₆₇	C₂²	semidirect
26. #493×C₆₇	C₂²	semidirect
27. #492×C₆₇	C₂²	semidirect
28. #473×C₆₇	C₂²	semidirect
29. #472×C₆₇	C₂²	semidirect
30. #471×C₆₇	C₂²	semidirect
31. #456×C₆₇	C₂²	semidirect
32. #455×C₆₇	C₂²	semidirect
33. #454×C₆₇	C₂²	semidirect
34. #453×C₆₇	C₂²	semidirect
35. #443×C₆₇	C₂²	semidirect
36. #443×C₆₇	C₂²	semidirect
37. #443×C₆₇	C₂²	semidirect
38. #443×C₆₇	C₂²	semidirect
39. C₂×#108×C₆₇	C₂²	semidirect
40. C₂×#104×C₆₇	C₂²	semidirect
41. #413×C₆₇	C₂²	semidirect
42. C₂×C₄×C₈×C₆₇	C₂²	semidirect
68. #135×C₆₇	D₄	semidirect
69. #135×C₆₇	D₄	semidirect
73. C₂×Q₁₆×C₆₇	D₄	semidirect
76. C₂×SD₁₆×C₆₇	D₄	semidirect
79. C₂×D₈×C₆₇	D₄	semidirect
220. #61105	C₆₇	direct (N × H)
226. #3584	C₂×C₆₇	semidirect
227. #3293	C₂×C₆₇	semidirect
228. #3292	C₂×C₆₇	semidirect
229. #3291	C₂×C₆₇	semidirect
230. #3290	C₂×C₆₇	semidirect
231. #3147	C₂×C₆₇	semidirect
232. #3067	C₂×C₆₇	semidirect
233. #3018	C₂×C₆₇	semidirect
234. C₂×#443	C₂×C₆₇	semidirect
235. C₆₇	#61105	direct (N × H)
236. #585	C₂²×C₆₇	semidirect
238. #578	C₂²×C₆₇	semidirect
240. #577	C₂²×C₆₇	semidirect
242. #576	C₂²×C₆₇	semidirect
244. #575	C₂²×C₆₇	semidirect
247. C₂×#135	C₂²×C₆₇	semidirect
248. C₂×#131	C₂²×C₆₇	semidirect
249. #532	C₂²×C₆₇	semidirect
250. #495	C₂²×C₆₇	semidirect
251. #494	C₂²×C₆₇	semidirect
252. #493	C₂²×C₆₇	semidirect
253. #492	C₂²×C₆₇	semidirect
254. #473	C₂²×C₆₇	semidirect
255. #472	C₂²×C₆₇	semidirect
256. #471	C₂²×C₆₇	semidirect
257. #456	C₂²×C₆₇	semidirect
258. #455	C₂²×C₆₇	semidirect
259. #454	C₂²×C₆₇	semidirect
260. #453	C₂²×C₆₇	semidirect
261. #443	C₂²×C₆₇	semidirect
262. #443	C₂²×C₆₇	semidirect
263. #443	C₂²×C₆₇	semidirect
264. #443	C₂²×C₆₇	semidirect
265. C₂×#108	C₂²×C₆₇	semidirect
266. C₂×#104	C₂²×C₆₇	semidirect
267. #413	C₂²×C₆₇	semidirect
268. C₂×C₄×C₈	C₂²×C₆₇	semidirect
294. #135	D₄×C₆₇	semidirect
295. #135	D₄×C₆₇	semidirect
299. C₂×Q₁₆	D₄×C₆₇	semidirect
302. C₂×SD₁₆	D₄×C₆₇	semidirect
305. C₂×D₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3584	2
C₆₇×#3293	2
C₆₇×#3292	2
C₆₇×#3291	2
C₆₇×#3290	2
C₆₇×#3147	2
C₆₇×#3067	2
C₆₇×#3018	2
C₂×#443×C₆₇	2
#61105	67

Subgroups — C67×#61105 #30740356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61105 #30740356